Basic Definitions and Operations
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Vector Multiplication in Three Dimensions
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Gradient
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Scalar Multiplication
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Unit Vector
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Vectors in Three Dimension
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Curl (Also Called Rotor)
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[{"Name":"Basic Definitions and Operations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vectors vs Scalars","Duration":"3m 6s","ChapterTopicVideoID":8922,"CourseChapterTopicPlaylistID":85317,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8922.jpeg","UploadDate":"2017-05-06T09:54:40.9370000","DurationForVideoObject":"PT3M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"Hello. In this lecture I want to talk about something very basic but important,"},{"Start":"00:04.200 ","End":"00:08.085","Text":"the difference between scalars and vectors. Let\u0027s start with vectors."},{"Start":"00:08.085 ","End":"00:11.895","Text":"A vector is a physical magnitude with a given direction to it."},{"Start":"00:11.895 ","End":"00:14.340","Text":"It\u0027s important for describing something where I need to know"},{"Start":"00:14.340 ","End":"00:17.250","Text":"both the size and the direction that it\u0027s headed."},{"Start":"00:17.250 ","End":"00:19.770","Text":"For example, let\u0027s say I have a car,"},{"Start":"00:19.770 ","End":"00:23.070","Text":"and that car is driving 30 kilometers an hour."},{"Start":"00:23.070 ","End":"00:26.190","Text":"It\u0027s not only important for me to know how"},{"Start":"00:26.190 ","End":"00:29.055","Text":"fast that car is driving but also what direction it\u0027s going."},{"Start":"00:29.055 ","End":"00:30.435","Text":"Is it going to the right?"},{"Start":"00:30.435 ","End":"00:32.655","Text":"Is it going up into the left?"},{"Start":"00:32.655 ","End":"00:34.780","Text":"Is it going a different direction altogether?"},{"Start":"00:34.780 ","End":"00:36.060","Text":"All these things are important to me,"},{"Start":"00:36.060 ","End":"00:39.120","Text":"so I need to use a vector which allows me to know both the size,"},{"Start":"00:39.120 ","End":"00:41.525","Text":"the magnitude, and the direction."},{"Start":"00:41.525 ","End":"00:44.120","Text":"If I don\u0027t have both a magnitude and a direction,"},{"Start":"00:44.120 ","End":"00:47.660","Text":"I can\u0027t properly describe the velocity of this car."},{"Start":"00:47.660 ","End":"00:50.510","Text":"Now, as opposed to a vector,"},{"Start":"00:50.510 ","End":"00:54.395","Text":"a scalar is a magnitude without direction."},{"Start":"00:54.395 ","End":"01:01.130","Text":"It could be measuring something like weight or mass or temperature or something else."},{"Start":"01:01.130 ","End":"01:04.600","Text":"For example, if I have something that\u0027s 30 degrees Celsius or Fahrenheit,"},{"Start":"01:04.600 ","End":"01:07.280","Text":"it doesn\u0027t really matter, it doesn\u0027t have a direction, it\u0027s not moving anywhere."},{"Start":"01:07.280 ","End":"01:10.005","Text":"It\u0027s not important for me to know which direction it\u0027s headed."},{"Start":"01:10.005 ","End":"01:12.950","Text":"There are many things that I\u0027m going to measure only in scalar terms,"},{"Start":"01:12.950 ","End":"01:14.395","Text":"only in terms of the magnitude."},{"Start":"01:14.395 ","End":"01:17.360","Text":"The sign for a vector is an arrow."},{"Start":"01:17.360 ","End":"01:20.285","Text":"The arrow direction tells me what direction"},{"Start":"01:20.285 ","End":"01:23.765","Text":"the object is moving and the length will tell me the magnitude of it,"},{"Start":"01:23.765 ","End":"01:26.125","Text":"how fast it\u0027s going if it\u0027s moving."},{"Start":"01:26.125 ","End":"01:29.555","Text":"For example, with this car going 30 kilometers an hour,"},{"Start":"01:29.555 ","End":"01:32.945","Text":"if it\u0027s going to the right, the direction of the vector would be towards the right."},{"Start":"01:32.945 ","End":"01:36.710","Text":"Now, the length of it is supposed to represent the magnitude, of course,"},{"Start":"01:36.710 ","End":"01:37.820","Text":"there\u0027s no direct connection"},{"Start":"01:37.820 ","End":"01:41.750","Text":"between 30 kilometers an hour and a certain lengths on my paper,"},{"Start":"01:41.750 ","End":"01:43.220","Text":"but I can do it quantitatively."},{"Start":"01:43.220 ","End":"01:44.449","Text":"I can do it relatively."},{"Start":"01:44.449 ","End":"01:46.040","Text":"If I had a second car,"},{"Start":"01:46.040 ","End":"01:50.375","Text":"let\u0027s say this car was going 50 kilometers an hour in the same direction,"},{"Start":"01:50.375 ","End":"01:52.280","Text":"I would draw a different vector that would be 1 and"},{"Start":"01:52.280 ","End":"01:56.275","Text":"2/3 times as long as the first vector."},{"Start":"01:56.275 ","End":"02:01.220","Text":"I can\u0027t really know the size of 1 vector by looking at 1 arrow, it\u0027s all relative."},{"Start":"02:01.220 ","End":"02:04.055","Text":"I have to look at both arrows to understand what I\u0027m seeing."},{"Start":"02:04.055 ","End":"02:07.520","Text":"Now, if we want to write out a vector in an equation,"},{"Start":"02:07.520 ","End":"02:09.200","Text":"for example, there are couple of ways to do it."},{"Start":"02:09.200 ","End":"02:13.510","Text":"The most common is to use a letter with an arrow on the top, a capital letter,"},{"Start":"02:13.510 ","End":"02:19.315","Text":"so we\u0027d have A with an arrow on the top to symbolize the vector A if that vector were A,"},{"Start":"02:19.315 ","End":"02:21.725","Text":"and we wanted to write it in notation."},{"Start":"02:21.725 ","End":"02:24.380","Text":"Some people use bolded letters,"},{"Start":"02:24.380 ","End":"02:27.065","Text":"so it\u0027ll just be a bold A with no arrow on top."},{"Start":"02:27.065 ","End":"02:30.050","Text":"Additionally, some people use small letters,"},{"Start":"02:30.050 ","End":"02:32.660","Text":"lowercase letters with an arrow on top."},{"Start":"02:32.660 ","End":"02:34.745","Text":"That can also be a vector."},{"Start":"02:34.745 ","End":"02:40.520","Text":"Of course, we can also have a lowercase letter in bold symbolizing a vector."},{"Start":"02:40.520 ","End":"02:43.750","Text":"Now, the beauty of vector is it doesn\u0027t matter where it starts or"},{"Start":"02:43.750 ","End":"02:46.790","Text":"where it ends as long as it\u0027s of the same length and the same direction,"},{"Start":"02:46.790 ","End":"02:49.265","Text":"it\u0027s the same vector no matter what position it\u0027s in."},{"Start":"02:49.265 ","End":"02:51.320","Text":"That\u0027s the same vector as the one before."},{"Start":"02:51.320 ","End":"02:53.960","Text":"If we were to switch it back or to any other point on the screen,"},{"Start":"02:53.960 ","End":"02:56.120","Text":"as long as those are the same length and same direction,"},{"Start":"02:56.120 ","End":"02:58.510","Text":"it would continue to be the same vector."},{"Start":"02:58.510 ","End":"03:00.350","Text":"That\u0027s the end of our introduction."},{"Start":"03:00.350 ","End":"03:01.505","Text":"In the next video,"},{"Start":"03:01.505 ","End":"03:03.710","Text":"we\u0027re going to look at what you can do with vectors,"},{"Start":"03:03.710 ","End":"03:06.750","Text":"starting with connecting vectors."}],"ID":9545},{"Watched":false,"Name":"Adding Vectors","Duration":"2m 32s","ChapterTopicVideoID":8923,"CourseChapterTopicPlaylistID":85317,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8923.jpeg","UploadDate":"2017-03-21T08:31:08.2370000","DurationForVideoObject":"PT2M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.825","Text":"In this bit, I want to talk about how we connect two vectors."},{"Start":"00:03.825 ","End":"00:06.165","Text":"They can be of different angles, of different sizes."},{"Start":"00:06.165 ","End":"00:08.520","Text":"But if I connect them, I want to see what happens."},{"Start":"00:08.520 ","End":"00:10.476","Text":"If I connect vectors A and B,"},{"Start":"00:10.476 ","End":"00:11.640","Text":"what do I get?"},{"Start":"00:11.640 ","End":"00:15.615","Text":"The first thing you need to realize is that you\u0027re going to get another vector."},{"Start":"00:15.615 ","End":"00:18.420","Text":"This is important to understand and it\u0027s not obvious,"},{"Start":"00:18.420 ","End":"00:19.860","Text":"but when you connect two vectors,"},{"Start":"00:19.860 ","End":"00:21.810","Text":"you get a third vector."},{"Start":"00:21.810 ","End":"00:25.815","Text":"I want to find what happens when I connect vector A and vector B"},{"Start":"00:25.815 ","End":"00:29.715","Text":"or another way to put it would be I want to find what happens to vector C,"},{"Start":"00:29.715 ","End":"00:31.515","Text":"the result of these two vectors."},{"Start":"00:31.515 ","End":"00:33.435","Text":"There\u0027s a couple of ways of finding this."},{"Start":"00:33.435 ","End":"00:39.110","Text":"The first is if we take vector A and lay it out on our graphing paper,"},{"Start":"00:39.110 ","End":"00:43.655","Text":"and move vector B so that the tail of vector B is at the head of vector A."},{"Start":"00:43.655 ","End":"00:47.120","Text":"Then you can see we have this imaginary line that would"},{"Start":"00:47.120 ","End":"00:50.945","Text":"connect the tail of vector A and the head of vector B."},{"Start":"00:50.945 ","End":"00:52.700","Text":"If we draw out that line,"},{"Start":"00:52.700 ","End":"00:55.370","Text":"that will give us vector C,"},{"Start":"00:55.370 ","End":"00:59.240","Text":"it\u0027s going to have both the length, the magnitude that is,"},{"Start":"00:59.240 ","End":"01:03.635","Text":"and the angle that I want for vector C. This is the first way to do it."},{"Start":"01:03.635 ","End":"01:08.810","Text":"The second way is if we take the tails of vectors A and B and connect them together,"},{"Start":"01:08.810 ","End":"01:13.700","Text":"as you can see on the second half of the page here and I have both vectors pointing in"},{"Start":"01:13.700 ","End":"01:16.130","Text":"the same angles that they always point of the same magnitude they"},{"Start":"01:16.130 ","End":"01:19.435","Text":"always are and then I turn them into parallelograms."},{"Start":"01:19.435 ","End":"01:24.790","Text":"What I can do is take a parallel to B and put it at the head of A and"},{"Start":"01:24.790 ","End":"01:26.990","Text":"a parallel of A and put it at the head of B of"},{"Start":"01:26.990 ","End":"01:30.815","Text":"the same magnitude of the original A and of the original B."},{"Start":"01:30.815 ","End":"01:34.865","Text":"Then I have myself a parallelogram and if I connect"},{"Start":"01:34.865 ","End":"01:37.850","Text":"the diagonal angle from"},{"Start":"01:37.850 ","End":"01:42.065","Text":"the tail points to the head points of the imaginary lines that I\u0027ve added."},{"Start":"01:42.065 ","End":"01:47.315","Text":"That will also give me the vector C. If I fill that in,"},{"Start":"01:47.315 ","End":"01:51.530","Text":"that\u0027s also vector C. An important characteristic of"},{"Start":"01:51.530 ","End":"01:56.960","Text":"adding together vectors is that it doesn\u0027t matter if I add A and then B or B and then A."},{"Start":"01:56.960 ","End":"01:59.449","Text":"You can see, for example,"},{"Start":"01:59.449 ","End":"02:01.900","Text":"in the second chart,"},{"Start":"02:01.900 ","End":"02:05.120","Text":"that if I were to add A first or B first, it doesn\u0027t really matter."},{"Start":"02:05.120 ","End":"02:06.290","Text":"They\u0027re touching each other and the tails,"},{"Start":"02:06.290 ","End":"02:08.105","Text":"it doesn\u0027t matter the order there."},{"Start":"02:08.105 ","End":"02:09.960","Text":"Is a little harder to see,"},{"Start":"02:09.960 ","End":"02:13.475","Text":"but on the first way we did things this way."},{"Start":"02:13.475 ","End":"02:17.765","Text":"If I put down, let\u0027s say B first instead of A,"},{"Start":"02:17.765 ","End":"02:22.175","Text":"then I added a to the end of B, like so."},{"Start":"02:22.175 ","End":"02:27.305","Text":"I would also get vector C. Let me add an A, there you go."},{"Start":"02:27.305 ","End":"02:32.820","Text":"I would also get vector C. That\u0027s addition now let\u0027s move on to subtraction."}],"ID":9197},{"Watched":false,"Name":"Subtracting Vectors","Duration":"1m 19s","ChapterTopicVideoID":8924,"CourseChapterTopicPlaylistID":85317,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8924.jpeg","UploadDate":"2017-03-21T08:31:16.2170000","DurationForVideoObject":"PT1M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"When we approach subtraction of vectors,"},{"Start":"00:02.670 ","End":"00:07.320","Text":"it\u0027s important to think about the concept of negative or opposite vectors."},{"Start":"00:07.320 ","End":"00:11.550","Text":"If we\u0027re trying to subtract vector B from vector A, first,"},{"Start":"00:11.550 ","End":"00:15.510","Text":"let\u0027s look at the negative vector B as opposed to the vector B."},{"Start":"00:15.510 ","End":"00:18.240","Text":"Opposite or negative vector B is the same vector,"},{"Start":"00:18.240 ","End":"00:21.240","Text":"the same magnitude just with the exact opposite angle"},{"Start":"00:21.240 ","End":"00:24.585","Text":"as if we took the head of the arrow and put on the opposite side."},{"Start":"00:24.585 ","End":"00:27.120","Text":"You can see here that negative B is the same magnitude"},{"Start":"00:27.120 ","End":"00:30.240","Text":"and the exact opposite angle of the vector B."},{"Start":"00:30.240 ","End":"00:33.375","Text":"If we\u0027re going to do a subtraction of vectors,"},{"Start":"00:33.375 ","End":"00:36.660","Text":"the easiest way to do it is instead of subtracting the initial vector B,"},{"Start":"00:36.660 ","End":"00:40.020","Text":"we can add the opposite vector, negative B."},{"Start":"00:40.020 ","End":"00:45.270","Text":"C equals A minus B or A plus opposite B."},{"Start":"00:45.270 ","End":"00:49.825","Text":"The way that we do that with the head-to-tail method is we put down our vector A."},{"Start":"00:49.825 ","End":"00:51.250","Text":"Then at the head of A,"},{"Start":"00:51.250 ","End":"00:53.480","Text":"we put down the tail of vector negative B,"},{"Start":"00:53.480 ","End":"00:55.565","Text":"which is the opposite of vector B."},{"Start":"00:55.565 ","End":"00:58.580","Text":"At the head or the tail of vector A,"},{"Start":"00:58.580 ","End":"01:02.525","Text":"we put down the tail of vector C going towards the head of vector negative B,"},{"Start":"01:02.525 ","End":"01:07.100","Text":"and that\u0027s our new vector C. We can also do this with the parallelogram by putting"},{"Start":"01:07.100 ","End":"01:12.110","Text":"down A and the original vector B tail to tail and instead of using the main diagonal,"},{"Start":"01:12.110 ","End":"01:13.925","Text":"we use the secondary diagonal."},{"Start":"01:13.925 ","End":"01:16.850","Text":"In this case it\u0027s going from the head of B to the head of A."},{"Start":"01:16.850 ","End":"01:19.440","Text":"That\u0027s the method. Thanks."}],"ID":9198},{"Watched":false,"Name":"Breaking Down Vectors","Duration":"4m 39s","ChapterTopicVideoID":8925,"CourseChapterTopicPlaylistID":85317,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8925.jpeg","UploadDate":"2017-03-21T08:31:44.6230000","DurationForVideoObject":"PT4M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:02.880","Text":"Hello. In this part, I\u0027d like to talk about breaking"},{"Start":"00:02.880 ","End":"00:05.040","Text":"down your vector into its different components."},{"Start":"00:05.040 ","End":"00:08.520","Text":"Let\u0027s assume we have the vector A and that\u0027s drawn here."},{"Start":"00:08.520 ","End":"00:15.093","Text":"I have the magnitude of the vector written this way and the direction of the vector."},{"Start":"00:15.093 ","End":"00:16.380","Text":"We can call it the angle Theta,"},{"Start":"00:16.380 ","End":"00:19.530","Text":"which is the angle from A with respect to the x-axis."},{"Start":"00:19.530 ","End":"00:22.170","Text":"Now if I want to write out the x component and"},{"Start":"00:22.170 ","End":"00:25.635","Text":"the y component of my vector, I can do the following."},{"Start":"00:25.635 ","End":"00:29.550","Text":"Imagine that I take the end of my vector and transpose it down,"},{"Start":"00:29.550 ","End":"00:32.565","Text":"projected down to the x-axis by drawing a vertical line."},{"Start":"00:32.565 ","End":"00:33.885","Text":"I get a right triangle,"},{"Start":"00:33.885 ","End":"00:35.370","Text":"and the base of this triangle,"},{"Start":"00:35.370 ","End":"00:36.989","Text":"which is along the x-axis,"},{"Start":"00:36.989 ","End":"00:39.455","Text":"we\u0027re going to call that A_x or A_x."},{"Start":"00:39.455 ","End":"00:45.275","Text":"Now, the magnitude of A_x has a relationship with magnitude of the vector A."},{"Start":"00:45.275 ","End":"00:48.875","Text":"We can write that relationship as follows."},{"Start":"00:48.875 ","End":"00:53.120","Text":"A_x over the magnitude of the vector A"},{"Start":"00:53.120 ","End":"00:57.855","Text":"equals the cosine of Theta and we can also write that"},{"Start":"00:57.855 ","End":"01:02.030","Text":"as A_x equals the cosine of Theta times"},{"Start":"01:02.030 ","End":"01:07.165","Text":"the magnitude of the vector A or the magnitude of vector A times the cosine of Theta."},{"Start":"01:07.165 ","End":"01:13.130","Text":"Basically, the projection of A vector onto the x-axis or"},{"Start":"01:13.130 ","End":"01:15.320","Text":"the x element of the vector equals"},{"Start":"01:15.320 ","End":"01:18.710","Text":"the magnitude of the vector A times the cosine of Theta."},{"Start":"01:18.710 ","End":"01:21.065","Text":"Now of course, I can do the same thing for y,"},{"Start":"01:21.065 ","End":"01:22.535","Text":"which is a different relationship."},{"Start":"01:22.535 ","End":"01:26.480","Text":"The y element of A equals the vector A,"},{"Start":"01:26.480 ","End":"01:29.105","Text":"the magnitude of the vector A times the sine of Theta."},{"Start":"01:29.105 ","End":"01:33.635","Text":"Now, this is only for measuring Theta as the angle between the x-axis and the vector A."},{"Start":"01:33.635 ","End":"01:37.369","Text":"If that angle changes then my equations will change accordingly."},{"Start":"01:37.369 ","End":"01:39.050","Text":"Sometimes we want to do the opposite."},{"Start":"01:39.050 ","End":"01:41.600","Text":"Instead of finding the x element and the y element,"},{"Start":"01:41.600 ","End":"01:44.030","Text":"I\u0027m given the x element and the y element and I need to"},{"Start":"01:44.030 ","End":"01:47.030","Text":"find the magnitude of the vector so 2 ways to do this."},{"Start":"01:47.030 ","End":"01:49.850","Text":"The first is if you look at the x and the y element,"},{"Start":"01:49.850 ","End":"01:52.190","Text":"they make a right-angle triangle with"},{"Start":"01:52.190 ","End":"01:56.045","Text":"the vector magnitude so we can do the Pythagorean theorem and so"},{"Start":"01:56.045 ","End":"01:58.280","Text":"the magnitude of the vector equals"},{"Start":"01:58.280 ","End":"02:02.900","Text":"the square root of the x element squared plus the y elements squared."},{"Start":"02:02.900 ","End":"02:05.630","Text":"The other way to do it is tangent of the angle"},{"Start":"02:05.630 ","End":"02:10.040","Text":"Theta equals the y element over the x element."},{"Start":"02:10.040 ","End":"02:12.740","Text":"Let\u0027s assume for a moment that the magnitude of"},{"Start":"02:12.740 ","End":"02:16.115","Text":"my vector is 4 and the angle of Theta is 30 degrees."},{"Start":"02:16.115 ","End":"02:19.100","Text":"Then I can use my formula to find the x-component,"},{"Start":"02:19.100 ","End":"02:22.625","Text":"which would equal 4 times the cosine of 30 degrees."},{"Start":"02:22.625 ","End":"02:30.170","Text":"That equals 4 times the square root of 3 over 2 and my end result is 2 square root of 3."},{"Start":"02:30.170 ","End":"02:33.740","Text":"For the y element, I find that it\u0027s equal to"},{"Start":"02:33.740 ","End":"02:39.160","Text":"4 times the sine of 30 degrees and 4 times sine of 30 degrees equals 2,"},{"Start":"02:39.160 ","End":"02:44.630","Text":"so that I now know my x and y coordinates of the vector."},{"Start":"02:44.630 ","End":"02:49.174","Text":"If the vector starts at the origin and follows its trajectory,"},{"Start":"02:49.174 ","End":"02:54.620","Text":"it will reach the points 2 square root of 3 on the x-axis,"},{"Start":"02:54.620 ","End":"02:58.685","Text":"and it will reach the 0.2 on the y-axis."},{"Start":"02:58.685 ","End":"03:03.400","Text":"Those are the coordinates of the end of my vector if it starts at the origin."},{"Start":"03:03.400 ","End":"03:08.115","Text":"These 2 coordinate points 2 square root of 3,"},{"Start":"03:08.115 ","End":"03:13.130","Text":"and 2 are the transpositions that projections of the end of the vector onto"},{"Start":"03:13.130 ","End":"03:15.710","Text":"the x-axis and the y-axis and that tells me"},{"Start":"03:15.710 ","End":"03:18.980","Text":"the coordinates that the vector ends at if it starts at the origin."},{"Start":"03:18.980 ","End":"03:20.960","Text":"Now if I were to do this in reverse and we\u0027re"},{"Start":"03:20.960 ","End":"03:24.140","Text":"given those 2 coordinates to find the magnitude,"},{"Start":"03:24.140 ","End":"03:28.130","Text":"we can say that the magnitude equals the square root of 2,"},{"Start":"03:28.130 ","End":"03:32.745","Text":"square root 3 squared, which is 4 times 3 plus 2 squared, which is 4,"},{"Start":"03:32.745 ","End":"03:37.005","Text":"so the square root of that is 12 plus 4 is square root of 16,"},{"Start":"03:37.005 ","End":"03:39.605","Text":"which equals 4 which we already know is our answer."},{"Start":"03:39.605 ","End":"03:47.830","Text":"If we\u0027re trying to find the angle of Theta. Tangent of Theta equals 2 square root 3,"},{"Start":"03:47.830 ","End":"03:53.120","Text":"which gives us tangent Theta equals 1 over square root of 3."},{"Start":"03:53.120 ","End":"03:56.605","Text":"If you were to do that on your calculator and you do shift tan,"},{"Start":"03:56.605 ","End":"04:03.385","Text":"which does the inverse of a tangent and you\u0027ll find that Theta is equal to 30 degrees."},{"Start":"04:03.385 ","End":"04:05.705","Text":"These 2 systems are interchangeable."},{"Start":"04:05.705 ","End":"04:08.959","Text":"You can display something as magnitude and direction,"},{"Start":"04:08.959 ","End":"04:11.390","Text":"or you can display something as x and y coordinates."},{"Start":"04:11.390 ","End":"04:13.775","Text":"It depends on the system that you\u0027re trying to use,"},{"Start":"04:13.775 ","End":"04:16.895","Text":"so something using magnitude and"},{"Start":"04:16.895 ","End":"04:21.425","Text":"a direction would be using a polar coordinate or cylindrical coordinate system,"},{"Start":"04:21.425 ","End":"04:24.170","Text":"we would call that a polar or cylindrical"},{"Start":"04:24.170 ","End":"04:28.460","Text":"projection and something using x and y uses your Cartesian coordinate system."},{"Start":"04:28.460 ","End":"04:30.620","Text":"We call that a Cartesian projection."},{"Start":"04:30.620 ","End":"04:33.305","Text":"Because the 2 are mathematically equivalent,"},{"Start":"04:33.305 ","End":"04:36.260","Text":"I can choose which system of coordinates I want to use for"},{"Start":"04:36.260 ","End":"04:40.620","Text":"my question depending on what seems easiest for that question."}],"ID":9199},{"Watched":false,"Name":"More Notes on Breaking Down Vectors","Duration":"7m 5s","ChapterTopicVideoID":8926,"CourseChapterTopicPlaylistID":85317,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8926.jpeg","UploadDate":"2021-11-18T08:53:29.2330000","DurationForVideoObject":"PT7M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:03.840","Text":"I wanted to add a few more notes about breaking down vectors."},{"Start":"00:03.840 ","End":"00:07.140","Text":"The first thing is when you\u0027re looking for your angle Theta by doing"},{"Start":"00:07.140 ","End":"00:11.235","Text":"shift tan of your y element A_y over x element A_x,"},{"Start":"00:11.235 ","End":"00:13.320","Text":"you\u0027re going to get 2 options for Theta."},{"Start":"00:13.320 ","End":"00:15.445","Text":"The first is what your calculator is going to give you,"},{"Start":"00:15.445 ","End":"00:18.805","Text":"and the second is what your calculator gives you plus 180 degrees."},{"Start":"00:18.805 ","End":"00:20.885","Text":"We\u0027ll call that Theta 2 and Theta 1."},{"Start":"00:20.885 ","End":"00:22.590","Text":"Theta 2 is what your calculator gives you."},{"Start":"00:22.590 ","End":"00:26.265","Text":"Theta 1 is 180 degrees more on top of that."},{"Start":"00:26.265 ","End":"00:29.640","Text":"The question is, how do you choose which 1 is correct because they\u0027ll both"},{"Start":"00:29.640 ","End":"00:34.185","Text":"be correct answers for the shift tan of A_y over A_x."},{"Start":"00:34.185 ","End":"00:37.930","Text":"First, you have to know what quadrant your vector is in."},{"Start":"00:37.930 ","End":"00:41.270","Text":"For example, if we look at the vector on our left here,"},{"Start":"00:41.270 ","End":"00:43.100","Text":"we know that that\u0027s in the first quadrant"},{"Start":"00:43.100 ","End":"00:46.190","Text":"so x has to be positive and y has to be positive."},{"Start":"00:46.190 ","End":"00:49.955","Text":"That means the angle should be between 0 and 90 degrees."},{"Start":"00:49.955 ","End":"00:54.290","Text":"But if we had, for example, the opposite vector,"},{"Start":"00:54.290 ","End":"00:56.555","Text":"the negative vector of that,"},{"Start":"00:56.555 ","End":"01:01.685","Text":"we know it\u0027s in the third quadrants that it should be between 180 and 270 degrees."},{"Start":"01:01.685 ","End":"01:04.625","Text":"We know which of the 2 to choose."},{"Start":"01:04.625 ","End":"01:07.115","Text":"With your negative or opposite vector,"},{"Start":"01:07.115 ","End":"01:08.480","Text":"instead of being x,"},{"Start":"01:08.480 ","End":"01:09.650","Text":"y, it\u0027d be negative x,"},{"Start":"01:09.650 ","End":"01:12.080","Text":"negative y, and if you put that in your calculator,"},{"Start":"01:12.080 ","End":"01:14.600","Text":"you\u0027ll find the exact same solution for Theta."},{"Start":"01:14.600 ","End":"01:19.175","Text":"But we know that if we\u0027re looking for an angle that is 0-90 degrees,"},{"Start":"01:19.175 ","End":"01:23.210","Text":"that we need to choose then that would be a first quadrant vector."},{"Start":"01:23.210 ","End":"01:26.495","Text":"If we\u0027re looking for an angle between 180 and 270 degrees,"},{"Start":"01:26.495 ","End":"01:31.690","Text":"which is your angle Theta 2 plus 180 degrees,"},{"Start":"01:31.690 ","End":"01:38.825","Text":"then you\u0027re going to look for something in the third quadrant or your opposite vector."},{"Start":"01:38.825 ","End":"01:41.945","Text":"Just to repeat that, if I know that I\u0027m looking for,"},{"Start":"01:41.945 ","End":"01:44.240","Text":"let\u0027s say, again, the vector on the left,"},{"Start":"01:44.240 ","End":"01:45.785","Text":"then I would know that I\u0027m looking for something between"},{"Start":"01:45.785 ","End":"01:48.830","Text":"0 and 90 degrees so choose my positive vector,"},{"Start":"01:48.830 ","End":"01:50.090","Text":"the vector in the first quadrant."},{"Start":"01:50.090 ","End":"01:53.120","Text":"If I\u0027m looking for something that\u0027s the opposite of that,"},{"Start":"01:53.120 ","End":"01:58.865","Text":"I would know to add 180 degrees to my calculator results and get that negative angle."},{"Start":"01:58.865 ","End":"02:00.710","Text":"Now, here\u0027s a second example."},{"Start":"02:00.710 ","End":"02:05.925","Text":"Second case is if we\u0027re given the points negative 3 and 5."},{"Start":"02:05.925 ","End":"02:10.685","Text":"Now, if I want to do shift tan of negative 3 and 5 in my calculator,"},{"Start":"02:10.685 ","End":"02:14.965","Text":"it will come out negative 5/3,"},{"Start":"02:14.965 ","End":"02:21.530","Text":"and the answer will be negative 59.03 degrees more or less."},{"Start":"02:21.530 ","End":"02:22.700","Text":"What does that tell me?"},{"Start":"02:22.700 ","End":"02:24.920","Text":"That\u0027s a vector in"},{"Start":"02:24.920 ","End":"02:30.275","Text":"the fourth quadrant with an angle of negative 59 degrees from the x-axis."},{"Start":"02:30.275 ","End":"02:34.168","Text":"Now, what I can do is add 180 degrees to that,"},{"Start":"02:34.168 ","End":"02:39.750","Text":"and I get something in the neighborhood of 120.96 degrees,"},{"Start":"02:39.750 ","End":"02:45.320","Text":"that will be something between 90 and 180 degrees 120.96,"},{"Start":"02:45.320 ","End":"02:47.195","Text":"that would be in the second quadrant."},{"Start":"02:47.195 ","End":"02:50.750","Text":"That would be with a positive y and a negative x. whereas"},{"Start":"02:50.750 ","End":"02:54.710","Text":"our first initial result from the calculator gives us a negative y and a positive x."},{"Start":"02:54.710 ","End":"02:56.810","Text":"How do I figure out which I need?"},{"Start":"02:56.810 ","End":"03:00.020","Text":"Well, if I look at my coordinates negative 3 and 5,"},{"Start":"03:00.020 ","End":"03:03.620","Text":"I know that my negative 3, my x that is,"},{"Start":"03:03.620 ","End":"03:07.680","Text":"is negative, and my y, 5 is positive."},{"Start":"03:07.680 ","End":"03:10.940","Text":"I know that I\u0027m looking for something in the second quadrant."},{"Start":"03:10.940 ","End":"03:13.505","Text":"That means that I\u0027m not going to use my first result."},{"Start":"03:13.505 ","End":"03:16.610","Text":"I\u0027m going to use the second result where I added 180 degrees so that"},{"Start":"03:16.610 ","End":"03:21.925","Text":"my real answer is not negative 59, it\u0027s 120.96."},{"Start":"03:21.925 ","End":"03:26.930","Text":"The vector that we\u0027re not using has the coordinates 3 negative 5,"},{"Start":"03:26.930 ","End":"03:28.520","Text":"and if I put that into my calculator,"},{"Start":"03:28.520 ","End":"03:30.065","Text":"I\u0027m going to get the exact same result."},{"Start":"03:30.065 ","End":"03:32.330","Text":"My calculator doesn\u0027t know if I\u0027m using negative 3,"},{"Start":"03:32.330 ","End":"03:33.860","Text":"5 or 3 negative 5."},{"Start":"03:33.860 ","End":"03:37.940","Text":"The lesson here is that you need to look at the points are given because you might"},{"Start":"03:37.940 ","End":"03:42.200","Text":"be given 2 opposite vectors that will give you the exact same results in your calculator."},{"Start":"03:42.200 ","End":"03:44.810","Text":"You need to be weary of what the actual coordinates"},{"Start":"03:44.810 ","End":"03:47.970","Text":"are so you can find your correct answer for the angle that is."},{"Start":"03:47.970 ","End":"03:51.380","Text":"Another problem you might encounter is breaking down an angle relative"},{"Start":"03:51.380 ","End":"03:54.845","Text":"to the y-axis or any other axis that\u0027s not the x-axis."},{"Start":"03:54.845 ","End":"03:58.175","Text":"When I\u0027m given an angle like Theta here for the vector b,"},{"Start":"03:58.175 ","End":"04:03.590","Text":"what I want to do is first I\u0027m going to find my magnitudes in terms of x and y,"},{"Start":"04:03.590 ","End":"04:04.880","Text":"the x and the y components."},{"Start":"04:04.880 ","End":"04:08.800","Text":"Then I\u0027ll add my symbols either plus or minus on my own."},{"Start":"04:08.800 ","End":"04:16.228","Text":"For B_y, I know that it\u0027s equal to the magnitude of B times the cosine of Theta."},{"Start":"04:16.228 ","End":"04:17.890","Text":"For B_x, my x component,"},{"Start":"04:17.890 ","End":"04:22.250","Text":"I know that it\u0027s equal to the magnitude of the vector b times the sine of Theta."},{"Start":"04:22.250 ","End":"04:23.480","Text":"Now, I look and say, okay,"},{"Start":"04:23.480 ","End":"04:25.430","Text":"my vectors in the second quadrant,"},{"Start":"04:25.430 ","End":"04:27.860","Text":"I know that my y component has to be positive,"},{"Start":"04:27.860 ","End":"04:29.630","Text":"so I can keep that positive."},{"Start":"04:29.630 ","End":"04:34.910","Text":"I also know that my x component has to be negative in the second quadrant, x is negative."},{"Start":"04:34.910 ","End":"04:37.595","Text":"On my own, adding the negative sign,"},{"Start":"04:37.595 ","End":"04:44.040","Text":"the minus sign before the magnitude of the vector B is that my end result is negative."},{"Start":"04:44.040 ","End":"04:49.685","Text":"To recap, if I\u0027m breaking down a vector using an angle that\u0027s not relative to the x-axis."},{"Start":"04:49.685 ","End":"04:51.260","Text":"I\u0027ll do the geometry first."},{"Start":"04:51.260 ","End":"04:53.705","Text":"Then once I\u0027ve found the magnitudes,"},{"Start":"04:53.705 ","End":"04:58.670","Text":"I can add in the positive or negative attribute on my own."},{"Start":"04:58.670 ","End":"05:00.710","Text":"That\u0027s the first way to do things."},{"Start":"05:00.710 ","End":"05:05.465","Text":"The second way is to make the angle relative to the x-axis."},{"Start":"05:05.465 ","End":"05:07.993","Text":"Then the plus and minus should work itself out,"},{"Start":"05:07.993 ","End":"05:10.010","Text":"and I don\u0027t have to deal with it myself."},{"Start":"05:10.010 ","End":"05:13.580","Text":"In this case, because the angle Theta is relative to the y-axis,"},{"Start":"05:13.580 ","End":"05:16.390","Text":"relative to the x-axis will just be 90 degrees more."},{"Start":"05:16.390 ","End":"05:20.555","Text":"Let\u0027s call it Theta squiggle equals our initial Theta plus 90 degrees."},{"Start":"05:20.555 ","End":"05:27.964","Text":"The x element equals the magnitude of B times the cosine of Theta plus 90 degrees,"},{"Start":"05:27.964 ","End":"05:33.890","Text":"and the y element is the magnitude of B times the sine of Theta plus 90 degrees."},{"Start":"05:33.890 ","End":"05:35.533","Text":"That way it\u0027ll work itself out,"},{"Start":"05:35.533 ","End":"05:37.320","Text":"and you don\u0027t have to deal with positive and negative,"},{"Start":"05:37.320 ","End":"05:42.320","Text":"you\u0027ll find your correct results by repairing it so that it\u0027s relative to the x-axis."},{"Start":"05:42.320 ","End":"05:45.050","Text":"Let\u0027s say that Theta is 30 degrees in this case,"},{"Start":"05:45.050 ","End":"05:47.345","Text":"and B\u0027s magnitude is 2."},{"Start":"05:47.345 ","End":"05:50.990","Text":"If I plug it in using our first method,"},{"Start":"05:50.990 ","End":"05:55.290","Text":"the y element equals 2 times the cosine of 30 degrees,"},{"Start":"05:55.290 ","End":"05:58.905","Text":"that equals 2 times the square root of 3 over 2, or square root of 3."},{"Start":"05:58.905 ","End":"06:00.950","Text":"The x element will be negative,"},{"Start":"06:00.950 ","End":"06:02.420","Text":"which we add in ourselves,"},{"Start":"06:02.420 ","End":"06:05.030","Text":"times 2 times the sine of 30,"},{"Start":"06:05.030 ","End":"06:07.495","Text":"which will give us a result of negative 1."},{"Start":"06:07.495 ","End":"06:09.800","Text":"If I use my second method,"},{"Start":"06:09.800 ","End":"06:14.270","Text":"I\u0027ll do for the x component 2 times the cosine of 120 degrees,"},{"Start":"06:14.270 ","End":"06:15.950","Text":"that\u0027s 30 plus 90,"},{"Start":"06:15.950 ","End":"06:17.705","Text":"and the result is negative 1,"},{"Start":"06:17.705 ","End":"06:19.775","Text":"just like I had with my first method."},{"Start":"06:19.775 ","End":"06:24.680","Text":"The y component is going to equal 2 times the sine of 120,"},{"Start":"06:24.680 ","End":"06:26.405","Text":"again, 30 plus 90."},{"Start":"06:26.405 ","End":"06:29.990","Text":"Just like 2 times square root of 3 over 2 equals square root of 3."},{"Start":"06:29.990 ","End":"06:33.010","Text":"This will give me a result of square root of 3."},{"Start":"06:33.010 ","End":"06:35.060","Text":"To give a little summary,"},{"Start":"06:35.060 ","End":"06:39.260","Text":"if I make sure that my angle is always relative to the x-axis,"},{"Start":"06:39.260 ","End":"06:44.023","Text":"I can always have my x be a cosine and my y be sine,"},{"Start":"06:44.023 ","End":"06:48.950","Text":"and my negative or positive value will sort itself out through the geometry itself."},{"Start":"06:48.950 ","End":"06:52.160","Text":"If I want to keep it relative to the y-axis,"},{"Start":"06:52.160 ","End":"06:57.230","Text":"I need to then add in the symbols myself and do the geometry just to find the magnitude."},{"Start":"06:57.230 ","End":"06:58.970","Text":"But it will also work itself out as long as I\u0027m"},{"Start":"06:58.970 ","End":"07:01.130","Text":"diligent with my signs positive or negative."},{"Start":"07:01.130 ","End":"07:05.340","Text":"Those are extra notes for breaking down vectors."}],"ID":9200},{"Watched":false,"Name":"Adding Vectors Algebraically","Duration":"1m 33s","ChapterTopicVideoID":8927,"CourseChapterTopicPlaylistID":85317,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8927.jpeg","UploadDate":"2021-11-18T08:53:35.4700000","DurationForVideoObject":"PT1M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"I want to take this opportunity to show you how we add vectors algebraically."},{"Start":"00:04.890 ","End":"00:07.830","Text":"Usually, this is the preferred method for adding vectors."},{"Start":"00:07.830 ","End":"00:11.790","Text":"We\u0027re not going to draw out 2 vectors like we did geometrically and draw lines."},{"Start":"00:11.790 ","End":"00:13.829","Text":"We can just calculate this mathematically."},{"Start":"00:13.829 ","End":"00:15.540","Text":"We don\u0027t need to do tail to head,"},{"Start":"00:15.540 ","End":"00:17.160","Text":"head to head, tail to tail."},{"Start":"00:17.160 ","End":"00:19.260","Text":"This is easier, I think."},{"Start":"00:19.260 ","End":"00:21.680","Text":"When we add vectors A and vector B,"},{"Start":"00:21.680 ","End":"00:25.220","Text":"we basically add the x elements and then we add the y elements,"},{"Start":"00:25.220 ","End":"00:28.470","Text":"and then we come up with new x and y elements."},{"Start":"00:28.470 ","End":"00:32.565","Text":"If vector A plus vector B equals vector C,"},{"Start":"00:32.565 ","End":"00:36.930","Text":"Then we can suppose that A_x plus B_x equals"},{"Start":"00:36.930 ","End":"00:42.470","Text":"C_x and the components A_y and B_y equals the new components C_y."},{"Start":"00:42.470 ","End":"00:44.630","Text":"Let\u0027s do a numerical example."},{"Start":"00:44.630 ","End":"00:49.445","Text":"Vector A is 2,3 and vector B is negative 1,3."},{"Start":"00:49.445 ","End":"00:53.375","Text":"I can put those together 2 minus 1 and 3 plus 3."},{"Start":"00:53.375 ","End":"00:57.490","Text":"I end up with 1/6 as my C coordinates."},{"Start":"00:57.490 ","End":"01:02.195","Text":"In the end, C equals 1 for x and 6 for y."},{"Start":"01:02.195 ","End":"01:07.144","Text":"Now a second example is if I\u0027m given my vector in terms of magnitude and degrees,"},{"Start":"01:07.144 ","End":"01:11.350","Text":"I need to first break it down into x and y and then I can add them."},{"Start":"01:11.350 ","End":"01:15.185","Text":"For example, if A is a magnitude 6 angle of 30,"},{"Start":"01:15.185 ","End":"01:18.395","Text":"B has a magnitude of 2 and an angle of 120 degrees."},{"Start":"01:18.395 ","End":"01:21.500","Text":"I can break down my A_x and A_y as you see here,"},{"Start":"01:21.500 ","End":"01:24.460","Text":"and I get 3 root 3 and 3."},{"Start":"01:24.460 ","End":"01:29.000","Text":"For B, I perform the same operation and get 1 and root 3."},{"Start":"01:29.000 ","End":"01:30.620","Text":"If I add those together,"},{"Start":"01:30.620 ","End":"01:33.420","Text":"I get my results as you can see below."}],"ID":9201},{"Watched":false,"Name":"Multiplying a Vector by a Scalar","Duration":"1m 51s","ChapterTopicVideoID":8928,"CourseChapterTopicPlaylistID":85317,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8928.jpeg","UploadDate":"2017-03-21T08:32:51.8930000","DurationForVideoObject":"PT1M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"Hello. Now I want to show you how to multiply a vector by a scalar."},{"Start":"00:04.725 ","End":"00:06.960","Text":"Now the way we do that is if you recall,"},{"Start":"00:06.960 ","End":"00:09.840","Text":"a scalar only has magnitude and doesn\u0027t have direction."},{"Start":"00:09.840 ","End":"00:14.100","Text":"We\u0027ll only change the magnitude of our vector and not the direction or the angle of it."},{"Start":"00:14.100 ","End":"00:18.060","Text":"If we\u0027re doing this in Cartesian coordinates, let\u0027s say we\u0027re multiplying,"},{"Start":"00:18.060 ","End":"00:23.715","Text":"c by the vector A will end up with c times each of the coordinates."},{"Start":"00:23.715 ","End":"00:25.650","Text":"C times A_x, times A_y,"},{"Start":"00:25.650 ","End":"00:29.430","Text":"times A_z or cA_x, cA_y, cA_z."},{"Start":"00:29.430 ","End":"00:31.305","Text":"If in our example,"},{"Start":"00:31.305 ","End":"00:33.354","Text":"A has the coordinates 1,"},{"Start":"00:33.354 ","End":"00:37.325","Text":"2, 3, then when we find our new vector B,"},{"Start":"00:37.325 ","End":"00:40.165","Text":"which is A times the scalar c,"},{"Start":"00:40.165 ","End":"00:42.420","Text":"will have 5 times 1,"},{"Start":"00:42.420 ","End":"00:46.185","Text":"5 times 2, 5 times 3."},{"Start":"00:46.185 ","End":"00:48.197","Text":"Our result ends up being 5,"},{"Start":"00:48.197 ","End":"00:49.500","Text":"10, 15,"},{"Start":"00:49.500 ","End":"00:53.420","Text":"so that we end up with 5 times the magnitude,"},{"Start":"00:53.420 ","End":"00:55.100","Text":"but no change in direction."},{"Start":"00:55.100 ","End":"00:59.195","Text":"Again, our result will only affect the magnitude and not the angle or the direction."},{"Start":"00:59.195 ","End":"01:02.120","Text":"Now if we\u0027re doing this in polar or cylindrical coordinates,"},{"Start":"01:02.120 ","End":"01:04.955","Text":"let\u0027s say we\u0027re not given the magnitude,"},{"Start":"01:04.955 ","End":"01:08.180","Text":"and we have to find it, or if we\u0027re given the magnitude doesn\u0027t matter."},{"Start":"01:08.180 ","End":"01:11.075","Text":"Either way, the magnitude is going to be root 14."},{"Start":"01:11.075 ","End":"01:13.805","Text":"When we multiply by the magnitude,"},{"Start":"01:13.805 ","End":"01:18.495","Text":"it\u0027s going to be 5 times magnitude A as you can see below,"},{"Start":"01:18.495 ","End":"01:20.600","Text":"and that\u0027s 5 times root 14."},{"Start":"01:20.600 ","End":"01:22.880","Text":"But again, that only will affect"},{"Start":"01:22.880 ","End":"01:26.510","Text":"our magnitude and will not affect the direction of our vector."},{"Start":"01:26.510 ","End":"01:30.560","Text":"The magnitude of vector B is 5 times the magnitude of vector A,"},{"Start":"01:30.560 ","End":"01:33.380","Text":"but the direction, the angle Theta is the same."},{"Start":"01:33.380 ","End":"01:36.110","Text":"For a quick recap in Cartesian coordinates,"},{"Start":"01:36.110 ","End":"01:38.720","Text":"you multiply it by each of the coordinates."},{"Start":"01:38.720 ","End":"01:41.525","Text":"In cylindrical or polar coordinates,"},{"Start":"01:41.525 ","End":"01:44.675","Text":"you multiply by the magnitude as a whole."},{"Start":"01:44.675 ","End":"01:48.870","Text":"Again, it\u0027s going to change only your magnitude and not your direction."},{"Start":"01:48.870 ","End":"01:51.930","Text":"That\u0027s how you multiply a vector by a scalar."}],"ID":9202},{"Watched":false,"Name":"Exercise- Adding And Subtracting Vectors","Duration":"4m 5s","ChapterTopicVideoID":8929,"CourseChapterTopicPlaylistID":85317,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8929.jpeg","UploadDate":"2017-03-21T08:33:27.7870000","DurationForVideoObject":"PT4M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"Hello. We are given 3 vectors, A,"},{"Start":"00:03.300 ","End":"00:06.375","Text":"B, and C, each with an x and a y component."},{"Start":"00:06.375 ","End":"00:07.650","Text":"This is the x-component,"},{"Start":"00:07.650 ","End":"00:08.835","Text":"this is the y component,"},{"Start":"00:08.835 ","End":"00:10.905","Text":"x, y, x, and y."},{"Start":"00:10.905 ","End":"00:12.615","Text":"Now, in our first question,"},{"Start":"00:12.615 ","End":"00:17.175","Text":"we\u0027re being asked to calculate what a plus b plus c is."},{"Start":"00:17.175 ","End":"00:19.395","Text":"Let\u0027s see how we do this."},{"Start":"00:19.395 ","End":"00:22.560","Text":"Now, what we have to do in vector addition is"},{"Start":"00:22.560 ","End":"00:25.220","Text":"we have to add up each of the components separately."},{"Start":"00:25.220 ","End":"00:28.220","Text":"First, we add up all of the x components and then we add"},{"Start":"00:28.220 ","End":"00:32.160","Text":"up all of the y components. Let\u0027s answer this."},{"Start":"00:32.160 ","End":"00:35.579","Text":"For question Number 1 of the x components,"},{"Start":"00:35.579 ","End":"00:36.690","Text":"let\u0027s call this new vector,"},{"Start":"00:36.690 ","End":"00:42.550","Text":"Vector D equals 1 plus 4 plus 3,"},{"Start":"00:42.550 ","End":"00:46.205","Text":"which is just all the x components from vectors a, b, and c,"},{"Start":"00:46.205 ","End":"00:47.480","Text":"and now all the y\u0027s,"},{"Start":"00:47.480 ","End":"00:51.580","Text":"so we\u0027re going to have 3 plus 2 plus 5,"},{"Start":"00:51.580 ","End":"00:58.890","Text":"which makes this vector equal to 8, 10."},{"Start":"00:58.910 ","End":"01:01.155","Text":"To the next question."},{"Start":"01:01.155 ","End":"01:02.310","Text":"In the next question,"},{"Start":"01:02.310 ","End":"01:07.015","Text":"we\u0027re being asked to calculate what a minus b minus c is."},{"Start":"01:07.015 ","End":"01:10.085","Text":"Exactly like in the previous question,"},{"Start":"01:10.085 ","End":"01:13.640","Text":"all we have to do is minus all of"},{"Start":"01:13.640 ","End":"01:18.800","Text":"the x components and then minus all of the y components."},{"Start":"01:18.800 ","End":"01:25.404","Text":"Let\u0027s call this Vector E. Then we\u0027re going to subtract all the x components."},{"Start":"01:25.404 ","End":"01:28.740","Text":"We have 1 minus 4 minus 3."},{"Start":"01:28.740 ","End":"01:35.450","Text":"It\u0027s the exact same thing but with a minus and then comma 3 minus 2, minus 5."},{"Start":"01:35.450 ","End":"01:45.220","Text":"Then this in total equals negative 6 and negative 4."},{"Start":"01:45.220 ","End":"01:47.345","Text":"That\u0027s it for that question."},{"Start":"01:47.345 ","End":"01:49.265","Text":"Now, in the third question,"},{"Start":"01:49.265 ","End":"01:54.395","Text":"we\u0027re being asked to calculate what 2a plus 3b minus 4c is."},{"Start":"01:54.395 ","End":"02:00.230","Text":"Let\u0027s call this F. Notice I\u0027ve put arrows, sorry,"},{"Start":"02:00.230 ","End":"02:04.220","Text":"like this at the top of each letter vector,"},{"Start":"02:04.220 ","End":"02:08.495","Text":"because it\u0027s the more correct way to symbolize a vector."},{"Start":"02:08.495 ","End":"02:11.885","Text":"You write it with an arrow on top or bold."},{"Start":"02:11.885 ","End":"02:14.330","Text":"There\u0027s a lesson about this if you don\u0027t remember that,"},{"Start":"02:14.330 ","End":"02:16.380","Text":"go back to the lesson."},{"Start":"02:16.850 ","End":"02:19.695","Text":"As you notice in this question,"},{"Start":"02:19.695 ","End":"02:22.240","Text":"there is a coefficient in front of each letter,"},{"Start":"02:22.240 ","End":"02:25.550","Text":"which means that each component be it x and"},{"Start":"02:25.550 ","End":"02:29.105","Text":"the y components is multiplied by this coefficient."},{"Start":"02:29.105 ","End":"02:35.779","Text":"With a, we\u0027ll have 2 times the x coefficient and 2 times the y coefficient."},{"Start":"02:35.779 ","End":"02:39.260","Text":"Let\u0027s see how this works. Let\u0027s start."},{"Start":"02:39.260 ","End":"02:41.550","Text":"We have 2a,"},{"Start":"02:41.550 ","End":"02:43.140","Text":"so that\u0027s 2 times 1,"},{"Start":"02:43.140 ","End":"02:48.720","Text":"which is 2 plus 3b so that\u0027s 3 times 4,"},{"Start":"02:48.720 ","End":"02:56.310","Text":"which is plus 12 minus 4 times c,"},{"Start":"02:56.310 ","End":"02:57.900","Text":"which is minus 4 times 3,"},{"Start":"02:57.900 ","End":"03:00.585","Text":"which is minus 12."},{"Start":"03:00.585 ","End":"03:03.270","Text":"Now for the y components,"},{"Start":"03:03.270 ","End":"03:06.850","Text":"2a is here 2 times"},{"Start":"03:06.920 ","End":"03:13.785","Text":"6 then we have plus 3 times b."},{"Start":"03:13.785 ","End":"03:20.055","Text":"It\u0027s 3 times 2 plus 6 and then we have minus 4 times c,"},{"Start":"03:20.055 ","End":"03:21.885","Text":"which is minus 4 times 5,"},{"Start":"03:21.885 ","End":"03:24.480","Text":"which is minus 20."},{"Start":"03:24.480 ","End":"03:31.185","Text":"Then this equals grand total of 2 and negative 8."},{"Start":"03:31.185 ","End":"03:33.935","Text":"There we go, this is the end of our question."},{"Start":"03:33.935 ","End":"03:38.660","Text":"Now, it\u0027s important to remember when working with vector addition, subtraction,"},{"Start":"03:38.660 ","End":"03:46.685","Text":"and scalar multiplication is that you work on each component,"},{"Start":"03:46.685 ","End":"03:50.915","Text":"the x, y, if you have z components or however many components you have,"},{"Start":"03:50.915 ","End":"03:54.425","Text":"you work on each component separately."},{"Start":"03:54.425 ","End":"03:56.750","Text":"If it\u0027s an addition, first,"},{"Start":"03:56.750 ","End":"03:58.130","Text":"you add all the x components,"},{"Start":"03:58.130 ","End":"03:59.600","Text":"then you add all the y components."},{"Start":"03:59.600 ","End":"04:03.395","Text":"Subtraction is the same as addition and with multiplication."},{"Start":"04:03.395 ","End":"04:06.030","Text":"That\u0027s the end of the question."}],"ID":9203}],"Thumbnail":null,"ID":85317},{"Name":"Vector Multiplication in Three Dimensions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vector Multiplication Using the Determinant","Duration":"6m 41s","ChapterTopicVideoID":8930,"CourseChapterTopicPlaylistID":85318,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8930.jpeg","UploadDate":"2021-11-18T08:58:11.2170000","DurationForVideoObject":"PT6M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.545","Text":"I want to take this opportunity to talk to you about vector multiplication."},{"Start":"00:04.545 ","End":"00:08.355","Text":"Now, there are 2 main ways to do vector multiplication."},{"Start":"00:08.355 ","End":"00:10.770","Text":"The first is using a determinant and the"},{"Start":"00:10.770 ","End":"00:14.220","Text":"second is to find the magnitude and then the angle."},{"Start":"00:14.220 ","End":"00:18.675","Text":"In this video, we\u0027ll talk about how to do this using a determinant."},{"Start":"00:18.675 ","End":"00:21.255","Text":"I\u0027m going to talk about this purely in mathematic terms,"},{"Start":"00:21.255 ","End":"00:23.775","Text":"ignoring any discussion of torque,"},{"Start":"00:23.775 ","End":"00:27.645","Text":"so anyone who feels they are already comfortable with"},{"Start":"00:27.645 ","End":"00:32.640","Text":"multiplying vectors can move on to the next video."},{"Start":"00:32.640 ","End":"00:39.630","Text":"If I want to multiply the vector a times the vector b using a vector multiplication,"},{"Start":"00:39.630 ","End":"00:42.435","Text":"what I can do is make a determinant."},{"Start":"00:42.435 ","End":"00:44.610","Text":"It depends on the coordinate system I\u0027m using,"},{"Start":"00:44.610 ","End":"00:46.820","Text":"but assuming I\u0027m in Cartesian coordinates,"},{"Start":"00:46.820 ","End":"00:49.325","Text":"I\u0027ll write out in my top row,"},{"Start":"00:49.325 ","End":"00:54.360","Text":"x hat, y hat, and z hat."},{"Start":"00:54.360 ","End":"00:58.770","Text":"The next row, I\u0027ll write out my a components."},{"Start":"00:58.770 ","End":"01:02.715","Text":"There\u0027ll be a_x, a_y, and a_z,"},{"Start":"01:02.715 ","End":"01:05.780","Text":"and then the bottom row, I\u0027ll do the same for b, I\u0027ll write out b_x,"},{"Start":"01:05.780 ","End":"01:09.740","Text":"b_y, and b_z."},{"Start":"01:09.820 ","End":"01:13.174","Text":"When I\u0027m performing the determinant operation,"},{"Start":"01:13.174 ","End":"01:17.660","Text":"what I\u0027ll do is go component by component x, then y, then z."},{"Start":"01:17.660 ","End":"01:19.370","Text":"When I\u0027m working on my x component,"},{"Start":"01:19.370 ","End":"01:20.990","Text":"what I want to do is first,"},{"Start":"01:20.990 ","End":"01:22.790","Text":"read out that it\u0027s my x component,"},{"Start":"01:22.790 ","End":"01:24.170","Text":"put x hat at the end,"},{"Start":"01:24.170 ","End":"01:28.740","Text":"and then I will cross out the row of x,"},{"Start":"01:28.740 ","End":"01:32.955","Text":"y, z, and the row of x components a_x and b_x."},{"Start":"01:32.955 ","End":"01:37.120","Text":"Now I have 4 components left and I\u0027ll multiply them to find my answer,"},{"Start":"01:37.120 ","End":"01:41.030","Text":"so the way I do that is multiply a_y by b_z,"},{"Start":"01:41.030 ","End":"01:42.530","Text":"that\u0027s my main diagonal."},{"Start":"01:42.530 ","End":"01:44.105","Text":"I\u0027ll do that first,"},{"Start":"01:44.105 ","End":"01:48.815","Text":"a_y times b_z, and put that into my solution."},{"Start":"01:48.815 ","End":"01:50.600","Text":"A_y times z,"},{"Start":"01:50.600 ","End":"01:53.930","Text":"and subtract from that the secondary diagonal."},{"Start":"01:53.930 ","End":"01:56.730","Text":"That it\u0027ll be a_z times b_y."},{"Start":"01:56.730 ","End":"01:58.380","Text":"My solution, for now,"},{"Start":"01:58.380 ","End":"02:02.475","Text":"is a_y times b_z minus a_z times b_y."},{"Start":"02:02.475 ","End":"02:06.020","Text":"I have my x component and I can move on to my y component."},{"Start":"02:06.020 ","End":"02:10.820","Text":"Now the way that it works with the determinant is I\u0027ll subtract the next term."},{"Start":"02:10.820 ","End":"02:13.760","Text":"I would go positive, negative, positive, negative."},{"Start":"02:13.760 ","End":"02:16.640","Text":"I do plus x minus y plus z,"},{"Start":"02:16.640 ","End":"02:18.790","Text":"if I had a 4th term, I would subtract that,"},{"Start":"02:18.790 ","End":"02:21.050","Text":"so I\u0027m going to subtract my y term now."},{"Start":"02:21.050 ","End":"02:24.030","Text":"Again, I\u0027m going to do the same procedure just with my y term,"},{"Start":"02:24.030 ","End":"02:26.465","Text":"so I cross out my x, my y, and my Z,"},{"Start":"02:26.465 ","End":"02:28.670","Text":"put my y hat on the end of course,"},{"Start":"02:28.670 ","End":"02:33.095","Text":"and I cross out my y elements a_y, and b_y."},{"Start":"02:33.095 ","End":"02:36.080","Text":"Now, I\u0027ll multiply based on the main diagonal"},{"Start":"02:36.080 ","End":"02:39.125","Text":"and the secondary diagonal and write out my solution."},{"Start":"02:39.125 ","End":"02:44.685","Text":"The main diagonal is a_x times b_z, and I can write that in."},{"Start":"02:44.685 ","End":"02:51.070","Text":"I\u0027m going to subtract from that the secondary diagonal, a_z times b_x."},{"Start":"02:51.070 ","End":"02:53.480","Text":"Now that I\u0027ve solved for my y,"},{"Start":"02:53.480 ","End":"02:59.915","Text":"my y portion of the solution is a_x times b_z minus a_z times b_xy hat."},{"Start":"02:59.915 ","End":"03:02.450","Text":"Now, I\u0027m going to add if you recall, we do positive,"},{"Start":"03:02.450 ","End":"03:05.455","Text":"negative, positive, negative, so on and so forth."},{"Start":"03:05.455 ","End":"03:07.515","Text":"I\u0027ll add my z component,"},{"Start":"03:07.515 ","End":"03:09.440","Text":"and I\u0027m going to do the same procedure again,"},{"Start":"03:09.440 ","End":"03:11.480","Text":"just crossing out my z components."},{"Start":"03:11.480 ","End":"03:13.250","Text":"I cross out my z components,"},{"Start":"03:13.250 ","End":"03:15.080","Text":"I cross out my x, y,"},{"Start":"03:15.080 ","End":"03:17.360","Text":"and z, I\u0027m left with 4 terms,"},{"Start":"03:17.360 ","End":"03:19.250","Text":"and I add my z hat in the end."},{"Start":"03:19.250 ","End":"03:22.940","Text":"Now, my main diagonal is a_x times b_y, and from that,"},{"Start":"03:22.940 ","End":"03:24.845","Text":"I subtract my secondary diagonal,"},{"Start":"03:24.845 ","End":"03:27.930","Text":"which is a_y times b_x."},{"Start":"03:28.250 ","End":"03:31.665","Text":"This is your answer using determinants."},{"Start":"03:31.665 ","End":"03:33.505","Text":"For those who have room,"},{"Start":"03:33.505 ","End":"03:36.250","Text":"this is a formula worth remembering,"},{"Start":"03:36.250 ","End":"03:40.550","Text":"and it might be worthwhile to write it down on your formula sheet."},{"Start":"03:41.390 ","End":"03:46.370","Text":"Now we can do an example to see if we\u0027ve mastered the knowledge."},{"Start":"03:46.370 ","End":"03:50.415","Text":"Vector a is 1, 2,"},{"Start":"03:50.415 ","End":"03:52.710","Text":"3 in terms of x, y, and z,"},{"Start":"03:52.710 ","End":"03:56.910","Text":"Cartesian coordinates, and vector b equals 1,"},{"Start":"03:56.910 ","End":"04:01.570","Text":"negative 1, 3 in Cartesian coordinates again, x, y,"},{"Start":"04:01.570 ","End":"04:06.740","Text":"and z. Vector c equals vector a times vector b,"},{"Start":"04:06.740 ","End":"04:10.730","Text":"so we need to do a vector multiplication and we\u0027ll use a determinant."},{"Start":"04:10.850 ","End":"04:14.110","Text":"We do a determinant, we can then skip"},{"Start":"04:14.110 ","End":"04:18.055","Text":"the matrices stage and write out what our solution might be."},{"Start":"04:18.055 ","End":"04:22.650","Text":"A solution is 2 times 3 minus 3 times"},{"Start":"04:22.650 ","End":"04:27.495","Text":"negative 1 for our x component minus our y component."},{"Start":"04:27.495 ","End":"04:37.170","Text":"Our y component will be 1 times 3 minus 3 times 1 y hat plus our z component,"},{"Start":"04:37.170 ","End":"04:42.540","Text":"and our z-component is 1 times negative 1 minus 2 times 1."},{"Start":"04:42.540 ","End":"04:44.575","Text":"If we calculate this,"},{"Start":"04:44.575 ","End":"04:48.300","Text":"our c vector ends up equaling 9,"},{"Start":"04:48.300 ","End":"04:51.135","Text":"0, negative 3,"},{"Start":"04:51.135 ","End":"04:53.688","Text":"and that\u0027s our solution."},{"Start":"04:53.688 ","End":"04:58.070","Text":"One characteristic that is important to remember is that our c vector,"},{"Start":"04:58.070 ","End":"05:04.860","Text":"the solution of our multiplication will be perpendicular to our first 2 terms."},{"Start":"05:05.240 ","End":"05:07.815","Text":"Remember, we\u0027re in a 3-dimensional field."},{"Start":"05:07.815 ","End":"05:09.585","Text":"If a and b are in some plane,"},{"Start":"05:09.585 ","End":"05:12.200","Text":"then c will be perpendicular to that plane."},{"Start":"05:12.200 ","End":"05:14.600","Text":"This is the best I can do to draw it out 3 dimensions."},{"Start":"05:14.600 ","End":"05:16.804","Text":"It\u0027s obviously hard to imagine sometimes."},{"Start":"05:16.804 ","End":"05:20.825","Text":"A second characteristic that we should talk about"},{"Start":"05:20.825 ","End":"05:25.520","Text":"is that a times b gives you 1 solution,"},{"Start":"05:25.520 ","End":"05:27.785","Text":"but if you did b cross a,"},{"Start":"05:27.785 ","End":"05:30.030","Text":"then you would end up with the negative solution of that,"},{"Start":"05:30.030 ","End":"05:32.105","Text":"so c would face the opposite direction,"},{"Start":"05:32.105 ","End":"05:36.160","Text":"and it\u0027d be the same as negative a_xb."},{"Start":"05:36.160 ","End":"05:38.265","Text":"Instead of the answer, in this case,"},{"Start":"05:38.265 ","End":"05:40.040","Text":"9, 0, negative 3,"},{"Start":"05:40.040 ","End":"05:41.900","Text":"you would end up with the answer negative 9,"},{"Start":"05:41.900 ","End":"05:45.840","Text":"0, positive 3."},{"Start":"05:45.840 ","End":"05:48.350","Text":"One last thing to mention is if we\u0027re looking for"},{"Start":"05:48.350 ","End":"05:51.695","Text":"the magnitude of the vector c. We\u0027re not interested in the direction,"},{"Start":"05:51.695 ","End":"05:53.510","Text":"but rather only the magnitude,"},{"Start":"05:53.510 ","End":"05:56.015","Text":"there is an operation we can do for that as well."},{"Start":"05:56.015 ","End":"05:59.105","Text":"The magnitude of the vector c is of course,"},{"Start":"05:59.105 ","End":"06:05.125","Text":"equal to the magnitude of the vector multiplication a vector times b vector."},{"Start":"06:05.125 ","End":"06:09.080","Text":"What that\u0027s equal to is the magnitude of the vector a times"},{"Start":"06:09.080 ","End":"06:12.770","Text":"the magnitude of the vector b times sine Alpha,"},{"Start":"06:12.770 ","End":"06:16.120","Text":"Alpha being the angle between vector a and vector b."},{"Start":"06:16.120 ","End":"06:17.710","Text":"If that\u0027s the angle Alpha,"},{"Start":"06:17.710 ","End":"06:21.200","Text":"we can do magnitude a times magnitude of b times sine Alpha,"},{"Start":"06:21.200 ","End":"06:24.890","Text":"and we\u0027ll get the same magnitude of c. You can do it this way,"},{"Start":"06:24.890 ","End":"06:26.360","Text":"you can do it the other way,"},{"Start":"06:26.360 ","End":"06:28.700","Text":"and what we\u0027ll do in our next video is talk more"},{"Start":"06:28.700 ","End":"06:30.650","Text":"about how we use the second method that I"},{"Start":"06:30.650 ","End":"06:34.670","Text":"just brought up and can find the information we need to do from there."},{"Start":"06:34.670 ","End":"06:36.320","Text":"To be honest, in general, we will use"},{"Start":"06:36.320 ","End":"06:38.730","Text":"the second method more often than the determinant,"},{"Start":"06:38.730 ","End":"06:42.150","Text":"but the determinant is a great way to introduce the topic."}],"ID":9204},{"Watched":false,"Name":"Vector Multiplication Using Length and Direction (The Right Hand Rule)","Duration":"6m 33s","ChapterTopicVideoID":8931,"CourseChapterTopicPlaylistID":85318,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8931.jpeg","UploadDate":"2017-03-21T08:35:46.8330000","DurationForVideoObject":"PT6M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.525","Text":"In the last video, we talked about vector multiplication using a determinant."},{"Start":"00:04.525 ","End":"00:07.240","Text":"In this video, we\u0027ll talk about how you can use length and"},{"Start":"00:07.240 ","End":"00:09.925","Text":"direction to find the same result."},{"Start":"00:09.925 ","End":"00:13.540","Text":"Let\u0027s say I have 2 vector multiplication between vectors a and b,"},{"Start":"00:13.540 ","End":"00:18.580","Text":"my solution will be vector c. I\u0027m going to get the same solution of course,"},{"Start":"00:18.580 ","End":"00:22.090","Text":"if I use a determinant or if I use length and direction,"},{"Start":"00:22.090 ","End":"00:26.720","Text":"but perhaps I want to use length and direction because I find it more comfortable."},{"Start":"00:26.720 ","End":"00:32.260","Text":"If we want to start, we can find first the magnitude of the vector c,"},{"Start":"00:32.260 ","End":"00:35.035","Text":"which is again our solution by saying that"},{"Start":"00:35.035 ","End":"00:40.150","Text":"the magnitude of c equals the magnitude of a times the magnitude of b,"},{"Start":"00:40.150 ","End":"00:42.365","Text":"times the sine of the angle Alpha."},{"Start":"00:42.365 ","End":"00:47.210","Text":"The sine of the angle Alpha is a sign between vector a and vector b."},{"Start":"00:47.210 ","End":"00:48.575","Text":"Let\u0027s say that\u0027s vector a,"},{"Start":"00:48.575 ","End":"00:49.970","Text":"this is vector b."},{"Start":"00:49.970 ","End":"00:53.375","Text":"The angle between them would be angle Alpha."},{"Start":"00:53.375 ","End":"00:56.600","Text":"This method finding magnitude or length and then"},{"Start":"00:56.600 ","End":"00:59.570","Text":"direction has the advantage of being much shorter."},{"Start":"00:59.570 ","End":"01:01.250","Text":"But it has 2 disadvantages."},{"Start":"01:01.250 ","End":"01:04.250","Text":"Firstly, I\u0027m not always given the sine of the angle Alpha,"},{"Start":"01:04.250 ","End":"01:05.630","Text":"and it can be very hard to find,"},{"Start":"01:05.630 ","End":"01:07.515","Text":"especially in 3 dimensions."},{"Start":"01:07.515 ","End":"01:11.360","Text":"The second disadvantage is that I don\u0027t always know"},{"Start":"01:11.360 ","End":"01:17.060","Text":"the direction that my vector is headed and for that I\u0027ll use the right-hand rule."},{"Start":"01:17.060 ","End":"01:20.225","Text":"You can find your magnitude using this formula,"},{"Start":"01:20.225 ","End":"01:21.640","Text":"your length through your magnitude,"},{"Start":"01:21.640 ","End":"01:24.580","Text":"but to find the direction, you\u0027re going to have to use the right-hand rule."},{"Start":"01:24.580 ","End":"01:27.170","Text":"There\u0027s a few ways of doing the right-hand rule."},{"Start":"01:27.170 ","End":"01:30.965","Text":"The one I\u0027m going to show you is what I find most comfortable or easiest."},{"Start":"01:30.965 ","End":"01:32.750","Text":"The first thing you\u0027re going to do is extend"},{"Start":"01:32.750 ","End":"01:35.345","Text":"your right hand in the direction of the vector a."},{"Start":"01:35.345 ","End":"01:40.310","Text":"You\u0027re going to have to extend your hand in the direction of vector a and not vector b,"},{"Start":"01:40.310 ","End":"01:43.280","Text":"or in the case of any other equation,"},{"Start":"01:43.280 ","End":"01:45.290","Text":"the first vector, not the second vector."},{"Start":"01:45.290 ","End":"01:50.570","Text":"This is rather important and your thumb should be at 90 degrees with your fingers."},{"Start":"01:50.570 ","End":"01:53.840","Text":"The next thing you\u0027re going to do is you\u0027re going to spin or swivel"},{"Start":"01:53.840 ","End":"01:57.665","Text":"your fingers in the direction of the vector b."},{"Start":"01:57.665 ","End":"02:01.300","Text":"Your fingers should look something like this."},{"Start":"02:01.300 ","End":"02:08.320","Text":"Great. Now, your thumb should still be pointing in the direction of the vector c,"},{"Start":"02:08.320 ","End":"02:11.495","Text":"meaning it should be 90 degrees with"},{"Start":"02:11.495 ","End":"02:14.600","Text":"a and it should be 90 degrees with b, as we can see here."},{"Start":"02:14.600 ","End":"02:16.520","Text":"Do you see how your thumb is pointing the picture?"},{"Start":"02:16.520 ","End":"02:18.800","Text":"If we apply that to our drawing from before,"},{"Start":"02:18.800 ","End":"02:22.490","Text":"it should still have a 90-degree angle with b and a 90-degree angle with c,"},{"Start":"02:22.490 ","End":"02:26.225","Text":"which fulfills our property of it being perpendicular to both."},{"Start":"02:26.225 ","End":"02:31.483","Text":"We can consider c to be perpendicular to the ab plane,"},{"Start":"02:31.483 ","End":"02:34.565","Text":"an imaginary plane that a and b rest on."},{"Start":"02:34.565 ","End":"02:36.980","Text":"Now we know that any solution for"},{"Start":"02:36.980 ","End":"02:39.740","Text":"a vector multiplication has to be perpendicular to this plane,"},{"Start":"02:39.740 ","End":"02:42.965","Text":"but we don\u0027t know which way it\u0027s perpendicular because there\u0027s always 2 options."},{"Start":"02:42.965 ","End":"02:45.815","Text":"Using the right-hand rule, we find the correct option."},{"Start":"02:45.815 ","End":"02:49.166","Text":"If we suppose for a minute that vectors are switched,"},{"Start":"02:49.166 ","End":"02:52.805","Text":"that our vector a was the far vector, the short vector,"},{"Start":"02:52.805 ","End":"02:57.080","Text":"and vector b was the vector that seems parallel to us to the bottom of the page,"},{"Start":"02:57.080 ","End":"03:00.905","Text":"then I would have to spin my fingers in an awkward way."},{"Start":"03:00.905 ","End":"03:03.530","Text":"I turn my hand upside down to spin my fingers"},{"Start":"03:03.530 ","End":"03:07.470","Text":"towards the vector b and my thumb would be pointing downwards,"},{"Start":"03:07.470 ","End":"03:15.245","Text":"so I\u0027d know that my vector c is pointing downwards and not upwards from our perspective."},{"Start":"03:15.245 ","End":"03:18.305","Text":"An additional way to do the right-hand rule is,"},{"Start":"03:18.305 ","End":"03:19.775","Text":"as you see in this picture,"},{"Start":"03:19.775 ","End":"03:22.175","Text":"you\u0027re going to make your thumb the a vector,"},{"Start":"03:22.175 ","End":"03:24.410","Text":"your pointer finger, the b vector,"},{"Start":"03:24.410 ","End":"03:26.540","Text":"and your middle finger, the c vector."},{"Start":"03:26.540 ","End":"03:30.080","Text":"Make sure not to confuse between your middle and your pointer fingers."},{"Start":"03:30.080 ","End":"03:33.050","Text":"Sometimes it helps to put your thumb and"},{"Start":"03:33.050 ","End":"03:36.725","Text":"pointer finger in a gun shape first and then extend your middle finger."},{"Start":"03:36.725 ","End":"03:40.910","Text":"But as you can see, when you extend your fingers like this, your c angle,"},{"Start":"03:40.910 ","End":"03:45.140","Text":"your middle finger has 90-degree angle with both the b and the a vectors,"},{"Start":"03:45.140 ","End":"03:47.585","Text":"meaning your pointer finger and your thumb."},{"Start":"03:47.585 ","End":"03:52.740","Text":"Again, it\u0027s important to remember that your thumb is the first component, in this case,"},{"Start":"03:52.740 ","End":"03:55.190","Text":"a, your pointer finger, the second component,"},{"Start":"03:55.190 ","End":"03:56.570","Text":"in this case, b,"},{"Start":"03:56.570 ","End":"03:58.175","Text":"and your middle finger, the solution,"},{"Start":"03:58.175 ","End":"04:03.155","Text":"in this case, c. If we return to our graph from before,"},{"Start":"04:03.155 ","End":"04:08.775","Text":"we can use this example to solve as well and this method of the right-hand rule."},{"Start":"04:08.775 ","End":"04:12.320","Text":"If we go back to our original version where a is on the bottom,"},{"Start":"04:12.320 ","End":"04:13.832","Text":"b is a little above it,"},{"Start":"04:13.832 ","End":"04:17.450","Text":"we can draw it out with their hands so our thumb would be a,"},{"Start":"04:17.450 ","End":"04:19.240","Text":"so we can draw that."},{"Start":"04:19.240 ","End":"04:21.735","Text":"Excuse my drawing skills."},{"Start":"04:21.735 ","End":"04:25.160","Text":"There\u0027s your thumb. As you can see,"},{"Start":"04:25.160 ","End":"04:26.600","Text":"my wrist is below it."},{"Start":"04:26.600 ","End":"04:32.800","Text":"Then you have your pointer finger pointing towards b something like this."},{"Start":"04:32.800 ","End":"04:35.310","Text":"Where my middle finger is pointing,"},{"Start":"04:35.310 ","End":"04:39.570","Text":"notice that the palm of my hand is up,"},{"Start":"04:39.570 ","End":"04:42.405","Text":"the top of my hand is facing the ground,"},{"Start":"04:42.405 ","End":"04:45.235","Text":"so my middle finger will be pointing more or less vertically."},{"Start":"04:45.235 ","End":"04:51.140","Text":"In this sense, I know that c has to be pointing up and not down."},{"Start":"04:53.450 ","End":"04:56.005","Text":"Well, the drawing is not the best."},{"Start":"04:56.005 ","End":"04:58.090","Text":"I think you can see how this method would work,"},{"Start":"04:58.090 ","End":"05:00.220","Text":"where if your thumb is pointing out,"},{"Start":"05:00.220 ","End":"05:04.405","Text":"your finger is pointing out and your middle finger is pointing up,"},{"Start":"05:04.405 ","End":"05:05.710","Text":"then c must be going up,"},{"Start":"05:05.710 ","End":"05:07.590","Text":"and of course, there\u0027s the reverse, which could be true."},{"Start":"05:07.590 ","End":"05:09.725","Text":"In our case, c is pointing up."},{"Start":"05:09.725 ","End":"05:12.100","Text":"Again just to reiterate, this is a,"},{"Start":"05:12.100 ","End":"05:13.690","Text":"your pointer finger is b,"},{"Start":"05:13.690 ","End":"05:15.844","Text":"your middle finger is c. Really,"},{"Start":"05:15.844 ","End":"05:19.210","Text":"what I\u0027m doing here is not using the real angle of b,"},{"Start":"05:19.210 ","End":"05:23.275","Text":"but rather an imagined angle of b that is perpendicular to a,"},{"Start":"05:23.275 ","End":"05:27.040","Text":"but in the same direction as my vector b,"},{"Start":"05:27.040 ","End":"05:29.635","Text":"which that angle Alpha as opposed to a 90-degree angle."},{"Start":"05:29.635 ","End":"05:31.645","Text":"I\u0027m not using the real vector b,"},{"Start":"05:31.645 ","End":"05:33.250","Text":"I\u0027m using an imaginary vector."},{"Start":"05:33.250 ","End":"05:39.015","Text":"We can call it b_1 or sub anything really,"},{"Start":"05:39.015 ","End":"05:43.650","Text":"and I use 90 degrees instead of the angle Theta."},{"Start":"05:45.430 ","End":"05:48.350","Text":"1 property that I want to bring up,"},{"Start":"05:48.350 ","End":"05:50.149","Text":"again, just to emphasize,"},{"Start":"05:50.149 ","End":"05:54.980","Text":"is that vector multiplication is not commutative. What does that mean?"},{"Start":"05:54.980 ","End":"06:00.260","Text":"It means that vector a times vector b does not equal vector b times vector a."},{"Start":"06:00.260 ","End":"06:02.125","Text":"In fact, it equals the opposite."},{"Start":"06:02.125 ","End":"06:07.955","Text":"Vector a times vector b equals negative vector b times the vector a."},{"Start":"06:07.955 ","End":"06:11.690","Text":"This is, of course different than scalar multiplication,"},{"Start":"06:11.690 ","End":"06:16.230","Text":"where a.b equals b.a."},{"Start":"06:16.790 ","End":"06:21.350","Text":"But with vector multiplication, it is not the case."},{"Start":"06:21.350 ","End":"06:23.150","Text":"There is no commutative property."},{"Start":"06:23.150 ","End":"06:27.740","Text":"That\u0027s the end of our lecture on finding the solution of vector multiplication"},{"Start":"06:27.740 ","End":"06:33.030","Text":"using the length and the direction and the discussion of the right-hand rule in general."}],"ID":9205},{"Watched":false,"Name":"Choosing the Direction of the Z Axis","Duration":"1m 5s","ChapterTopicVideoID":8932,"CourseChapterTopicPlaylistID":85318,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8932.jpeg","UploadDate":"2017-03-21T08:35:55.2930000","DurationForVideoObject":"PT1M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.100","Text":"In this segment, I want to talk about how we find the direction of the z-axis."},{"Start":"00:05.100 ","End":"00:06.419","Text":"Now it\u0027s not arbitrary."},{"Start":"00:06.419 ","End":"00:09.060","Text":"It turns out that the z-axis is directly relative to"},{"Start":"00:09.060 ","End":"00:12.570","Text":"the x-axis and the y-axis and we can\u0027t just choose a direction."},{"Start":"00:12.570 ","End":"00:14.790","Text":"But the best way to think about it is as if"},{"Start":"00:14.790 ","End":"00:19.305","Text":"the x-axis was a vector and the y-axis was a vector, like so."},{"Start":"00:19.305 ","End":"00:22.425","Text":"Then the z-axis is the vector."},{"Start":"00:22.425 ","End":"00:25.290","Text":"That\u0027s the solution of their vector multiplication."},{"Start":"00:25.290 ","End":"00:27.290","Text":"In the same way that we did in the last video,"},{"Start":"00:27.290 ","End":"00:30.680","Text":"you can think of the z vector as equal to or"},{"Start":"00:30.680 ","End":"00:35.660","Text":"the z-axis as equal to the vector multiplication of x and y."},{"Start":"00:35.660 ","End":"00:38.525","Text":"If x is my hand,"},{"Start":"00:38.525 ","End":"00:40.760","Text":"and y is the direction I turned my fingers,"},{"Start":"00:40.760 ","End":"00:42.270","Text":"then we\u0027re using the right-hand rule,"},{"Start":"00:42.270 ","End":"00:45.355","Text":"z would be where the thumb goes in this case up."},{"Start":"00:45.355 ","End":"00:47.205","Text":"That would be your z-axis."},{"Start":"00:47.205 ","End":"00:50.854","Text":"In the same way, if you switched around the x and the y axis,"},{"Start":"00:50.854 ","End":"00:52.640","Text":"let\u0027s say the x-axis is going there,"},{"Start":"00:52.640 ","End":"00:55.925","Text":"your y-axis is pointing towards our bottom-left corner,"},{"Start":"00:55.925 ","End":"00:57.680","Text":"in that case, using the right-hand rule,"},{"Start":"00:57.680 ","End":"01:00.220","Text":"the z-axis would point downwards."},{"Start":"01:00.220 ","End":"01:05.190","Text":"In that sense, we use the right-hand rule to find our z-axis as well."}],"ID":9206},{"Watched":false,"Name":"Exercise- Vector Multiplication","Duration":"9m 29s","ChapterTopicVideoID":12206,"CourseChapterTopicPlaylistID":85318,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12206.jpeg","UploadDate":"2018-07-08T12:14:16.6700000","DurationForVideoObject":"PT9M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"Hello, in this exercise,"},{"Start":"00:02.400 ","End":"00:04.410","Text":"we\u0027re given 4 vectors,"},{"Start":"00:04.410 ","End":"00:06.585","Text":"A and B, which are 2-dimensional,"},{"Start":"00:06.585 ","End":"00:09.060","Text":"and C and D, which are 3-dimensional."},{"Start":"00:09.060 ","End":"00:11.130","Text":"We\u0027re asked to find 3 different things."},{"Start":"00:11.130 ","End":"00:13.290","Text":"The first, let\u0027s start with A,"},{"Start":"00:13.290 ","End":"00:18.180","Text":"is we\u0027re supposed to find the scalar multiplication of vectors A and B."},{"Start":"00:18.180 ","End":"00:23.925","Text":"If you recall a scalar multiplication or A vector dot B vector,"},{"Start":"00:23.925 ","End":"00:27.615","Text":"remember the dot is equal to A_x,"},{"Start":"00:27.615 ","End":"00:30.465","Text":"the x component of A times the x component of B,"},{"Start":"00:30.465 ","End":"00:33.930","Text":"B_x plus A_y times B_y."},{"Start":"00:33.930 ","End":"00:36.085","Text":"If we multiply that out,"},{"Start":"00:36.085 ","End":"00:41.300","Text":"1 times 1 plus 2 times negative 3,"},{"Start":"00:41.300 ","End":"00:44.900","Text":"that\u0027s 1 times 1 plus 2 times negative 3."},{"Start":"00:44.900 ","End":"00:49.175","Text":"Our result is 1 minus 6,"},{"Start":"00:49.175 ","End":"00:51.280","Text":"which is negative 5."},{"Start":"00:51.280 ","End":"00:52.965","Text":"There you\u0027ve solved part a."},{"Start":"00:52.965 ","End":"00:54.650","Text":"Of course, there is a second way as well."},{"Start":"00:54.650 ","End":"00:58.910","Text":"We could have found the magnitude of the vectors A and B."},{"Start":"00:58.910 ","End":"01:01.070","Text":"If we knew the angle between the 2 of them,"},{"Start":"01:01.070 ","End":"01:05.720","Text":"we could have solved multiplying the magnitude of vector A times"},{"Start":"01:05.720 ","End":"01:07.640","Text":"the magnitude of vector B times"},{"Start":"01:07.640 ","End":"01:10.865","Text":"the cosine of the angle between them, which we call Alpha."},{"Start":"01:10.865 ","End":"01:12.360","Text":"Of course, in this case,"},{"Start":"01:12.360 ","End":"01:13.845","Text":"we didn\u0027t have the angle,"},{"Start":"01:13.845 ","End":"01:15.180","Text":"and we did have the points."},{"Start":"01:15.180 ","End":"01:16.250","Text":"As soon as you have the points,"},{"Start":"01:16.250 ","End":"01:19.160","Text":"it can often be easier to just multiply the x components,"},{"Start":"01:19.160 ","End":"01:21.560","Text":"multiply the y components and the z components if there"},{"Start":"01:21.560 ","End":"01:24.820","Text":"are z components and find the answer that way."},{"Start":"01:24.820 ","End":"01:27.860","Text":"Keep in mind that negative 5 is only a magnitude,"},{"Start":"01:27.860 ","End":"01:30.865","Text":"is a scalar, there\u0027s no direction, it\u0027s not a vector."},{"Start":"01:30.865 ","End":"01:33.040","Text":"Moving on to part b,"},{"Start":"01:33.040 ","End":"01:37.565","Text":"it asks us to do a vector multiplication of the vectors A and B."},{"Start":"01:37.565 ","End":"01:39.290","Text":"Again, we\u0027re still in 2-dimensions,"},{"Start":"01:39.290 ","End":"01:41.270","Text":"so we can just use our formula."},{"Start":"01:41.270 ","End":"01:44.750","Text":"If you recall the formula A cross B,"},{"Start":"01:44.750 ","End":"01:49.145","Text":"which is the vector multiplication of A and B,"},{"Start":"01:49.145 ","End":"01:53.785","Text":"is equal to A_x, the x component of A."},{"Start":"01:53.785 ","End":"02:00.805","Text":"Again, A_x times by minus A_y times B_x."},{"Start":"02:00.805 ","End":"02:03.290","Text":"Remember, the order here is important."},{"Start":"02:03.290 ","End":"02:08.210","Text":"This isn\u0027t commutative, so A cross B is not the same as B cross A."},{"Start":"02:08.210 ","End":"02:10.265","Text":"In fact, it\u0027s equal to the opposite."},{"Start":"02:10.265 ","End":"02:14.725","Text":"If we had B cross A, we would have the opposite result."},{"Start":"02:14.725 ","End":"02:16.935","Text":"If we solve this,"},{"Start":"02:16.935 ","End":"02:22.110","Text":"A_x is 1 and by is negative 3."},{"Start":"02:22.110 ","End":"02:27.850","Text":"If we write that out, our first portion is 1 times negative 3."},{"Start":"02:27.850 ","End":"02:33.080","Text":"A_y is 2 and b_x is 1."},{"Start":"02:33.080 ","End":"02:35.090","Text":"Minus 2 times 1,"},{"Start":"02:35.090 ","End":"02:36.965","Text":"we get negative 3 minus 2,"},{"Start":"02:36.965 ","End":"02:39.125","Text":"and our answer is again minus 5."},{"Start":"02:39.125 ","End":"02:40.775","Text":"This is just a coincidence."},{"Start":"02:40.775 ","End":"02:42.200","Text":"It won\u0027t always happen that"},{"Start":"02:42.200 ","End":"02:45.410","Text":"your scalar multiplication and your vector multiplication are the same."},{"Start":"02:45.410 ","End":"02:46.520","Text":"This is pure coincidence."},{"Start":"02:46.520 ","End":"02:48.245","Text":"In fact, it\u0027s rather unlikely."},{"Start":"02:48.245 ","End":"02:49.760","Text":"Now this is our magnitude,"},{"Start":"02:49.760 ","End":"02:51.514","Text":"but if we\u0027re looking for the direction,"},{"Start":"02:51.514 ","End":"02:53.510","Text":"because this is a vector multiplication,"},{"Start":"02:53.510 ","End":"02:57.455","Text":"and it needs both magnitude and direction, length and direction."},{"Start":"02:57.455 ","End":"03:02.270","Text":"We need to remember because we\u0027re only given the x and the y coordinates"},{"Start":"03:02.270 ","End":"03:07.055","Text":"that the vector will end up going in the direction of the z-axis."},{"Start":"03:07.055 ","End":"03:09.560","Text":"We can write out that the magnitude is negative 5,"},{"Start":"03:09.560 ","End":"03:12.275","Text":"and it\u0027s in the direction z-hat."},{"Start":"03:12.275 ","End":"03:16.580","Text":"Of course, there\u0027s a second way to do this multiplication as well."},{"Start":"03:16.580 ","End":"03:21.560","Text":"We could have taken the magnitude of A times the magnitude of"},{"Start":"03:21.560 ","End":"03:26.575","Text":"the vector B and multiply that by the sine of the angle Alpha,"},{"Start":"03:26.575 ","End":"03:29.095","Text":"the angle between the 2 vectors."},{"Start":"03:29.095 ","End":"03:30.910","Text":"Of course that\u0027s if we\u0027re given that angle,"},{"Start":"03:30.910 ","End":"03:31.930","Text":"in this case we weren\u0027t."},{"Start":"03:31.930 ","End":"03:35.215","Text":"The easier path was what we did earlier."},{"Start":"03:35.215 ","End":"03:36.490","Text":"We would have had the same answer,"},{"Start":"03:36.490 ","End":"03:38.485","Text":"negative 5 in the direction of z."},{"Start":"03:38.485 ","End":"03:42.765","Text":"But the way we did it, given our data, was easier."},{"Start":"03:42.765 ","End":"03:45.125","Text":"For part c,"},{"Start":"03:45.125 ","End":"03:51.925","Text":"we are given the vector multiplication of C and D. Now these are 3-dimensional vectors."},{"Start":"03:51.925 ","End":"03:53.860","Text":"We can\u0027t do the operation we did above."},{"Start":"03:53.860 ","End":"03:55.705","Text":"We need to use the determinant."},{"Start":"03:55.705 ","End":"03:57.430","Text":"Now, if you recall,"},{"Start":"03:57.430 ","End":"03:59.960","Text":"if we\u0027re doing C cross D,"},{"Start":"03:59.960 ","End":"04:02.020","Text":"the way we do a determinant is created"},{"Start":"04:02.020 ","End":"04:06.055","Text":"a matrix in 3-dimensions that will have 3 rows and 3 columns."},{"Start":"04:06.055 ","End":"04:09.440","Text":"In the first row, assuming that we\u0027re using Cartesian coordinates,"},{"Start":"04:09.440 ","End":"04:11.180","Text":"which I can assume because the points here are"},{"Start":"04:11.180 ","End":"04:13.850","Text":"given in coordinates that seem to be Cartesian,"},{"Start":"04:13.850 ","End":"04:15.155","Text":"I\u0027ll write in my first row,"},{"Start":"04:15.155 ","End":"04:17.695","Text":"x-hat, y-hat, and z-hat."},{"Start":"04:17.695 ","End":"04:19.320","Text":"In the second row,"},{"Start":"04:19.320 ","End":"04:21.185","Text":"remember the order here is important."},{"Start":"04:21.185 ","End":"04:23.780","Text":"I\u0027m going to write my C coordinates,"},{"Start":"04:23.780 ","End":"04:25.055","Text":"the x coordinate first,"},{"Start":"04:25.055 ","End":"04:26.150","Text":"the y coordinate second,"},{"Start":"04:26.150 ","End":"04:27.470","Text":"and the z-coordinate third."},{"Start":"04:27.470 ","End":"04:29.345","Text":"Again, the order is important."},{"Start":"04:29.345 ","End":"04:32.510","Text":"C_x, C_y and C_z."},{"Start":"04:32.510 ","End":"04:35.390","Text":"Again, my first component of the multiplication."},{"Start":"04:35.390 ","End":"04:36.950","Text":"In the bottom row,"},{"Start":"04:36.950 ","End":"04:40.765","Text":"D_x, D_y, D_z."},{"Start":"04:40.765 ","End":"04:43.940","Text":"Of course, I have to put my 2 lines bracketing this in,"},{"Start":"04:43.940 ","End":"04:45.920","Text":"so I know that there\u0027s a determinant."},{"Start":"04:45.920 ","End":"04:49.700","Text":"If we put in our values x,"},{"Start":"04:49.700 ","End":"04:51.650","Text":"y, and z stay the same of course."},{"Start":"04:51.650 ","End":"04:55.070","Text":"You have x-hat, y-hat, and z-hat."},{"Start":"04:55.070 ","End":"04:59.315","Text":"For C_x, our value is negative 1."},{"Start":"04:59.315 ","End":"05:00.965","Text":"C_y is 2,"},{"Start":"05:00.965 ","End":"05:03.065","Text":"and C_z is negative 2."},{"Start":"05:03.065 ","End":"05:04.775","Text":"D_x is 2,"},{"Start":"05:04.775 ","End":"05:07.235","Text":"D_y is 0 and D_z is 1,"},{"Start":"05:07.235 ","End":"05:08.975","Text":"and we can close our determinant."},{"Start":"05:08.975 ","End":"05:13.580","Text":"The procedure for determinant is to go component by component first x,"},{"Start":"05:13.580 ","End":"05:16.920","Text":"then y, then z, and do a special operation for each one."},{"Start":"05:16.920 ","End":"05:19.880","Text":"What we\u0027re going to do is cross out the x, the y,"},{"Start":"05:19.880 ","End":"05:22.175","Text":"and the z every time and then take the row,"},{"Start":"05:22.175 ","End":"05:25.205","Text":"the column that is of the variable we\u0027re working with."},{"Start":"05:25.205 ","End":"05:27.170","Text":"First, we\u0027ll do x, and we cross out"},{"Start":"05:27.170 ","End":"05:30.480","Text":"that column and do cross multiplications on the diagonals."},{"Start":"05:30.480 ","End":"05:35.465","Text":"We\u0027re going to do x, but x-hat in cross out the row and the x column,"},{"Start":"05:35.465 ","End":"05:39.725","Text":"and we\u0027re left with the y column and the z column. What do we do?"},{"Start":"05:39.725 ","End":"05:41.750","Text":"We go on our main diagonal first."},{"Start":"05:41.750 ","End":"05:45.065","Text":"We multiply C_y by D_z,"},{"Start":"05:45.065 ","End":"05:47.665","Text":"which gives us 2 times 1."},{"Start":"05:47.665 ","End":"05:52.670","Text":"Then we\u0027re going to multiply on the secondary diagonal C_z,"},{"Start":"05:52.670 ","End":"05:56.515","Text":"by D_y negative 2 times 0."},{"Start":"05:56.515 ","End":"06:01.730","Text":"In a parenthesis first we\u0027ll write x-hat outside of the parenthesis and then within"},{"Start":"06:01.730 ","End":"06:08.840","Text":"the parentheses will write 2 times 1 minus negative 2 times 0."},{"Start":"06:08.840 ","End":"06:11.900","Text":"Now I\u0027m going to subtract my y element."},{"Start":"06:11.900 ","End":"06:14.030","Text":"If you recall, we do positive and negative,"},{"Start":"06:14.030 ","End":"06:17.180","Text":"than positive and negative, so on and so forth with the determinant."},{"Start":"06:17.180 ","End":"06:20.465","Text":"For my y component,"},{"Start":"06:20.465 ","End":"06:22.420","Text":"I\u0027ll be subtracting it."},{"Start":"06:22.420 ","End":"06:26.195","Text":"Negative y-hat will be,"},{"Start":"06:26.195 ","End":"06:27.365","Text":"remember we\u0027re left with,"},{"Start":"06:27.365 ","End":"06:30.185","Text":"once we cross out my y row and my y column,"},{"Start":"06:30.185 ","End":"06:32.780","Text":"I\u0027m left with D_x and C_x,"},{"Start":"06:32.780 ","End":"06:35.815","Text":"and I\u0027m left with C_z and D_z."},{"Start":"06:35.815 ","End":"06:37.875","Text":"First I do my main diagonal,"},{"Start":"06:37.875 ","End":"06:40.065","Text":"C_x times D_z,"},{"Start":"06:40.065 ","End":"06:46.370","Text":"which is negative 1 times 1 and subtract from that my secondary diagonal,"},{"Start":"06:46.370 ","End":"06:49.300","Text":"which is negative 2 times 2."},{"Start":"06:50.000 ","End":"06:54.490","Text":"Now if I move on to my z component,"},{"Start":"06:54.490 ","End":"06:57.080","Text":"I\u0027m going to add that because again,"},{"Start":"06:57.080 ","End":"06:58.910","Text":"positive, negative, positive, negative."},{"Start":"06:58.910 ","End":"07:01.955","Text":"I put in my z-hat and once again,"},{"Start":"07:01.955 ","End":"07:03.770","Text":"I\u0027m going to cross out a different component."},{"Start":"07:03.770 ","End":"07:05.735","Text":"I\u0027m going to cross out the same row,"},{"Start":"07:05.735 ","End":"07:06.830","Text":"the x, y, z row,"},{"Start":"07:06.830 ","End":"07:08.360","Text":"but this time the z column."},{"Start":"07:08.360 ","End":"07:12.365","Text":"I\u0027m left with D_x and C_x, D_y and C_y."},{"Start":"07:12.365 ","End":"07:15.560","Text":"My main diagonals negative 1 times 0, I subtract from that,"},{"Start":"07:15.560 ","End":"07:19.760","Text":"my secondary diagonal negative 2 times 2 and that\u0027s the z-hat element."},{"Start":"07:19.760 ","End":"07:21.275","Text":"If I solve here,"},{"Start":"07:21.275 ","End":"07:23.940","Text":"my x element will be 2,"},{"Start":"07:23.940 ","End":"07:29.315","Text":"2x-hat minus y-hat,"},{"Start":"07:29.315 ","End":"07:31.490","Text":"but my element is actually negative 5,"},{"Start":"07:31.490 ","End":"07:35.600","Text":"so it\u0027s plus 5y-hat and I add my z element,"},{"Start":"07:35.600 ","End":"07:36.860","Text":"which actually is negative."},{"Start":"07:36.860 ","End":"07:42.080","Text":"I subtract 4z. Once again,"},{"Start":"07:42.080 ","End":"07:45.575","Text":"keep in mind our solution here is a vector, it has direction."},{"Start":"07:45.575 ","End":"07:48.230","Text":"Vector multiplication will always give you a vector,"},{"Start":"07:48.230 ","End":"07:51.005","Text":"a scalar multiplication will give you a magnitude or length,"},{"Start":"07:51.005 ","End":"07:53.570","Text":"not a vector, it will not have direction."},{"Start":"07:53.570 ","End":"07:56.690","Text":"Now let\u0027s assume for a moment that I did want only the magnitude,"},{"Start":"07:56.690 ","End":"07:58.310","Text":"only the length of this vector."},{"Start":"07:58.310 ","End":"07:59.840","Text":"We\u0027ll call that vector e,"},{"Start":"07:59.840 ","End":"08:02.735","Text":"or new vector that we\u0027ve found is a solution of"},{"Start":"08:02.735 ","End":"08:07.080","Text":"C cross D. But if I want the magnitude of that,"},{"Start":"08:07.120 ","End":"08:10.355","Text":"signify the magnitude like so."},{"Start":"08:10.355 ","End":"08:14.900","Text":"What I do is take the magnitude of my result here, and how do I do that?"},{"Start":"08:14.900 ","End":"08:19.385","Text":"I take the square root of the square values of each of the components."},{"Start":"08:19.385 ","End":"08:24.725","Text":"That means that the magnitude of the vector e will equal the square root of 2^2"},{"Start":"08:24.725 ","End":"08:30.240","Text":"plus 5^2 plus 4^2 or negative 4^2."},{"Start":"08:30.240 ","End":"08:32.085","Text":"Of course, my solution will be the same."},{"Start":"08:32.085 ","End":"08:37.080","Text":"The square root of 2^2 is 4 plus 25,"},{"Start":"08:37.080 ","End":"08:39.500","Text":"which is 5^2 plus 16,"},{"Start":"08:39.500 ","End":"08:46.250","Text":"gives me the square root of 45 and that is the magnitude of the vector e. Of course,"},{"Start":"08:46.250 ","End":"08:47.825","Text":"I can always do the second way,"},{"Start":"08:47.825 ","End":"08:53.750","Text":"just like we did above in part b. I can find the magnitude of the vector e. Again,"},{"Start":"08:53.750 ","End":"08:55.415","Text":"my new vector, the solution,"},{"Start":"08:55.415 ","End":"09:00.440","Text":"the same way that I did in the vector in part b. I could take"},{"Start":"09:00.440 ","End":"09:07.550","Text":"the magnitude of the vector C multiplied by the magnitude of the vector D,"},{"Start":"09:07.550 ","End":"09:12.370","Text":"multiplied by the sine of the angle Theta between these 2."},{"Start":"09:12.370 ","End":"09:14.065","Text":"Now I don\u0027t have that angle Theta."},{"Start":"09:14.065 ","End":"09:16.670","Text":"In 3-dimensions, it\u0027s rather hard to come by that angle."},{"Start":"09:16.670 ","End":"09:18.275","Text":"I did this in the first method,"},{"Start":"09:18.275 ","End":"09:20.095","Text":"but that\u0027s always possible."},{"Start":"09:20.095 ","End":"09:21.980","Text":"You\u0027ve solved parts a, b,"},{"Start":"09:21.980 ","End":"09:25.290","Text":"and c, and that\u0027s the end of the exercise."}],"ID":12676}],"Thumbnail":null,"ID":85318},{"Name":"Gradient","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Intro to Gradients","Duration":"4m 44s","ChapterTopicVideoID":8942,"CourseChapterTopicPlaylistID":85319,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8942.jpeg","UploadDate":"2017-03-21T08:38:39.5430000","DurationForVideoObject":"PT4M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"Hello. In this video,"},{"Start":"00:01.830 ","End":"00:03.390","Text":"I\u0027m going to talk about a gradient."},{"Start":"00:03.390 ","End":"00:06.930","Text":"A gradient is a mathematical operation that we\u0027re going to need"},{"Start":"00:06.930 ","End":"00:11.595","Text":"to know in this course and we\u0027re going to need to perform a couple of times."},{"Start":"00:11.595 ","End":"00:17.610","Text":"We\u0027re going to talk about it in a very basic sense on mathematical sense."},{"Start":"00:17.610 ","End":"00:20.550","Text":"We\u0027re going to talk a little bit more about the meaning of"},{"Start":"00:20.550 ","End":"00:22.980","Text":"it and why we use it in the next lecture,"},{"Start":"00:22.980 ","End":"00:24.240","Text":"but to be quite honest with you,"},{"Start":"00:24.240 ","End":"00:26.505","Text":"it\u0027s not essential for this course."},{"Start":"00:26.505 ","End":"00:28.230","Text":"However, it might be good for you for"},{"Start":"00:28.230 ","End":"00:32.280","Text":"an enrichment course or for further learning to know that."},{"Start":"00:32.280 ","End":"00:36.180","Text":"For now, we\u0027re going to talk about it in Cartesian coordinates and other coordinates in"},{"Start":"00:36.180 ","End":"00:40.600","Text":"a basic sense because it\u0027s important that you know how it works."},{"Start":"00:40.600 ","End":"00:42.720","Text":"To begin, this is a nabla."},{"Start":"00:42.720 ","End":"00:47.060","Text":"A nabla is an upside-down triangle with a vector sign above it, an arrow."},{"Start":"00:47.060 ","End":"00:50.945","Text":"That symbolizes that we need to do a gradient function."},{"Start":"00:50.945 ","End":"00:54.100","Text":"Now, the gradient function is performed on f the function,"},{"Start":"00:54.100 ","End":"00:56.780","Text":"and the function must be a scalar function."},{"Start":"00:56.780 ","End":"00:57.860","Text":"What does that mean?"},{"Start":"00:57.860 ","End":"00:59.869","Text":"If we look below at the example,"},{"Start":"00:59.869 ","End":"01:01.610","Text":"we have a function of x, y,"},{"Start":"01:01.610 ","End":"01:05.630","Text":"and z, 2x^2 times y minus y times z."},{"Start":"01:05.630 ","End":"01:07.220","Text":"We have 3 variables,"},{"Start":"01:07.220 ","End":"01:09.500","Text":"but it only produces a magnitude,"},{"Start":"01:09.500 ","End":"01:12.320","Text":"it only produces a length or a number or a value,"},{"Start":"01:12.320 ","End":"01:13.730","Text":"it has no direction."},{"Start":"01:13.730 ","End":"01:16.010","Text":"This could describe some number."},{"Start":"01:16.010 ","End":"01:19.100","Text":"It could describe a point on a coordinate grid,"},{"Start":"01:19.100 ","End":"01:21.170","Text":"but it does not have a direction."},{"Start":"01:21.170 ","End":"01:23.390","Text":"The difference between a scalar function,"},{"Start":"01:23.390 ","End":"01:28.670","Text":"and for example below we have a vector function is a vector function has a direction."},{"Start":"01:28.670 ","End":"01:30.380","Text":"You see the x hat, y hat,"},{"Start":"01:30.380 ","End":"01:33.955","Text":"and z hat, the scalar function has no direction."},{"Start":"01:33.955 ","End":"01:37.010","Text":"What a gradient function does is it takes"},{"Start":"01:37.010 ","End":"01:40.880","Text":"a scalar function and converts it to a vector function with direction."},{"Start":"01:40.880 ","End":"01:43.970","Text":"To get down to nuts and bolts of how we do this,"},{"Start":"01:43.970 ","End":"01:46.850","Text":"what we\u0027re going to do is take partial derivatives of each of"},{"Start":"01:46.850 ","End":"01:50.135","Text":"our components and add them together to reach our solution."},{"Start":"01:50.135 ","End":"01:55.244","Text":"We take a partial derivative of f/x times x hat,"},{"Start":"01:55.244 ","End":"02:01.112","Text":"we add that to a partial derivative of f/y times y hat,"},{"Start":"02:01.112 ","End":"02:04.930","Text":"and we add that to partial derivative of f/z times z hat."},{"Start":"02:04.930 ","End":"02:07.010","Text":"If we do that with this example,"},{"Start":"02:07.010 ","End":"02:10.295","Text":"what we get is 2x^2 times y."},{"Start":"02:10.295 ","End":"02:12.920","Text":"This is the only part of the equation that deals in x,"},{"Start":"02:12.920 ","End":"02:14.300","Text":"so everything else drops out."},{"Start":"02:14.300 ","End":"02:17.330","Text":"We end up with 4xy times x hat."},{"Start":"02:17.330 ","End":"02:19.040","Text":"In terms of our y component,"},{"Start":"02:19.040 ","End":"02:21.910","Text":"both parts of our initial equation deal in y,"},{"Start":"02:21.910 ","End":"02:23.840","Text":"so 2x^2 times y,"},{"Start":"02:23.840 ","End":"02:27.035","Text":"the y will drop out, you\u0027re left with (2x^2), y times z,"},{"Start":"02:27.035 ","End":"02:29.060","Text":"the y drops out and you\u0027re left with z,"},{"Start":"02:29.060 ","End":"02:32.125","Text":"so it\u0027s 2x^2 minus z times y hat."},{"Start":"02:32.125 ","End":"02:35.965","Text":"With z element, you have y times z,"},{"Start":"02:35.965 ","End":"02:38.425","Text":"the z drops out, and you have yz hat,"},{"Start":"02:38.425 ","End":"02:40.270","Text":"so minus yz hat."},{"Start":"02:40.270 ","End":"02:42.400","Text":"If you\u0027re interested in what this means,"},{"Start":"02:42.400 ","End":"02:45.720","Text":"you should listen to the next lecture and we\u0027ll talk about it more."},{"Start":"02:45.720 ","End":"02:48.850","Text":"That\u0027s your solution for Cartesian coordinates."},{"Start":"02:48.850 ","End":"02:51.970","Text":"But you\u0027re also going to want to know how to do the same operation"},{"Start":"02:51.970 ","End":"02:55.500","Text":"in cylindrical or polar coordinates as well as spherical coordinates."},{"Start":"02:55.500 ","End":"02:57.820","Text":"The reason you want to know this is you might be given"},{"Start":"02:57.820 ","End":"03:01.030","Text":"a problem where your coordinates are given in spherical coordinates,"},{"Start":"03:01.030 ","End":"03:03.415","Text":"they are given in polar or cylindrical coordinates,"},{"Start":"03:03.415 ","End":"03:06.325","Text":"and you\u0027re going to need to find a solution."},{"Start":"03:06.325 ","End":"03:10.450","Text":"You could always transfer it back to Cartesian coordinates,"},{"Start":"03:10.450 ","End":"03:12.515","Text":"but oftentimes that\u0027s not worth the effort,"},{"Start":"03:12.515 ","End":"03:14.900","Text":"it\u0027s just going to make your problem more complicated."},{"Start":"03:14.900 ","End":"03:16.540","Text":"If we look at our example here,"},{"Start":"03:16.540 ","End":"03:19.735","Text":"this is given in terms of r, Theta, and z."},{"Start":"03:19.735 ","End":"03:21.820","Text":"That would be polar coordinates in"},{"Start":"03:21.820 ","End":"03:24.190","Text":"2 dimensions or cylindrical coordinates in 3 dimensions,"},{"Start":"03:24.190 ","End":"03:27.175","Text":"there is no z here, so we\u0027ll call this polar coordinates."},{"Start":"03:27.175 ","End":"03:30.820","Text":"We want to solve this, so we\u0027re going to do the exact same thing we did before."},{"Start":"03:30.820 ","End":"03:34.215","Text":"Partial derivatives over r, Theta, and z."},{"Start":"03:34.215 ","End":"03:37.960","Text":"But we\u0027re going to do a small difference here to account for"},{"Start":"03:37.960 ","End":"03:42.220","Text":"the fact that we\u0027re dealing with degrees and angles as opposed to distances."},{"Start":"03:42.220 ","End":"03:45.250","Text":"I know that you might not all know what r hat,"},{"Start":"03:45.250 ","End":"03:47.665","Text":"Theta hat, and z hat mean."},{"Start":"03:47.665 ","End":"03:51.940","Text":"Suffice it to say that they indicate the direction in the same way that x, y,"},{"Start":"03:51.940 ","End":"03:53.380","Text":"and z hat do above,"},{"Start":"03:53.380 ","End":"03:57.875","Text":"and that we\u0027ll talk about them more when we talk about the different coordinate systems."},{"Start":"03:57.875 ","End":"04:02.375","Text":"For now, you can add those in at the end and we can just go through the formula."},{"Start":"04:02.375 ","End":"04:06.305","Text":"If we have to do a partial derivative based on r,"},{"Start":"04:06.305 ","End":"04:11.735","Text":"then minus will drop out and you end up with cosine Theta over r^2 times r hat."},{"Start":"04:11.735 ","End":"04:13.055","Text":"We\u0027ve put that back in."},{"Start":"04:13.055 ","End":"04:17.270","Text":"Now if we\u0027re going to do a partial derivative of Theta,"},{"Start":"04:17.270 ","End":"04:19.100","Text":"we\u0027re going to multiply that by 1/r,"},{"Start":"04:19.100 ","End":"04:20.300","Text":"as you can see in the formula,"},{"Start":"04:20.300 ","End":"04:25.640","Text":"so we get 1/r times sine Theta over r times Theta hat because,"},{"Start":"04:25.640 ","End":"04:27.125","Text":"again, we have to put in Theta hat,"},{"Start":"04:27.125 ","End":"04:29.390","Text":"and there is no z element right now,"},{"Start":"04:29.390 ","End":"04:30.530","Text":"so we don\u0027t have to worry about it."},{"Start":"04:30.530 ","End":"04:34.925","Text":"Then you go through the same procedure when you\u0027re doing this with spherical coordinates,"},{"Start":"04:34.925 ","End":"04:37.570","Text":"it\u0027s just that the coefficients are slightly different."},{"Start":"04:37.570 ","End":"04:42.020","Text":"That\u0027s it. That\u0027s your solution for the gradient and your 3 coordinate systems."},{"Start":"04:42.020 ","End":"04:45.540","Text":"In our next lecture, we\u0027ll talk about the meaning of the gradient."}],"ID":9208},{"Watched":false,"Name":"The Meaning of a Gradient Function","Duration":"10m 54s","ChapterTopicVideoID":8943,"CourseChapterTopicPlaylistID":85319,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8943.jpeg","UploadDate":"2017-03-21T08:40:22.8670000","DurationForVideoObject":"PT10M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this lecture, we\u0027re going to talk about the meaning of the gradient."},{"Start":"00:03.570 ","End":"00:07.730","Text":"This is a little bit above and beyond the requirements for the course,"},{"Start":"00:07.730 ","End":"00:11.190","Text":"but hopefully you\u0027ll find it interesting and I\u0027m sure that it will explain in"},{"Start":"00:11.190 ","End":"00:14.805","Text":"greater detail what\u0027s going on when we perform a gradient function."},{"Start":"00:14.805 ","End":"00:16.725","Text":"This lecture is optional,"},{"Start":"00:16.725 ","End":"00:18.900","Text":"but it might help you in calculus and it\u0027ll certainly help you"},{"Start":"00:18.900 ","End":"00:21.585","Text":"understand what\u0027s going on in this course a little bit better."},{"Start":"00:21.585 ","End":"00:24.500","Text":"To begin, when we look at a differential function,"},{"Start":"00:24.500 ","End":"00:26.720","Text":"it\u0027s measuring the difference in something over time."},{"Start":"00:26.720 ","End":"00:30.140","Text":"If we have a function of x, it\u0027s measuring the difference in x over time."},{"Start":"00:30.140 ","End":"00:32.720","Text":"It gives a quantity to our difference."},{"Start":"00:32.720 ","End":"00:34.100","Text":"If we have the following function,"},{"Start":"00:34.100 ","End":"00:35.990","Text":"x and that\u0027s our x-axis,"},{"Start":"00:35.990 ","End":"00:39.260","Text":"our differential function will measure how"},{"Start":"00:39.260 ","End":"00:43.100","Text":"much our initial function has changed over a miniscule amount of time."},{"Start":"00:43.100 ","End":"00:44.645","Text":"In our first point,"},{"Start":"00:44.645 ","End":"00:46.010","Text":"as opposed to our second point,"},{"Start":"00:46.010 ","End":"00:47.135","Text":"we\u0027ll have less change."},{"Start":"00:47.135 ","End":"00:48.680","Text":"In our second point, we\u0027ll have more change."},{"Start":"00:48.680 ","End":"00:50.435","Text":"Why? As the curve gets steeper,"},{"Start":"00:50.435 ","End":"00:52.775","Text":"there\u0027s more change per unit of time than"},{"Start":"00:52.775 ","End":"00:56.285","Text":"at our first point where the curve is less steep."},{"Start":"00:56.285 ","End":"01:00.335","Text":"Ultimately, this is what our differential function our derivative is showing us,"},{"Start":"01:00.335 ","End":"01:01.670","Text":"is the change over time."},{"Start":"01:01.670 ","End":"01:03.825","Text":"As our function gets steeper,"},{"Start":"01:03.825 ","End":"01:05.510","Text":"our change will be greater."},{"Start":"01:05.510 ","End":"01:07.610","Text":"As our function gets more gradual,"},{"Start":"01:07.610 ","End":"01:09.005","Text":"the change will be less."},{"Start":"01:09.005 ","End":"01:10.670","Text":"Now let\u0027s look at it in 3 dimensions."},{"Start":"01:10.670 ","End":"01:13.700","Text":"Let\u0027s say we have a 3-dimensional function,"},{"Start":"01:13.700 ","End":"01:17.300","Text":"f(x, y, z),"},{"Start":"01:17.300 ","End":"01:19.615","Text":"let\u0027s say we\u0027re measuring temperature."},{"Start":"01:19.615 ","End":"01:23.645","Text":"In this case, temperature is going to be dependent"},{"Start":"01:23.645 ","End":"01:28.130","Text":"upon the location in a room with coordinates x, y, and z."},{"Start":"01:28.130 ","End":"01:29.450","Text":"It\u0027s a 3-dimensional space."},{"Start":"01:29.450 ","End":"01:33.860","Text":"Here\u0027s our room. We can draw out a few points within our room."},{"Start":"01:33.860 ","End":"01:37.030","Text":"There\u0027s your x, your y, and your z axes."},{"Start":"01:37.030 ","End":"01:40.910","Text":"At one point, the temperature is going to be 27 degrees,"},{"Start":"01:40.910 ","End":"01:42.605","Text":"but at our second point,"},{"Start":"01:42.605 ","End":"01:44.000","Text":"a little higher up in the room,"},{"Start":"01:44.000 ","End":"01:45.545","Text":"because heat rises,"},{"Start":"01:45.545 ","End":"01:49.820","Text":"the temperature will be 28 degrees and a little higher still it\u0027ll be 29 degrees."},{"Start":"01:49.820 ","End":"01:50.990","Text":"But again, there\u0027s a 3-dimensional room."},{"Start":"01:50.990 ","End":"01:52.190","Text":"We\u0027re not going to go on 1 line."},{"Start":"01:52.190 ","End":"01:54.830","Text":"Another point lower in the room is 28 degrees,"},{"Start":"01:54.830 ","End":"01:56.495","Text":"another point 27,"},{"Start":"01:56.495 ","End":"01:59.330","Text":"29, another 28 degrees here or there,"},{"Start":"01:59.330 ","End":"02:02.365","Text":"another point might even be 26 degrees."},{"Start":"02:02.365 ","End":"02:06.695","Text":"Now I want to have a differential of my temperature as well."},{"Start":"02:06.695 ","End":"02:10.595","Text":"But I have a little bit of a problem because it depends on the direction."},{"Start":"02:10.595 ","End":"02:15.740","Text":"Now I can move up and my temperature might increase by 1 degree or I can"},{"Start":"02:15.740 ","End":"02:21.080","Text":"move to the right and my temperature might increase by nothing or decrease by a degree,"},{"Start":"02:21.080 ","End":"02:23.020","Text":"or it could do many things."},{"Start":"02:23.020 ","End":"02:28.520","Text":"I could go on a diagonal or a different diagonal and it will change in a different way."},{"Start":"02:28.520 ","End":"02:32.270","Text":"Because I have an infinite number of directions I"},{"Start":"02:32.270 ","End":"02:35.870","Text":"can move and an infinite number of points between which I can move,"},{"Start":"02:35.870 ","End":"02:38.555","Text":"I have a problem in determining where"},{"Start":"02:38.555 ","End":"02:41.815","Text":"I\u0027m measuring my derivative and what direction it\u0027s going in."},{"Start":"02:41.815 ","End":"02:43.720","Text":"Now I have a bit of a problem,"},{"Start":"02:43.720 ","End":"02:45.950","Text":"I don\u0027t know how I\u0027m going to do my"},{"Start":"02:45.950 ","End":"02:49.235","Text":"derivative and luckily this is where the gradient helps."},{"Start":"02:49.235 ","End":"02:51.725","Text":"If you recall, the gradient is the nabla,"},{"Start":"02:51.725 ","End":"02:54.365","Text":"which it can also be called a del,"},{"Start":"02:54.365 ","End":"02:59.690","Text":"the upside down triangle of f equals the partial differential of x in"},{"Start":"02:59.690 ","End":"03:02.330","Text":"the direction of x plus the partial differential of y in the direction"},{"Start":"03:02.330 ","End":"03:05.860","Text":"of y plus the partial differential of z in the direction of z."},{"Start":"03:05.860 ","End":"03:07.640","Text":"Just a quick reminder,"},{"Start":"03:07.640 ","End":"03:09.200","Text":"when we do a gradient function,"},{"Start":"03:09.200 ","End":"03:13.850","Text":"we\u0027re taking a scalar function and converting it into some vector function,"},{"Start":"03:13.850 ","End":"03:17.465","Text":"meaning that we give magnitude a direction."},{"Start":"03:17.465 ","End":"03:19.685","Text":"Now the next question, of course,"},{"Start":"03:19.685 ","End":"03:23.510","Text":"is what is going on when we do a gradient and what does that mean for us?"},{"Start":"03:23.510 ","End":"03:26.840","Text":"Here you see our gradient function."},{"Start":"03:26.840 ","End":"03:29.480","Text":"What we can do to it is if we take the magnitude,"},{"Start":"03:29.480 ","End":"03:33.020","Text":"meaning we take the x component squared plus"},{"Start":"03:33.020 ","End":"03:36.845","Text":"the y component squared plus the z component squared,"},{"Start":"03:36.845 ","End":"03:38.345","Text":"and take the square root of that,"},{"Start":"03:38.345 ","End":"03:45.260","Text":"we\u0027re left with the magnitude and the magnitude is the magnitude of the greatest change,"},{"Start":"03:45.260 ","End":"03:48.390","Text":"the maximal change in the function."},{"Start":"03:48.590 ","End":"03:53.680","Text":"If we take that magnitude and we divide our initial function by the magnitude,"},{"Start":"03:53.680 ","End":"03:55.795","Text":"we\u0027re left with the direction of the vector."},{"Start":"03:55.795 ","End":"04:00.070","Text":"Because if you recall, a vector is made up of its magnitude times its direction,"},{"Start":"04:00.070 ","End":"04:04.090","Text":"so if we take the initial vector and divide it by its magnitude,"},{"Start":"04:04.090 ","End":"04:06.235","Text":"we\u0027re left with the direction, and the direction is"},{"Start":"04:06.235 ","End":"04:10.220","Text":"the direction of the greatest change in the function."},{"Start":"04:10.220 ","End":"04:13.590","Text":"I\u0027ll try to explain this using the 3-dimensional drawing here"},{"Start":"04:13.590 ","End":"04:16.870","Text":"and a 2-dimensional function to help us understand a little easier."},{"Start":"04:16.870 ","End":"04:19.470","Text":"Our 2-dimensional function is f(x,"},{"Start":"04:19.470 ","End":"04:23.485","Text":"y) = x^2 times y^2."},{"Start":"04:23.485 ","End":"04:31.575","Text":"The gradient function of that initial function is going to be 2xy^2 in the x-direction,"},{"Start":"04:31.575 ","End":"04:39.315","Text":"x hat plus 2yx^2 in the direction of y, y hat."},{"Start":"04:39.315 ","End":"04:44.810","Text":"If you noticed, I just did a gradient function in 2 dimensions."},{"Start":"04:44.810 ","End":"04:47.660","Text":"It\u0027s the same as doing a gradient function 3 dimensions,"},{"Start":"04:47.660 ","End":"04:50.000","Text":"we just set z to 0 and it works."},{"Start":"04:50.000 ","End":"04:53.210","Text":"Now the reason we\u0027re using a 2-dimensional function here is because"},{"Start":"04:53.210 ","End":"04:56.470","Text":"it\u0027s really hard to imagine this or draw this in 3 dimensions."},{"Start":"04:56.470 ","End":"05:00.275","Text":"What we can say is if this is our 3-dimensional graph,"},{"Start":"05:00.275 ","End":"05:02.585","Text":"then we have our x-axis and our y-axis,"},{"Start":"05:02.585 ","End":"05:08.785","Text":"ultimately our x-axis is this axis and y is that axis."},{"Start":"05:08.785 ","End":"05:11.060","Text":"We can say is I can choose any x,"},{"Start":"05:11.060 ","End":"05:15.340","Text":"y point and then the value of the function is my z point."},{"Start":"05:15.340 ","End":"05:19.890","Text":"Ultimately, z equals the value of the function."},{"Start":"05:20.360 ","End":"05:24.830","Text":"In that sense, the value of my function, the output,"},{"Start":"05:24.830 ","End":"05:29.180","Text":"will give me some surface that rests on top of the x,"},{"Start":"05:29.180 ","End":"05:32.450","Text":"y axis in a shape as we have here."},{"Start":"05:32.450 ","End":"05:34.895","Text":"If I did this in x, y, and z,"},{"Start":"05:34.895 ","End":"05:36.890","Text":"it would be very hard for me to draw this because I can\u0027t"},{"Start":"05:36.890 ","End":"05:38.945","Text":"draw out the value of the function."},{"Start":"05:38.945 ","End":"05:44.735","Text":"The best substitute I can do is to have some 3-dimensional graph and use colors."},{"Start":"05:44.735 ","End":"05:48.935","Text":"I could say red is when the value is higher and blue is when the value"},{"Start":"05:48.935 ","End":"05:53.690","Text":"is reduced or when it\u0027s sloping downwards and it\u0027d be very confusing,"},{"Start":"05:53.690 ","End":"05:55.850","Text":"hard to read, it wouldn\u0027t be"},{"Start":"05:55.850 ","End":"06:00.245","Text":"interpretable and wouldn\u0027t fulfill the purpose of a graph in that sense."},{"Start":"06:00.245 ","End":"06:03.380","Text":"Anyways, we\u0027re going to do this in 2 dimensions."},{"Start":"06:03.380 ","End":"06:06.650","Text":"If that explanation with 3 dimensions didn\u0027t make much sense to you,"},{"Start":"06:06.650 ","End":"06:07.710","Text":"don\u0027t worry about it too much,"},{"Start":"06:07.710 ","End":"06:11.240","Text":"we can just move forward to understanding that we\u0027re working in 2 dimensions and"},{"Start":"06:11.240 ","End":"06:16.155","Text":"the value I get out of my function will fill in for our z in some senses."},{"Start":"06:16.155 ","End":"06:18.140","Text":"If I take my function,"},{"Start":"06:18.140 ","End":"06:19.400","Text":"I can look on the x,"},{"Start":"06:19.400 ","End":"06:21.740","Text":"y plane and when I get my value of the function,"},{"Start":"06:21.740 ","End":"06:23.960","Text":"I can fill that in as my z point,"},{"Start":"06:23.960 ","End":"06:25.990","Text":"that red dot there."},{"Start":"06:25.990 ","End":"06:30.185","Text":"If I take my gradient function and apply it to this point,"},{"Start":"06:30.185 ","End":"06:32.840","Text":"I get interesting findings about the slope."},{"Start":"06:32.840 ","End":"06:34.880","Text":"Let\u0027s do this with an example."},{"Start":"06:34.880 ","End":"06:41.505","Text":"Let\u0027s assume that the point we\u0027re talking about here is where x=1 and y=2."},{"Start":"06:41.505 ","End":"06:48.795","Text":"Our value of the function f(1,2) is going to be 1^2 times 2^2, it\u0027s 4."},{"Start":"06:48.795 ","End":"06:52.545","Text":"Our gradient function, if we plug in 1 and 2,"},{"Start":"06:52.545 ","End":"06:57.120","Text":"we\u0027re going to get 2 times 1 times 2^2 in the direction of x,"},{"Start":"06:57.120 ","End":"07:01.040","Text":"which is 8, in the direction of x and the direction of y we get 2"},{"Start":"07:01.040 ","End":"07:05.980","Text":"times 2 times 1^2 which is 4 in the direction of y."},{"Start":"07:05.980 ","End":"07:09.440","Text":"Now that I have this solution to my gradient function,"},{"Start":"07:09.440 ","End":"07:12.545","Text":"I can think of it as a vector in every way."},{"Start":"07:12.545 ","End":"07:16.300","Text":"I can write that instead of 8x hat y,"},{"Start":"07:16.300 ","End":"07:18.650","Text":"4y hat, I can write that as 8,"},{"Start":"07:18.650 ","End":"07:19.790","Text":"4. It is a vector."},{"Start":"07:19.790 ","End":"07:23.030","Text":"What it means, I have a vector pointing in the direction 8,"},{"Start":"07:23.030 ","End":"07:24.455","Text":"4 coming from the origin,"},{"Start":"07:24.455 ","End":"07:26.090","Text":"pointing towards 8, 4."},{"Start":"07:26.090 ","End":"07:29.150","Text":"In terms of direction, we can also think of that as 2, 1."},{"Start":"07:29.150 ","End":"07:33.315","Text":"We\u0027re talking about the same ratio of x to y, 2 to 1."},{"Start":"07:33.315 ","End":"07:36.705","Text":"We could think of that as 8, 4 or just in terms of direction,"},{"Start":"07:36.705 ","End":"07:38.760","Text":"2, 1. What does that mean?"},{"Start":"07:38.760 ","End":"07:41.975","Text":"It means that at our x, y 0.12,"},{"Start":"07:41.975 ","End":"07:45.080","Text":"we have a vector pointing in the direction of 8,"},{"Start":"07:45.080 ","End":"07:47.300","Text":"4 or 2, 1, something like that."},{"Start":"07:47.300 ","End":"07:49.235","Text":"Now I have to remind you now,"},{"Start":"07:49.235 ","End":"07:51.890","Text":"this function that we\u0027re using in"},{"Start":"07:51.890 ","End":"07:54.680","Text":"2 dimensions is not the same function that\u0027s plotted out here,"},{"Start":"07:54.680 ","End":"07:56.240","Text":"and the point is not the same point."},{"Start":"07:56.240 ","End":"08:00.860","Text":"I expect that my vector will actually point in the direction of the maximal change."},{"Start":"08:00.860 ","End":"08:02.090","Text":"In this particular example,"},{"Start":"08:02.090 ","End":"08:06.020","Text":"it point downwards towards the bottom of this dome, so to speak."},{"Start":"08:06.020 ","End":"08:09.530","Text":"Again, it\u0027s pointing in the direction of maximal change."},{"Start":"08:09.530 ","End":"08:12.650","Text":"My direction of maximal change can be called my direction of"},{"Start":"08:12.650 ","End":"08:15.575","Text":"greatest increase or my direction of steepest descent."},{"Start":"08:15.575 ","End":"08:18.035","Text":"In this case, it\u0027s the steepest descent."},{"Start":"08:18.035 ","End":"08:24.470","Text":"You can see that that is clearly where the change is happening most quickly."},{"Start":"08:24.470 ","End":"08:27.110","Text":"As opposed to this direction of the second arrow where there\u0027s"},{"Start":"08:27.110 ","End":"08:29.690","Text":"a slight change if we\u0027re going upwards is a slight change."},{"Start":"08:29.690 ","End":"08:32.210","Text":"If we go down towards the bottom of that cone of that dome,"},{"Start":"08:32.210 ","End":"08:33.755","Text":"there\u0027s a greater change."},{"Start":"08:33.755 ","End":"08:36.920","Text":"Now if I want to talk about the magnitude of my gradient function,"},{"Start":"08:36.920 ","End":"08:38.405","Text":"let\u0027s return to our example."},{"Start":"08:38.405 ","End":"08:40.625","Text":"We have 8^2 plus 4^2."},{"Start":"08:40.625 ","End":"08:42.710","Text":"We\u0027re going to take the square root of that and that\u0027ll give us"},{"Start":"08:42.710 ","End":"08:46.175","Text":"the magnitude that equals the square root of 80,"},{"Start":"08:46.175 ","End":"08:48.545","Text":"which is 64 plus 16."},{"Start":"08:48.545 ","End":"08:50.510","Text":"The square root of 80 is our magnitude."},{"Start":"08:50.510 ","End":"08:51.620","Text":"What is the magnitude?"},{"Start":"08:51.620 ","End":"08:56.015","Text":"The magnitude is the slope in the direction of greatest change,"},{"Start":"08:56.015 ","End":"08:57.880","Text":"in this case, steep descent."},{"Start":"08:57.880 ","End":"09:00.635","Text":"In the direction of the steepest descent where our arrow is pointing,"},{"Start":"09:00.635 ","End":"09:03.385","Text":"that is our slope and it gives us the slope."},{"Start":"09:03.385 ","End":"09:05.660","Text":"It\u0027s as though we took a derivative in"},{"Start":"09:05.660 ","End":"09:09.575","Text":"the direction of the greatest increase or steepest descent."},{"Start":"09:09.575 ","End":"09:11.780","Text":"Here\u0027s what you get out of your gradient."},{"Start":"09:11.780 ","End":"09:16.025","Text":"You can find your direction of the greatest change."},{"Start":"09:16.025 ","End":"09:17.540","Text":"You can find the magnitude of that,"},{"Start":"09:17.540 ","End":"09:18.980","Text":"the slope and if you want,"},{"Start":"09:18.980 ","End":"09:23.540","Text":"you can also do something called normalizing where you ignore the magnitude,"},{"Start":"09:23.540 ","End":"09:25.190","Text":"we just want to find the direction."},{"Start":"09:25.190 ","End":"09:27.455","Text":"Like in our formula above,"},{"Start":"09:27.455 ","End":"09:30.260","Text":"we take our initial function,"},{"Start":"09:30.260 ","End":"09:34.030","Text":"our initial gradient solution that is 8,"},{"Start":"09:34.030 ","End":"09:38.135","Text":"4 in this case and divide that by"},{"Start":"09:38.135 ","End":"09:41.180","Text":"the magnitude of the gradient function root"},{"Start":"09:41.180 ","End":"09:44.765","Text":"80 and then you can just find your direction without dealing with the magnitude."},{"Start":"09:44.765 ","End":"09:46.370","Text":"Of course, you don\u0027t need to do that."},{"Start":"09:46.370 ","End":"09:48.950","Text":"You can get the direction from your points and you can get"},{"Start":"09:48.950 ","End":"09:52.340","Text":"the magnitude obviously by doing the magnitude operation."},{"Start":"09:52.340 ","End":"09:54.875","Text":"I hope this has all been clear so far."},{"Start":"09:54.875 ","End":"09:59.420","Text":"By the way, the same can apply in 3 dimensions."},{"Start":"09:59.420 ","End":"10:02.840","Text":"Let\u0027s say we have our 3 axes here, z, y,"},{"Start":"10:02.840 ","End":"10:07.100","Text":"and x and we have some point given along these axes."},{"Start":"10:07.100 ","End":"10:09.185","Text":"If we do the gradient function,"},{"Start":"10:09.185 ","End":"10:13.720","Text":"we\u0027ll find the direction of greatest change in the same way and we"},{"Start":"10:13.720 ","End":"10:18.685","Text":"can find the magnitude of that as well by taking the magnitude of the gradient function."},{"Start":"10:18.685 ","End":"10:21.860","Text":"Let\u0027s say, in that direction where the arrow is pointing,"},{"Start":"10:21.860 ","End":"10:23.600","Text":"we find that that\u0027s where the greatest change is."},{"Start":"10:23.600 ","End":"10:27.920","Text":"We can then take the magnitude and find out what the slope of that change is."},{"Start":"10:27.920 ","End":"10:32.525","Text":"I hope this was clear and thank you guys for coming along for this long lecture."},{"Start":"10:32.525 ","End":"10:36.320","Text":"The next thing we\u0027re going to do in our next lecture is talk about how we can use"},{"Start":"10:36.320 ","End":"10:41.150","Text":"the gradient to find the slope of change,"},{"Start":"10:41.150 ","End":"10:47.120","Text":"that is, the derivative in the direction that\u0027s not the direction of maximal change."},{"Start":"10:47.120 ","End":"10:48.910","Text":"We don\u0027t always want to find the maximal change,"},{"Start":"10:48.910 ","End":"10:53.070","Text":"sometimes we want to find the slope in a different direction."}],"ID":9209},{"Watched":false,"Name":"Calculating the Slope in a Different Direction","Duration":"3m 15s","ChapterTopicVideoID":8944,"CourseChapterTopicPlaylistID":85319,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8944.jpeg","UploadDate":"2021-11-18T09:01:10.4570000","DurationForVideoObject":"PT3M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Now I want to show you how to use the gradient to find the value of"},{"Start":"00:03.570 ","End":"00:08.220","Text":"a slope in a different direction that\u0027s not the direction of maximal change."},{"Start":"00:08.220 ","End":"00:10.005","Text":"Technically it\u0027s not so difficult,"},{"Start":"00:10.005 ","End":"00:12.570","Text":"what we do is a scalar multiplication between"},{"Start":"00:12.570 ","End":"00:17.280","Text":"the gradient function and a unit vector in the direction that we want."},{"Start":"00:17.280 ","End":"00:19.920","Text":"Ultimately what that means is that we\u0027re taking the magnitude because it\u0027s"},{"Start":"00:19.920 ","End":"00:25.050","Text":"a scalar multiplication times the 1 because it\u0027s a unit vector,"},{"Start":"00:25.050 ","End":"00:26.640","Text":"times the cosine of Theta,"},{"Start":"00:26.640 ","End":"00:33.420","Text":"which is the angle between our maximal change and the direction that we want to go."},{"Start":"00:33.420 ","End":"00:36.030","Text":"It turns out that all you need to do is take the magnitude of"},{"Start":"00:36.030 ","End":"00:38.790","Text":"the gradient function and multiply that by the cosine of Theta,"},{"Start":"00:38.790 ","End":"00:41.115","Text":"which is the angle between the direction you want,"},{"Start":"00:41.115 ","End":"00:42.875","Text":"and the direction of greatest change,"},{"Start":"00:42.875 ","End":"00:44.330","Text":"maximal change, that is."},{"Start":"00:44.330 ","End":"00:46.550","Text":"Now this doesn\u0027t seem very intuitive because it\u0027s only"},{"Start":"00:46.550 ","End":"00:49.250","Text":"based on the angle and not based on the function itself."},{"Start":"00:49.250 ","End":"00:52.805","Text":"In fact, there are some things that confirm this weirdness, for example,"},{"Start":"00:52.805 ","End":"00:56.750","Text":"if you want to change to a perpendicular angle,"},{"Start":"00:56.750 ","End":"01:00.560","Text":"your answer will always be 0 because cosine of 90 degrees,"},{"Start":"01:00.560 ","End":"01:02.720","Text":"which will give you a perpendicular angle is 0,"},{"Start":"01:02.720 ","End":"01:04.775","Text":"so the whole function equals 0."},{"Start":"01:04.775 ","End":"01:08.645","Text":"Now I warn you that there are in fact functions that exist where"},{"Start":"01:08.645 ","End":"01:13.535","Text":"the perpendicular angle to the angle of greatest change does not equal 0."},{"Start":"01:13.535 ","End":"01:16.400","Text":"That is, the change in that angle,"},{"Start":"01:16.400 ","End":"01:17.885","Text":"the slope at that angle."},{"Start":"01:17.885 ","End":"01:20.870","Text":"Those functions are just not differentiable."},{"Start":"01:20.870 ","End":"01:22.865","Text":"You can\u0027t do a derivative of them."},{"Start":"01:22.865 ","End":"01:25.910","Text":"But when we\u0027re dealing with differentiable equations,"},{"Start":"01:25.910 ","End":"01:27.320","Text":"those where you can take a derivative."},{"Start":"01:27.320 ","End":"01:32.050","Text":"In fact, the angle perpendicular to that of greatest change will always equal 0,"},{"Start":"01:32.050 ","End":"01:34.880","Text":"and we can find it by the function we just talked about,"},{"Start":"01:34.880 ","End":"01:40.310","Text":"taking the magnitude of the gradient function multiplied the cosine of Theta."},{"Start":"01:40.310 ","End":"01:42.140","Text":"Let\u0027s do an example to figure this out."},{"Start":"01:42.140 ","End":"01:47.450","Text":"In the example we\u0027re supposed to calculate the value of the slope of the function f(x,"},{"Start":"01:47.450 ","End":"01:50.580","Text":"y) equals 2x^2 times y."},{"Start":"01:50.580 ","End":"01:52.215","Text":"We\u0027re doing that at the point 1,"},{"Start":"01:52.215 ","End":"01:55.125","Text":"2 in the direction 1 over root 2,"},{"Start":"01:55.125 ","End":"01:57.205","Text":"negative 1 over root 2."},{"Start":"01:57.205 ","End":"02:02.780","Text":"Now notice that n, the direction we\u0027re going for is a unit vector. It\u0027s been normalized."},{"Start":"02:02.780 ","End":"02:05.300","Text":"What we first want to do is do a gradient function,"},{"Start":"02:05.300 ","End":"02:10.535","Text":"and the result is 4xy in the direction of x plus 2x^2 and the direction of y."},{"Start":"02:10.535 ","End":"02:12.200","Text":"Notice that typo there."},{"Start":"02:12.200 ","End":"02:14.345","Text":"If we plug in our points 1 and 2,"},{"Start":"02:14.345 ","End":"02:21.080","Text":"our result is 8 in the direction of x and 2 in the direction of y."},{"Start":"02:21.080 ","End":"02:26.060","Text":"That 8x hat plus 2y hat is my vector in the direction of greatest change."},{"Start":"02:26.060 ","End":"02:30.530","Text":"But what I want is the slope in the direction n hat."},{"Start":"02:30.530 ","End":"02:33.965","Text":"What I do is a scalar multiplication of 8,"},{"Start":"02:33.965 ","End":"02:36.570","Text":"2 and 1 over root 2,"},{"Start":"02:36.570 ","End":"02:39.160","Text":"negative 1 over root 2."},{"Start":"02:39.160 ","End":"02:40.990","Text":"When I do that scalar multiplication,"},{"Start":"02:40.990 ","End":"02:44.680","Text":"the result I end up with is 8 over root 2 plus negative"},{"Start":"02:44.680 ","End":"02:50.050","Text":"2 over root 2 or 8 over root 2 minus 2 root 2."},{"Start":"02:50.050 ","End":"02:51.400","Text":"If we simplify the solution,"},{"Start":"02:51.400 ","End":"02:58.210","Text":"we find that 6 over root 2 is the value of the slope in the direction n at the 1, 2."},{"Start":"02:58.210 ","End":"02:59.980","Text":"This is the end of our lecture on this."},{"Start":"02:59.980 ","End":"03:01.720","Text":"I realize it\u0027s not so intuitive,"},{"Start":"03:01.720 ","End":"03:04.180","Text":"but technically it\u0027s rather simple and straightforward,"},{"Start":"03:04.180 ","End":"03:08.650","Text":"and this is how you can find the value of a slope in"},{"Start":"03:08.650 ","End":"03:15.080","Text":"a certain direction that is not the direction of greatest change using your gradient."}],"ID":9210},{"Watched":false,"Name":"Minimum and Maximum","Duration":"2m ","ChapterTopicVideoID":8945,"CourseChapterTopicPlaylistID":85319,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8945.jpeg","UploadDate":"2017-03-21T08:41:07.6570000","DurationForVideoObject":"PT2M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.800","Text":"In this lecture, we\u0027re going to talk about"},{"Start":"00:01.800 ","End":"00:05.265","Text":"some important points on your graph when your gradient function equal 0."},{"Start":"00:05.265 ","End":"00:06.990","Text":"When your gradient function equal 0,"},{"Start":"00:06.990 ","End":"00:09.120","Text":"it means you have 1 of the 4 following points."},{"Start":"00:09.120 ","End":"00:11.280","Text":"Either a minimum, a maximum,"},{"Start":"00:11.280 ","End":"00:13.665","Text":"a saddle, or a ridge."},{"Start":"00:13.665 ","End":"00:16.110","Text":"These 4 points are similar to the same things in"},{"Start":"00:16.110 ","End":"00:19.410","Text":"typography or geography and we can talk about them the same way."},{"Start":"00:19.410 ","End":"00:22.800","Text":"This is similar to a 1 dimensional equation where if"},{"Start":"00:22.800 ","End":"00:26.235","Text":"your derivative equal 0 and then you know you have the minimum or a maximum."},{"Start":"00:26.235 ","End":"00:28.230","Text":"In this case, it\u0027s a little more complicated."},{"Start":"00:28.230 ","End":"00:30.660","Text":"For example, let\u0027s start with the maximum."},{"Start":"00:30.660 ","End":"00:33.060","Text":"You can see this circled point on the left."},{"Start":"00:33.060 ","End":"00:34.440","Text":"This is clearly a maximum,"},{"Start":"00:34.440 ","End":"00:36.885","Text":"it\u0027s something that\u0027s higher than the rest of the graph."},{"Start":"00:36.885 ","End":"00:39.375","Text":"You can see that in any given direction,"},{"Start":"00:39.375 ","End":"00:43.470","Text":"your slope is going to be the same,"},{"Start":"00:43.470 ","End":"00:46.940","Text":"and in that sense your gradient is going to equal 0."},{"Start":"00:46.940 ","End":"00:49.820","Text":"Now we can also talk about a saddle."},{"Start":"00:49.820 ","End":"00:51.590","Text":"A saddle is when you\u0027re at a minimum from"},{"Start":"00:51.590 ","End":"00:53.870","Text":"2 sides and at a maximum from the other 2 sides."},{"Start":"00:53.870 ","End":"00:56.120","Text":"Imagine if 1 of these 3 points,"},{"Start":"00:56.120 ","End":"00:59.000","Text":"if the graph continued over the edge and went"},{"Start":"00:59.000 ","End":"01:02.010","Text":"in a downward slope like a parabola. This would be a saddle."},{"Start":"01:02.010 ","End":"01:03.890","Text":"A saddle again you\u0027re in a minimum from"},{"Start":"01:03.890 ","End":"01:06.655","Text":"2 sides and your a maximum from the other 2 sides."},{"Start":"01:06.655 ","End":"01:10.310","Text":"Again, your gradient will equal 0."},{"Start":"01:10.310 ","End":"01:11.900","Text":"A shoulder is a similar point,"},{"Start":"01:11.900 ","End":"01:15.155","Text":"but it\u0027s at a minimum in 3 sides and a maximum on 1 side."},{"Start":"01:15.155 ","End":"01:19.610","Text":"Imagine the graph continues like this and your shoulder is as you can see,"},{"Start":"01:19.610 ","End":"01:26.210","Text":"a minimum going left to right for us and going outside of the graph it\u0027s at a convulsion,"},{"Start":"01:26.210 ","End":"01:28.445","Text":"an inflection in the middle of this graph."},{"Start":"01:28.445 ","End":"01:32.800","Text":"Again, it\u0027s a minimum on 3 points and a maximum on 1 point."},{"Start":"01:32.800 ","End":"01:34.970","Text":"Of course, the minimum we can\u0027t really draw here but"},{"Start":"01:34.970 ","End":"01:37.325","Text":"it\u0027s obviously at the bottom of the cone."},{"Start":"01:37.325 ","End":"01:39.725","Text":"If we want to find 1 of these important points,"},{"Start":"01:39.725 ","End":"01:42.650","Text":"we know that minimums and maximums and also we can"},{"Start":"01:42.650 ","End":"01:46.220","Text":"say shoulders and saddles are important for us."},{"Start":"01:46.220 ","End":"01:50.915","Text":"If we want to find when we set our gradient equal to 0 and we see what our results are."},{"Start":"01:50.915 ","End":"01:53.915","Text":"If we set our gradient equal to 0 we\u0027ll find the results we want."},{"Start":"01:53.915 ","End":"01:57.785","Text":"We can do this exactly the same as we do with a normal derivative,"},{"Start":"01:57.785 ","End":"02:00.330","Text":"it just now we\u0027re using a gradient."}],"ID":9211}],"Thumbnail":null,"ID":85319},{"Name":"Scalar Multiplication","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Scalar Multiplication","Duration":"7m 2s","ChapterTopicVideoID":9248,"CourseChapterTopicPlaylistID":85320,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9248.jpeg","UploadDate":"2017-04-02T09:12:12.1070000","DurationForVideoObject":"PT7M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"Hello. In this video,"},{"Start":"00:01.920 ","End":"00:04.425","Text":"I want to talk about something called the scalar multiplication,"},{"Start":"00:04.425 ","End":"00:07.395","Text":"which is an operation you can do between 2 vectors."},{"Start":"00:07.395 ","End":"00:10.260","Text":"First, we going to talk about how you do the operation,"},{"Start":"00:10.260 ","End":"00:12.900","Text":"and then we\u0027ll talk about what it means."},{"Start":"00:12.900 ","End":"00:15.180","Text":"The first thing you should know is that"},{"Start":"00:15.180 ","End":"00:18.810","Text":"a scalar multiplication is symbolized with a dot between the 2 vectors,"},{"Start":"00:18.810 ","End":"00:21.832","Text":"so A vector dot B vector."},{"Start":"00:21.832 ","End":"00:27.480","Text":"It\u0027s a bolded dot and you can say A dot B when you\u0027re talking about it."},{"Start":"00:27.480 ","End":"00:29.040","Text":"When you do this operation,"},{"Start":"00:29.040 ","End":"00:31.620","Text":"you\u0027re going to multiply the different components of"},{"Start":"00:31.620 ","End":"00:34.305","Text":"the 2 vectors by each other and add them all together."},{"Start":"00:34.305 ","End":"00:36.750","Text":"For example, here we have A.B,"},{"Start":"00:36.750 ","End":"00:43.390","Text":"so it\u0027d be A_x times B_x plus A_y times B_y plus A_z times B_z."},{"Start":"00:43.390 ","End":"00:47.705","Text":"Now the result of a scalar multiplication will always be a scalar."},{"Start":"00:47.705 ","End":"00:48.980","Text":"It\u0027s not going to be a vector,"},{"Start":"00:48.980 ","End":"00:51.385","Text":"it\u0027s only going to be a magnitude or a number."},{"Start":"00:51.385 ","End":"00:54.125","Text":"When we do this, if we multiply this out,"},{"Start":"00:54.125 ","End":"01:00.410","Text":"let\u0027s say A vector was 1,2,3 and the B vector was 2,-1,1."},{"Start":"01:00.410 ","End":"01:04.560","Text":"Our result would be 1 times 2 plus 2 times negative"},{"Start":"01:04.560 ","End":"01:09.685","Text":"1 plus 3 times 1 and our result would be 3."},{"Start":"01:09.685 ","End":"01:13.490","Text":"Notice this is just a number with no direction and no angle."},{"Start":"01:13.490 ","End":"01:18.155","Text":"The second way that you can do with scalar multiplication is by multiplying"},{"Start":"01:18.155 ","End":"01:21.380","Text":"the magnitude of the first vector and the magnitude"},{"Start":"01:21.380 ","End":"01:25.220","Text":"of the second vector by the cosine of the angle between them."},{"Start":"01:25.220 ","End":"01:28.100","Text":"Let\u0027s say in this example we have again,"},{"Start":"01:28.100 ","End":"01:30.980","Text":"vectors A and B, and the angle between them is 30 degrees."},{"Start":"01:30.980 ","End":"01:37.015","Text":"We know that the magnitude of the vector A is 2 and the magnitude of the vector B is 3."},{"Start":"01:37.015 ","End":"01:42.545","Text":"A.B would be 2 times 3 times the cosine of 30 degrees,"},{"Start":"01:42.545 ","End":"01:46.235","Text":"which equals 6 times the square root of 3 over 2,"},{"Start":"01:46.235 ","End":"01:48.730","Text":"or 3 times the square root of 3."},{"Start":"01:48.730 ","End":"01:52.370","Text":"That would be our answer. Depending on the data you\u0027re given,"},{"Start":"01:52.370 ","End":"01:54.035","Text":"you might want to use 1 method or the other."},{"Start":"01:54.035 ","End":"01:56.390","Text":"If you\u0027re just given the Cartesian coordinates,"},{"Start":"01:56.390 ","End":"01:58.070","Text":"you might want to use the first method if you\u0027re given"},{"Start":"01:58.070 ","End":"02:00.570","Text":"the magnitude and the angle you might want to use the second method."},{"Start":"02:00.570 ","End":"02:04.340","Text":"You can combine these methods to find the angle"},{"Start":"02:04.340 ","End":"02:06.140","Text":"between the 2 vectors if that\u0027s what you\u0027re looking"},{"Start":"02:06.140 ","End":"02:09.080","Text":"for and find the scalar multiplication in the meantime."},{"Start":"02:09.080 ","End":"02:11.130","Text":"The way we want to do that is to find the angle."},{"Start":"02:11.130 ","End":"02:13.355","Text":"We\u0027re going to isolate the element that has the angle in it,"},{"Start":"02:13.355 ","End":"02:17.480","Text":"which is the cosine of the angle between the 2 vectors and how we do that is"},{"Start":"02:17.480 ","End":"02:22.055","Text":"set your first method for finding the scalar multiplication equal to the second method."},{"Start":"02:22.055 ","End":"02:26.735","Text":"We then isolate our cosine of the angle so that we end up with cosine of the angle"},{"Start":"02:26.735 ","End":"02:30.180","Text":"equals the scalar multiplication of A and"},{"Start":"02:30.180 ","End":"02:34.050","Text":"B over the magnitude of A times the magnitude of B."},{"Start":"02:34.050 ","End":"02:41.980","Text":"In this example where A equals 1,2,3 and B equals 1,2,-1, what we can do,"},{"Start":"02:41.980 ","End":"02:45.635","Text":"is if we do the math, you can see below,"},{"Start":"02:45.635 ","End":"02:50.150","Text":"we have 1 times 1 plus 2 times 2 plus 3 times negative 1"},{"Start":"02:50.150 ","End":"02:54.990","Text":"over the square root of 1 plus 2 plus 3 each squared"},{"Start":"02:54.990 ","End":"02:57.600","Text":"times the square root of 1 plus 2 plus"},{"Start":"02:57.600 ","End":"03:00.110","Text":"negative 1 each squared and the result that you get is"},{"Start":"03:00.110 ","End":"03:05.770","Text":"negative 1 over 21 equals the cosine of the angle between the 2 vectors."},{"Start":"03:05.770 ","End":"03:08.410","Text":"From here you can find your angle Alpha,"},{"Start":"03:08.410 ","End":"03:14.305","Text":"which equals the cosine to the power of negative 1 of 1 over the square root of 21."},{"Start":"03:14.305 ","End":"03:17.915","Text":"Now you have your 2 methods for solving a scalar multiplication,"},{"Start":"03:17.915 ","End":"03:23.540","Text":"as well as a third method that combines the 2 to find the angle between the 2 vectors,"},{"Start":"03:23.540 ","End":"03:27.825","Text":"that angle Alpha and this is all in 3 dimensions."},{"Start":"03:27.825 ","End":"03:29.770","Text":"If you were to do this in 2 dimensions,"},{"Start":"03:29.770 ","End":"03:31.880","Text":"the operation would be exactly the same."},{"Start":"03:31.880 ","End":"03:35.390","Text":"You just forget your third coordinate."},{"Start":"03:35.390 ","End":"03:37.490","Text":"Let\u0027s assume that that\u0027s z that drops off."},{"Start":"03:37.490 ","End":"03:41.810","Text":"You could either set z equal to 0 or incompletely take it out of the equation."},{"Start":"03:41.810 ","End":"03:50.960","Text":"In our equation above, you just get rid of A_z times B_z and in this example,"},{"Start":"03:50.960 ","End":"03:52.880","Text":"you would just get rid of again the z"},{"Start":"03:52.880 ","End":"03:55.715","Text":"coordinate and get rid of the multiplication there as well."},{"Start":"03:55.715 ","End":"03:59.375","Text":"When you\u0027re talking about magnitude and angle, the magnitude,"},{"Start":"03:59.375 ","End":"04:01.670","Text":"if it\u0027s already given to you then having no z"},{"Start":"04:01.670 ","End":"04:04.860","Text":"doesn\u0027t affect the magnitude, because again,"},{"Start":"04:04.860 ","End":"04:11.535","Text":"it\u0027s already given, but when you want to combine the 2 methods to find your angle,"},{"Start":"04:11.535 ","End":"04:15.680","Text":"you just going to drop your z coordinate as well."},{"Start":"04:15.680 ","End":"04:19.430","Text":"Get rid of the z there, get rid of your z there and within your equation down here,"},{"Start":"04:19.430 ","End":"04:23.580","Text":"when you\u0027re finding the magnitude and when you\u0027re doing your scalar multiplication."},{"Start":"04:24.290 ","End":"04:26.885","Text":"Now if we switch gears for a moment,"},{"Start":"04:26.885 ","End":"04:34.121","Text":"we can look at 1 special characteristic of scalar multiplications."},{"Start":"04:34.121 ","End":"04:39.200","Text":"1 important characteristic of scalar multiplication is that the order is not important."},{"Start":"04:39.200 ","End":"04:41.870","Text":"A.B equals B.A."},{"Start":"04:41.870 ","End":"04:43.280","Text":"I can do a first,"},{"Start":"04:43.280 ","End":"04:44.450","Text":"I can do B first,"},{"Start":"04:44.450 ","End":"04:45.860","Text":"it doesn\u0027t matter what the order is,"},{"Start":"04:45.860 ","End":"04:48.065","Text":"the result will be the same."},{"Start":"04:48.065 ","End":"04:51.320","Text":"Now I want to talk for a moment about what happens during"},{"Start":"04:51.320 ","End":"04:55.020","Text":"scalar multiplication and what that means for us."},{"Start":"04:55.090 ","End":"04:57.875","Text":"When I do a scalar multiplication,"},{"Start":"04:57.875 ","End":"04:59.135","Text":"what I\u0027m really getting is"},{"Start":"04:59.135 ","End":"05:06.505","Text":"1 vector times the component of another vector that is parallel to my initial vector."},{"Start":"05:06.505 ","End":"05:09.625","Text":"Let me put this in other terms in order to explain it."},{"Start":"05:09.625 ","End":"05:12.655","Text":"If you recall, A.B equals the magnitude of"},{"Start":"05:12.655 ","End":"05:14.620","Text":"A times the magnitude of B times"},{"Start":"05:14.620 ","End":"05:17.035","Text":"the cosine of the angle between them, we\u0027ll call it Alpha."},{"Start":"05:17.035 ","End":"05:19.705","Text":"This is the angle Alpha, and that\u0027s A and that\u0027s B."},{"Start":"05:19.705 ","End":"05:23.170","Text":"When we take the magnitude of B times the cosine of Alpha,"},{"Start":"05:23.170 ","End":"05:29.050","Text":"what we\u0027re really doing is projecting or transposing the vector B onto the A axis."},{"Start":"05:29.050 ","End":"05:31.840","Text":"For the sake of the problem, let\u0027s assume that\u0027s also the x axis."},{"Start":"05:31.840 ","End":"05:34.475","Text":"We take B and bring it down to that axis,"},{"Start":"05:34.475 ","End":"05:37.120","Text":"and what we\u0027re left with is the component of B that\u0027s in"},{"Start":"05:37.120 ","End":"05:40.195","Text":"the exact same direction as A that\u0027s parallel to A,"},{"Start":"05:40.195 ","End":"05:44.065","Text":"and what we really get is the magnitude of B times the cosine of Alpha"},{"Start":"05:44.065 ","End":"05:48.525","Text":"is the B element in the exact same direction as A times A,"},{"Start":"05:48.525 ","End":"05:50.795","Text":"so that the y element in this case,"},{"Start":"05:50.795 ","End":"05:52.130","Text":"if A is on the x axis,"},{"Start":"05:52.130 ","End":"05:55.010","Text":"the y element is not affecting our solution,"},{"Start":"05:55.010 ","End":"05:57.540","Text":"but only the x element."},{"Start":"05:57.560 ","End":"06:00.664","Text":"Assuming B is a constant magnitude,"},{"Start":"06:00.664 ","End":"06:05.240","Text":"we can say that our perpendicular element will not affect the solution,"},{"Start":"06:05.240 ","End":"06:06.905","Text":"only the parallel element."},{"Start":"06:06.905 ","End":"06:09.665","Text":"We can even say this in non-constant terms."},{"Start":"06:09.665 ","End":"06:11.990","Text":"If B was instead, this line,"},{"Start":"06:11.990 ","End":"06:14.000","Text":"it has a different y element,"},{"Start":"06:14.000 ","End":"06:15.980","Text":"but the exact same x element,"},{"Start":"06:15.980 ","End":"06:19.370","Text":"so the solution would be the exact same because, again,"},{"Start":"06:19.370 ","End":"06:22.070","Text":"we\u0027re transposing or projecting our B down to"},{"Start":"06:22.070 ","End":"06:26.120","Text":"the x axis at a 90-degree angle and that\u0027s what\u0027s affecting it."},{"Start":"06:26.120 ","End":"06:29.240","Text":"Now in the same way that we said,"},{"Start":"06:29.240 ","End":"06:32.645","Text":"it doesn\u0027t matter the order, we can do the exact operation in the opposite."},{"Start":"06:32.645 ","End":"06:36.110","Text":"Let\u0027s assume for a second that B is in fact on the x axis."},{"Start":"06:36.110 ","End":"06:37.910","Text":"What doesn\u0027t really matter what axis it\u0027s on,"},{"Start":"06:37.910 ","End":"06:40.520","Text":"but we transpose A down at 90 degrees,"},{"Start":"06:40.520 ","End":"06:43.490","Text":"still we have a projected onto our B axis."},{"Start":"06:43.490 ","End":"06:45.200","Text":"Now, only that line, again,"},{"Start":"06:45.200 ","End":"06:48.470","Text":"the element of A that is parallel to B or projected down to"},{"Start":"06:48.470 ","End":"06:53.580","Text":"B times the cosine Alpha is going to affect our end result."},{"Start":"06:53.580 ","End":"06:56.895","Text":"It\u0027s going to be B times that distance."},{"Start":"06:56.895 ","End":"06:58.965","Text":"We\u0027re using the same idea."},{"Start":"06:58.965 ","End":"07:00.090","Text":"The same thing is happening,"},{"Start":"07:00.090 ","End":"07:03.660","Text":"and this is the main idea behind scalar multiplication."}],"ID":9541}],"Thumbnail":null,"ID":85320},{"Name":"Unit Vector","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Unit Vector","Duration":"2m 41s","ChapterTopicVideoID":9250,"CourseChapterTopicPlaylistID":85321,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9250.jpeg","UploadDate":"2017-04-02T09:31:45.2000000","DurationForVideoObject":"PT2M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this video, we\u0027re going to talk about the unit vector."},{"Start":"00:03.630 ","End":"00:06.105","Text":"The unit vector is a vector in every way,"},{"Start":"00:06.105 ","End":"00:09.345","Text":"just like any other vector and its size,"},{"Start":"00:09.345 ","End":"00:12.435","Text":"its magnitude is 1 so for the vector a,"},{"Start":"00:12.435 ","End":"00:14.940","Text":"the unit vector would point in the exact same direction as a,"},{"Start":"00:14.940 ","End":"00:19.350","Text":"but would have a magnitude of 1 and we can draw it here."},{"Start":"00:19.350 ","End":"00:23.970","Text":"It would look something like this and it would be symbolized with a hat,"},{"Start":"00:23.970 ","End":"00:25.695","Text":"a with hat on top of it."},{"Start":"00:25.695 ","End":"00:27.540","Text":"Let me show you what that looks like."},{"Start":"00:27.540 ","End":"00:29.280","Text":"It should be like that,"},{"Start":"00:29.280 ","End":"00:33.480","Text":"looks like a little pointy top and so a hat is a vector in every way,"},{"Start":"00:33.480 ","End":"00:36.165","Text":"it points in the exact same direction as a,"},{"Start":"00:36.165 ","End":"00:38.910","Text":"it just has a magnitude of 1."},{"Start":"00:38.910 ","End":"00:42.560","Text":"The way that we find the magnitude of a unit vector is"},{"Start":"00:42.560 ","End":"00:46.220","Text":"we take the vector and divide it by its magnitude."},{"Start":"00:46.220 ","End":"00:49.040","Text":"That will give us a vector in the same direction because dividing"},{"Start":"00:49.040 ","End":"00:51.710","Text":"by scalar won\u0027t affect the direction and it will"},{"Start":"00:51.710 ","End":"00:53.990","Text":"give us the answer of 1 for magnitude because you\u0027re dividing"},{"Start":"00:53.990 ","End":"00:57.740","Text":"1 number by itself and you\u0027ll always get 1 in that situation."},{"Start":"00:57.740 ","End":"01:00.110","Text":"Practically, when I do this,"},{"Start":"01:00.110 ","End":"01:03.574","Text":"I also need to not just find the value,"},{"Start":"01:03.574 ","End":"01:06.730","Text":"but I also need to find the points if we\u0027re talking about x-y coordinates."},{"Start":"01:06.730 ","End":"01:11.405","Text":"In this example, we could say that a goes to the points 3 and 2,"},{"Start":"01:11.405 ","End":"01:18.780","Text":"x3 and 2y so we would divide 3,2 by the square root of 3^2 plus 2^2 the magnitude,"},{"Start":"01:18.780 ","End":"01:21.060","Text":"so that\u0027s 1 over square root of 13 and we\u0027re doing"},{"Start":"01:21.060 ","End":"01:24.770","Text":"a scalar multiplications so we\u0027re going to do the x value times that and the y"},{"Start":"01:24.770 ","End":"01:28.340","Text":"value times that so it\u0027ll be 3 times 1 over the square root"},{"Start":"01:28.340 ","End":"01:32.390","Text":"of 13 and 2 times the square root of 1 over 13."},{"Start":"01:32.390 ","End":"01:34.880","Text":"That is our new x and y coordinates,"},{"Start":"01:34.880 ","End":"01:38.260","Text":"so 3 over the square root of 13 is our x-coordinate,"},{"Start":"01:38.260 ","End":"01:41.655","Text":"2 over the square root of 13 is our y-coordinate,"},{"Start":"01:41.655 ","End":"01:46.690","Text":"and if we want to check to make sure that the magnitude of that is 1,"},{"Start":"01:46.690 ","End":"01:52.875","Text":"then all we need to do is take 3 over square root of 13^2,"},{"Start":"01:52.875 ","End":"01:55.035","Text":"2 over square root of 13^2,"},{"Start":"01:55.035 ","End":"02:02.325","Text":"and put them in a square root so we can do that and we find that 3^2 over 13 is 9,"},{"Start":"02:02.325 ","End":"02:05.130","Text":"2^2 over 13 is a 4/13,"},{"Start":"02:05.130 ","End":"02:06.620","Text":"so 9 plus 4 is 13."},{"Start":"02:06.620 ","End":"02:10.290","Text":"We get the square root of 13/13, which equals 1."},{"Start":"02:12.910 ","End":"02:15.905","Text":"As far as the use of the unit vector,"},{"Start":"02:15.905 ","End":"02:18.845","Text":"we\u0027re going to talk about it more later,"},{"Start":"02:18.845 ","End":"02:23.000","Text":"but for now you should know that it\u0027s useful when we want to talk about a direction"},{"Start":"02:23.000 ","End":"02:27.170","Text":"of a vector and we\u0027re not as concerned or interested in the magnitude of a vector."},{"Start":"02:27.170 ","End":"02:30.290","Text":"In addition to that, we have 3 special unit vectors,"},{"Start":"02:30.290 ","End":"02:32.570","Text":"x hat, y hat,"},{"Start":"02:32.570 ","End":"02:36.460","Text":"and z hat and we\u0027re going to talk about those in the next lecture,"},{"Start":"02:36.460 ","End":"02:39.569","Text":"but they have a special importance."}],"ID":9542},{"Watched":false,"Name":"x and y Unit Vectors","Duration":"2m 17s","ChapterTopicVideoID":9249,"CourseChapterTopicPlaylistID":85321,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9249.jpeg","UploadDate":"2017-04-02T09:31:33.2900000","DurationForVideoObject":"PT2M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:04.380","Text":"The x and y unit vectors are special unit vectors."},{"Start":"00:04.380 ","End":"00:06.660","Text":"They\u0027re special because they\u0027re unit vectors that go"},{"Start":"00:06.660 ","End":"00:09.300","Text":"along the x and the y axis respectively."},{"Start":"00:09.300 ","End":"00:15.900","Text":"The x unit vector is a unit vector that goes along the x-axis with a length of 1."},{"Start":"00:15.900 ","End":"00:22.570","Text":"The y unit vector is a unit vector that goes along the y-axis with a length of 1."},{"Start":"00:22.570 ","End":"00:26.750","Text":"These two are special because they have a special use for us as well."},{"Start":"00:26.750 ","End":"00:30.470","Text":"Using these two unit vectors and a linear combination,"},{"Start":"00:30.470 ","End":"00:32.405","Text":"we can describe any other vector."},{"Start":"00:32.405 ","End":"00:35.495","Text":"What does that mean? A linear combination is what you see here."},{"Start":"00:35.495 ","End":"00:38.330","Text":"It\u0027s when we put a coefficient in front of x hat and"},{"Start":"00:38.330 ","End":"00:41.765","Text":"a coefficient in front of y hat to get our vector a."},{"Start":"00:41.765 ","End":"00:43.850","Text":"If you think about it in a different way,"},{"Start":"00:43.850 ","End":"00:47.795","Text":"the coefficients equal the length of the components of the vector."},{"Start":"00:47.795 ","End":"00:51.155","Text":"If your vector a,"},{"Start":"00:51.155 ","End":"00:56.390","Text":"was 3,2, then your coefficients would be ax would be 3,"},{"Start":"00:56.390 ","End":"01:01.575","Text":"ay would be 2 and you would get the vector 3,2,"},{"Start":"01:01.575 ","End":"01:06.480","Text":"vector a being described by 3x hat plus 2y hat."},{"Start":"01:06.480 ","End":"01:09.275","Text":"If that\u0027s our vector a, you can see it drawn here."},{"Start":"01:09.275 ","End":"01:13.669","Text":"We can check that it\u0027s actually the correct distance,"},{"Start":"01:13.669 ","End":"01:15.230","Text":"the correct length, the correct magnitude,"},{"Start":"01:15.230 ","End":"01:17.005","Text":"and the correct direction."},{"Start":"01:17.005 ","End":"01:21.005","Text":"How do we check this? If it is 3x hat plus 2y hat,"},{"Start":"01:21.005 ","End":"01:24.980","Text":"we can draw it out rather simply using geometric addition of vectors."},{"Start":"01:24.980 ","End":"01:29.320","Text":"If we draw out 3x on our x-axis,"},{"Start":"01:29.320 ","End":"01:35.215","Text":"that\u0027s going to be a vector going along the x-axis traveling three units."},{"Start":"01:35.215 ","End":"01:38.970","Text":"We can draw it out. It\u0027s 3 times the unit vector."},{"Start":"01:38.970 ","End":"01:44.905","Text":"Then on top of that at the head of the x unit vector 3x,"},{"Start":"01:44.905 ","End":"01:48.795","Text":"we can then add 2y unit vector is 2y hat."},{"Start":"01:48.795 ","End":"01:52.025","Text":"If we add that, it\u0027ll bring us directly to the point we want,"},{"Start":"01:52.025 ","End":"01:56.930","Text":"which is 3,2 we can see here that this perfectly"},{"Start":"01:56.930 ","End":"02:02.960","Text":"describes our vector 3,2 using a linear combination of coefficients and the unit vectors."},{"Start":"02:02.960 ","End":"02:07.835","Text":"You could write the vector a as 3,2 in parentheses,"},{"Start":"02:07.835 ","End":"02:10.340","Text":"or as 3x hat plus 2y hat."},{"Start":"02:10.340 ","End":"02:12.005","Text":"It\u0027s really the same thing."},{"Start":"02:12.005 ","End":"02:14.675","Text":"That\u0027s how you use the unit vectors x and y,"},{"Start":"02:14.675 ","End":"02:17.670","Text":"which can describe any other vector."}],"ID":9543}],"Thumbnail":null,"ID":85321},{"Name":"Vectors in Three Dimension","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vectors In Three Dimensions","Duration":"31m 40s","ChapterTopicVideoID":9291,"CourseChapterTopicPlaylistID":85322,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9291.jpeg","UploadDate":"2017-05-06T16:57:41.9370000","DurationForVideoObject":"PT31M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.935","Text":"Hello. In this lesson,"},{"Start":"00:01.935 ","End":"00:05.625","Text":"we\u0027re going to speak about vectors in 3 dimensions."},{"Start":"00:05.625 ","End":"00:09.479","Text":"In order to explain what a 3-dimensional vector is,"},{"Start":"00:09.479 ","End":"00:12.810","Text":"let\u0027s first draw our 3 axes."},{"Start":"00:12.810 ","End":"00:16.004","Text":"Over here we can see our 3 axes."},{"Start":"00:16.004 ","End":"00:20.225","Text":"Now, these 2 axes are perpendicular to one another."},{"Start":"00:20.225 ","End":"00:26.129","Text":"These is the axes that we know as if we\u0027re looking at our screen like it\u0027s a page and"},{"Start":"00:26.129 ","End":"00:32.234","Text":"these are the axes that go along the page that represent our 2-dimensional page."},{"Start":"00:32.234 ","End":"00:36.180","Text":"This axis over here represents the third dimension."},{"Start":"00:36.180 ","End":"00:41.935","Text":"We can imagine it as coming out of our screen towards us."},{"Start":"00:41.935 ","End":"00:48.920","Text":"This axis is also perpendicular to these other 2 axes."},{"Start":"00:48.920 ","End":"00:56.179","Text":"Now the way that we\u0027re going to label this is that this is our x-axes."},{"Start":"00:56.179 ","End":"01:02.330","Text":"The axes that is pointing towards us out of the page."},{"Start":"01:02.330 ","End":"01:04.850","Text":"This is the y-axis."},{"Start":"01:04.850 ","End":"01:08.039","Text":"Sorry, I didn\u0027t mean to draw that hat."},{"Start":"01:08.200 ","End":"01:14.035","Text":"These axes up here is our z-axis."},{"Start":"01:14.035 ","End":"01:21.710","Text":"Now, let\u0027s imagine that I want to describe some point in space."},{"Start":"01:21.710 ","End":"01:24.799","Text":"When we\u0027re dealing with a 3-dimensional vector,"},{"Start":"01:24.799 ","End":"01:26.945","Text":"I write it like so."},{"Start":"01:26.945 ","End":"01:29.120","Text":"I\u0027m going to have my x value here,"},{"Start":"01:29.120 ","End":"01:30.650","Text":"my Y value here,"},{"Start":"01:30.650 ","End":"01:33.485","Text":"and my Z value over here."},{"Start":"01:33.485 ","End":"01:38.219","Text":"Let\u0027s say I want to describe the point 1,"},{"Start":"01:38.219 ","End":"01:40.620","Text":"2, and 3."},{"Start":"01:40.970 ","End":"01:46.160","Text":"Now let\u0027s see how I draw this point over here."},{"Start":"01:46.160 ","End":"01:50.105","Text":"What we can do is we can imagine that on our x-axis,"},{"Start":"01:50.105 ","End":"01:52.724","Text":"we have a scale like so."},{"Start":"01:52.724 ","End":"01:55.804","Text":"I\u0027m just reminding you that this is our x value,"},{"Start":"01:55.804 ","End":"01:57.724","Text":"this is our y value,"},{"Start":"01:57.724 ","End":"01:59.450","Text":"and this is our z value."},{"Start":"01:59.450 ","End":"02:02.615","Text":"In x, I\u0027m going to go along 1."},{"Start":"02:02.615 ","End":"02:08.055","Text":"Let\u0027s imagine that over here is my value 1."},{"Start":"02:08.055 ","End":"02:10.624","Text":"Now what I\u0027m going to do is I\u0027m going to draw"},{"Start":"02:10.624 ","End":"02:15.275","Text":"a dotted line along which my point can be and notice,"},{"Start":"02:15.275 ","End":"02:20.935","Text":"that I have a 90-degree angle between my x-axis and the dotted line."},{"Start":"02:20.935 ","End":"02:26.585","Text":"Or in other words, my dotted line is parallel to my y-axis."},{"Start":"02:26.585 ","End":"02:30.425","Text":"So my x-coordinate is going to be somewhere along here."},{"Start":"02:30.425 ","End":"02:33.800","Text":"Then, let\u0027s look at our y-axis."},{"Start":"02:33.800 ","End":"02:38.605","Text":"Now our y-axis is also going to have scales."},{"Start":"02:38.605 ","End":"02:42.615","Text":"We can see that our y value is 2."},{"Start":"02:42.615 ","End":"02:45.665","Text":"Let\u0027s say that this point over here is 2."},{"Start":"02:45.665 ","End":"02:47.810","Text":"What I\u0027m going to do is I\u0027m going to draw"},{"Start":"02:47.810 ","End":"02:51.815","Text":"a dotted line and see where it intercepts over here."},{"Start":"02:51.815 ","End":"02:55.579","Text":"Again, there is a 90-degree angle over here."},{"Start":"02:55.579 ","End":"03:03.110","Text":"This dotted line that I just drew now for the y value is parallel to my x-axis."},{"Start":"03:03.110 ","End":"03:05.930","Text":"That means that along the x,"},{"Start":"03:05.930 ","End":"03:10.160","Text":"y plane, my point is going to be here."},{"Start":"03:10.160 ","End":"03:13.220","Text":"Now we go into our third dimension."},{"Start":"03:13.220 ","End":"03:17.149","Text":"This is the 3, which means that we\u0027re going up."},{"Start":"03:17.149 ","End":"03:23.549","Text":"We go into our z-axis and again we have a scale."},{"Start":"03:24.940 ","End":"03:29.150","Text":"We can see that our z value here is 3."},{"Start":"03:29.150 ","End":"03:32.275","Text":"Let\u0027s say that that is this point over here."},{"Start":"03:32.275 ","End":"03:36.484","Text":"Now, what I\u0027m going to do is I\u0027m going to draw a point."},{"Start":"03:36.484 ","End":"03:45.319","Text":"Now notice that this also has a 90-degree angle between the dotted line and our z-axis."},{"Start":"03:45.319 ","End":"03:49.190","Text":"The reason I\u0027m drawing this a little bit slanted is because you have to"},{"Start":"03:49.190 ","End":"03:53.689","Text":"imagine that this is in a diagonal coming towards us."},{"Start":"03:53.689 ","End":"04:00.235","Text":"It\u0027s at 45 degrees to my x-axis and at 45 degrees to my y-axis."},{"Start":"04:00.235 ","End":"04:04.154","Text":"It\u0027s coming straight out towards us."},{"Start":"04:04.154 ","End":"04:07.790","Text":"Then we can see that if we draw this,"},{"Start":"04:07.790 ","End":"04:12.759","Text":"our point in 3D space is going to be here."},{"Start":"04:12.759 ","End":"04:19.570","Text":"So this is our point p. Of course,"},{"Start":"04:19.570 ","End":"04:26.540","Text":"this dotted line over here is at 90 degrees to the x, y plane."},{"Start":"04:27.290 ","End":"04:34.900","Text":"Now, we have our point P. But how do we define this vector when we\u0027re in 3D space?"},{"Start":"04:34.900 ","End":"04:39.069","Text":"The way we did it when we were working in 2D space, just some x,"},{"Start":"04:39.069 ","End":"04:45.204","Text":"y plane is that we would draw a line from the origin up until our points."},{"Start":"04:45.204 ","End":"04:49.660","Text":"We\u0027re going to do the exact same thing here in 3 dimensions."},{"Start":"04:49.660 ","End":"04:52.749","Text":"We\u0027re going to start from our origin and point"},{"Start":"04:52.749 ","End":"04:58.659","Text":"it towards our point P. It\u0027s an arrow because it\u0027s a vector."},{"Start":"04:58.659 ","End":"05:00.760","Text":"It has a direction."},{"Start":"05:00.760 ","End":"05:04.480","Text":"Now this arrow, as we said, is a vector."},{"Start":"05:04.480 ","End":"05:06.129","Text":"Let\u0027s call this vector,"},{"Start":"05:06.129 ","End":"05:08.100","Text":"let\u0027s give it a name. Let\u0027s call it a."},{"Start":"05:08.100 ","End":"05:11.559","Text":"Remember that when we\u0027re defining vectors,"},{"Start":"05:11.559 ","End":"05:17.199","Text":"we have to put an arrow on top of its name to denote that it\u0027s a vector."},{"Start":"05:17.199 ","End":"05:20.559","Text":"Then what will we call this vector?"},{"Start":"05:20.559 ","End":"05:24.625","Text":"So a vector A is given by"},{"Start":"05:24.625 ","End":"05:33.420","Text":"just the coordinates 1, 2, 3."},{"Start":"05:33.420 ","End":"05:39.739","Text":"Just like in 2D, we draw an arrow from the origin until our point in space."},{"Start":"05:39.739 ","End":"05:43.584","Text":"We then can give a name to that vector."},{"Start":"05:43.584 ","End":"05:46.429","Text":"Now, the components of this vector,"},{"Start":"05:46.429 ","End":"05:47.936","Text":"as we can see, we have 1,"},{"Start":"05:47.936 ","End":"05:51.994","Text":"2, 3. These are components."},{"Start":"05:51.994 ","End":"05:57.305","Text":"This component is our vector A \u0027s x component."},{"Start":"05:57.305 ","End":"06:01.699","Text":"This is our vector A\u0027s y component,"},{"Start":"06:01.699 ","End":"06:05.659","Text":"and this is our vector A\u0027s z component."},{"Start":"06:05.659 ","End":"06:15.289","Text":"This is of course exactly how we denoted the components of our vectors in 2 dimensions."},{"Start":"06:15.289 ","End":"06:21.030","Text":"The only difference is that now we have a third coordinate Az."},{"Start":"06:21.580 ","End":"06:26.480","Text":"Let\u0027s do another little practice in order to understand."},{"Start":"06:26.480 ","End":"06:29.884","Text":"Here we have our 3-dimensional axes again,"},{"Start":"06:29.884 ","End":"06:31.825","Text":"our x, our y, and our z."},{"Start":"06:31.825 ","End":"06:35.070","Text":"Here we have some kind of vector."},{"Start":"06:35.070 ","End":"06:38.864","Text":"Let\u0027s call this B vector."},{"Start":"06:38.864 ","End":"06:45.084","Text":"Our b vector is comprised of the x component,"},{"Start":"06:45.084 ","End":"06:50.769","Text":"xy component, and xz component."},{"Start":"06:52.610 ","End":"06:57.109","Text":"Let\u0027s imagine that I don\u0027t know what this point is."},{"Start":"06:57.109 ","End":"07:01.790","Text":"This point is obviously defined by these components."},{"Start":"07:01.790 ","End":"07:05.000","Text":"That means I don\u0027t know what these components are."},{"Start":"07:05.000 ","End":"07:12.359","Text":"How can I find out from this sketch what my components are?"},{"Start":"07:12.970 ","End":"07:16.610","Text":"Let\u0027s begin by finding our z component."},{"Start":"07:16.610 ","End":"07:22.715","Text":"All I have to do is I go from the tip of my vector,"},{"Start":"07:22.715 ","End":"07:28.804","Text":"and I\u0027m just going to draw a line which is perpendicular to my z-axis."},{"Start":"07:28.804 ","End":"07:34.410","Text":"Now of course, because we\u0027re dealing with a 3-dimensional drawing,"},{"Start":"07:34.410 ","End":"07:39.529","Text":"I\u0027m going to draw it that it looks horizontal."},{"Start":"07:39.529 ","End":"07:43.279","Text":"But this is of course, a 90-degree angle."},{"Start":"07:43.279 ","End":"07:52.330","Text":"Then, we can say that this whole height is bz."},{"Start":"07:53.710 ","End":"07:59.689","Text":"So this is our z value for a point over here,"},{"Start":"07:59.689 ","End":"08:06.119","Text":"and also our z value for our b vector z component."},{"Start":"08:06.310 ","End":"08:12.485","Text":"Now what I want to do is I wanted to find my bx and by components."},{"Start":"08:12.485 ","End":"08:18.275","Text":"What I do is I draw a line perpendicularly down."},{"Start":"08:18.275 ","End":"08:23.345","Text":"Now I know that this looks like the vector pointing above the y-axis."},{"Start":"08:23.345 ","End":"08:27.010","Text":"But it\u0027s just because this drawing is in 3 dimensions."},{"Start":"08:27.010 ","End":"08:29.720","Text":"So we draw it down and of course,"},{"Start":"08:29.720 ","End":"08:36.474","Text":"it has to be at 90 degrees to my 2-dimensional x, y plane."},{"Start":"08:36.474 ","End":"08:40.820","Text":"All I want to do is I draw a line which is"},{"Start":"08:40.820 ","End":"08:47.240","Text":"perpendicular to my x-axis or parallel to my y-axis."},{"Start":"08:47.240 ","End":"08:49.610","Text":"However you want to think of it."},{"Start":"08:49.610 ","End":"08:51.020","Text":"Then I get here,"},{"Start":"08:51.020 ","End":"08:53.914","Text":"and of course, this is at 90 degrees."},{"Start":"08:53.914 ","End":"08:59.480","Text":"I also draw a perpendicular line from my y-axis,"},{"Start":"08:59.480 ","End":"09:01.850","Text":"or in other words,"},{"Start":"09:01.850 ","End":"09:05.844","Text":"parallel to my x-axis, a dotted line."},{"Start":"09:05.844 ","End":"09:08.020","Text":"Then, of course,"},{"Start":"09:08.020 ","End":"09:14.574","Text":"this value up until this point over here is going to be my Bx."},{"Start":"09:14.574 ","End":"09:18.285","Text":"And this point specifically will be Bx."},{"Start":"09:18.285 ","End":"09:20.305","Text":"This value over here,"},{"Start":"09:20.305 ","End":"09:22.431","Text":"or all of this length that doesn\u0027t make a difference,"},{"Start":"09:22.431 ","End":"09:24.730","Text":"it\u0027s the same thing, is our by,"},{"Start":"09:24.730 ","End":"09:29.120","Text":"the y component of our b vector."},{"Start":"09:30.320 ","End":"09:34.464","Text":"When we represent a vector like so,"},{"Start":"09:34.464 ","End":"09:37.045","Text":"with an x, y, and z components,"},{"Start":"09:37.045 ","End":"09:41.770","Text":"just like when we were speaking about in 2D when we had an x and y component,"},{"Start":"09:41.770 ","End":"09:48.400","Text":"this is called the Cartesian format."},{"Start":"09:48.890 ","End":"09:53.139","Text":"Just like in 2 dimensions when we\u0027re dealing with"},{"Start":"09:53.139 ","End":"09:58.229","Text":"the Cartesian format, we have components."},{"Start":"09:58.229 ","End":"10:01.299","Text":"These components are, of course,"},{"Start":"10:01.299 ","End":"10:04.525","Text":"our x, y, and z components."},{"Start":"10:04.525 ","End":"10:07.900","Text":"We can remember that when we\u0027re dealing with 2D,"},{"Start":"10:07.900 ","End":"10:11.110","Text":"we have our polar coordinates,"},{"Start":"10:11.110 ","End":"10:14.124","Text":"which is our size and direction."},{"Start":"10:14.124 ","End":"10:17.620","Text":"The equivalent to our polar coordinates,"},{"Start":"10:17.620 ","End":"10:21.129","Text":"which is the exact same thing except now we\u0027re dealing with"},{"Start":"10:21.129 ","End":"10:28.870","Text":"3 dimensions is to use spherical coordinate system or the spherical format."},{"Start":"10:28.870 ","End":"10:33.555","Text":"So the spherical format is for 3 dimensions."},{"Start":"10:33.555 ","End":"10:35.725","Text":"If we\u0027re just working in 2 dimensions,"},{"Start":"10:35.725 ","End":"10:39.214","Text":"the equivalent is the polar format,"},{"Start":"10:39.214 ","End":"10:41.155","Text":"the polar coordinate system."},{"Start":"10:41.155 ","End":"10:46.429","Text":"We know that the spherical format or the polar format as well,"},{"Start":"10:46.429 ","End":"10:53.910","Text":"is comprised of our size and direction."},{"Start":"10:58.830 ","End":"11:07.134","Text":"We have 2 ways of showing our points in a 3-dimensional system."},{"Start":"11:07.134 ","End":"11:09.760","Text":"Now when we\u0027re dealing with the spherical format,"},{"Start":"11:09.760 ","End":"11:12.580","Text":"it\u0027s probably the most complicated thing that we\u0027re"},{"Start":"11:12.580 ","End":"11:16.580","Text":"going to have to go through in this lesson."},{"Start":"11:17.520 ","End":"11:20.530","Text":"In our spherical format,"},{"Start":"11:20.530 ","End":"11:22.795","Text":"we have our size,"},{"Start":"11:22.795 ","End":"11:24.939","Text":"the size of our arrow,"},{"Start":"11:24.939 ","End":"11:26.200","Text":"the size of our vector,"},{"Start":"11:26.200 ","End":"11:30.309","Text":"how long it is and we have our direction,"},{"Start":"11:30.309 ","End":"11:35.420","Text":"y direction is denoted by angles."},{"Start":"11:36.720 ","End":"11:39.805","Text":"In our polar format,"},{"Start":"11:39.805 ","End":"11:45.730","Text":"we\u0027re used to being in 2-dimensions with our x and y plane and we have 1 angle"},{"Start":"11:45.730 ","End":"11:52.164","Text":"usually denoted by Theta to show the direction that our vector is going in."},{"Start":"11:52.164 ","End":"11:55.195","Text":"However, because now we\u0027re in 3-dimensions,"},{"Start":"11:55.195 ","End":"12:00.700","Text":"we\u0027re going to need for our direction 2 angles,"},{"Start":"12:00.700 ","End":"12:05.660","Text":"2 denotes the direction of the arrow."},{"Start":"12:06.660 ","End":"12:09.804","Text":"Just like in our Cartesian format,"},{"Start":"12:09.804 ","End":"12:12.334","Text":"in 3-dimensions, we have 1,"},{"Start":"12:12.334 ","End":"12:14.319","Text":"2, 3 components."},{"Start":"12:14.319 ","End":"12:18.130","Text":"In our spherical format in 3-dimensions,"},{"Start":"12:18.130 ","End":"12:23.019","Text":"we also have 3 components our first component is the size,"},{"Start":"12:23.019 ","End":"12:31.580","Text":"and our second 2 or last 2 components are 2 angles denoting direction."},{"Start":"12:32.760 ","End":"12:37.809","Text":"Let\u0027s see what our size is."},{"Start":"12:37.809 ","End":"12:43.884","Text":"Our size, if this is our vector B, denoted like so,"},{"Start":"12:43.884 ","End":"12:50.185","Text":"the size is our absolute value."},{"Start":"12:50.185 ","End":"12:58.099","Text":"That is simply the length of the arrow."},{"Start":"12:59.850 ","End":"13:04.330","Text":"Now, let\u0027s talk about direction."},{"Start":"13:04.330 ","End":"13:08.395","Text":"Direction is given by our 2 angles."},{"Start":"13:08.395 ","End":"13:10.700","Text":"Let\u0027s see what they are."},{"Start":"13:11.010 ","End":"13:19.469","Text":"The first angle that we have is going to be our angle with"},{"Start":"13:19.469 ","End":"13:28.464","Text":"the z-axis and this angle"},{"Start":"13:28.464 ","End":"13:33.170","Text":"is usually denoted by the Greek letter Phi."},{"Start":"13:33.450 ","End":"13:37.719","Text":"If we go and look on our diagram,"},{"Start":"13:37.719 ","End":"13:44.379","Text":"we can see that this angle with the z-axis is our Phi over here."},{"Start":"13:44.379 ","End":"13:46.825","Text":"Now, it\u0027s a bit difficult to see."},{"Start":"13:46.825 ","End":"13:53.875","Text":"You have to just understand that between the z-axis and our vector over here,"},{"Start":"13:53.875 ","End":"13:58.460","Text":"we have some kind of angle which is called Phi."},{"Start":"13:58.950 ","End":"14:04.300","Text":"Similarly, if we draw our angle Phi for our A vector,"},{"Start":"14:04.300 ","End":"14:08.980","Text":"we\u0027ll see that our angle Phi is this over here,"},{"Start":"14:08.980 ","End":"14:13.190","Text":"between the z-axis and our arrow."},{"Start":"14:13.200 ","End":"14:17.064","Text":"Now let\u0027s take a look what happens to"},{"Start":"14:17.064 ","End":"14:22.119","Text":"a vector with some kind of angle Phi between x and the z-axis."},{"Start":"14:22.119 ","End":"14:24.940","Text":"If we change our angle Phi."},{"Start":"14:24.940 ","End":"14:29.930","Text":"Let\u0027s just draw a little sketch if this is our z,"},{"Start":"14:30.030 ","End":"14:36.700","Text":"this is our y, this is our x-axes."},{"Start":"14:36.700 ","End":"14:42.444","Text":"Then we have some vector going like this,"},{"Start":"14:42.444 ","End":"14:52.825","Text":"and let\u0027s call this vector D. Then we said that this angle over here is our Phi."},{"Start":"14:52.825 ","End":"14:58.340","Text":"Now let\u0027s see what happens when we play with the size of this angle Phi."},{"Start":"15:01.110 ","End":"15:03.340","Text":"Let\u0027s take a look,"},{"Start":"15:03.340 ","End":"15:06.175","Text":"so we can see that if I increase,"},{"Start":"15:06.175 ","End":"15:08.109","Text":"or decrease my angle,"},{"Start":"15:08.109 ","End":"15:12.654","Text":"my vector is going to move up and down."},{"Start":"15:12.654 ","End":"15:19.740","Text":"Imagine that the size is maintained and if my angle Phi becomes 0,"},{"Start":"15:19.740 ","End":"15:27.149","Text":"so my vector is going to be pointing exactly in the z-axis direction."},{"Start":"15:27.149 ","End":"15:32.425","Text":"If my angle Phi will be equal to 90 degrees,"},{"Start":"15:32.425 ","End":"15:38.830","Text":"so it will simply be on my x, y plane somewhere."},{"Start":"15:38.830 ","End":"15:42.595","Text":"Then the larger my angle Phi becomes,"},{"Start":"15:42.595 ","End":"15:48.954","Text":"the lower down my vector will appear and the smaller my angle Phi becomes,"},{"Start":"15:48.954 ","End":"15:54.589","Text":"the closer my vector will be to my z-axis."},{"Start":"15:57.150 ","End":"16:01.570","Text":"Of course, if my angle Phi is 180 degrees,"},{"Start":"16:01.570 ","End":"16:04.945","Text":"my orange arrow denoting my vector D,"},{"Start":"16:04.945 ","End":"16:09.589","Text":"will be pointing down in the negative z-direction."},{"Start":"16:09.660 ","End":"16:16.540","Text":"When my Phi is between 90 degrees and 180 degrees,"},{"Start":"16:16.540 ","End":"16:23.680","Text":"my vector will simply be pointing somewhere below my x, y plane."},{"Start":"16:23.680 ","End":"16:30.120","Text":"Now, of course, Phi can be bigger than 180,"},{"Start":"16:30.120 ","End":"16:36.550","Text":"and then that will simply mean that this vector is pointing backwards."},{"Start":"16:36.550 ","End":"16:39.324","Text":"Let\u0027s see if we can show it."},{"Start":"16:39.324 ","End":"16:42.550","Text":"If it\u0027s bigger than 180,"},{"Start":"16:42.550 ","End":"16:45.624","Text":"my vector will be like this."},{"Start":"16:45.624 ","End":"16:53.799","Text":"However, we never speak about Phi as being bigger than 180 simply because,"},{"Start":"16:53.799 ","End":"16:57.730","Text":"well, we\u0027ll show you afterwards that there\u0027s other ways"},{"Start":"16:57.730 ","End":"17:02.500","Text":"to denote that this vector pointing in the opposite direction."},{"Start":"17:02.500 ","End":"17:05.305","Text":"Whenever we\u0027re speaking about Phi,"},{"Start":"17:05.305 ","End":"17:15.620","Text":"we always say that Phi is somewhere between 0 and 180 degrees."},{"Start":"17:18.000 ","End":"17:24.079","Text":"The second angle now is the angle"},{"Start":"17:28.380 ","End":"17:36.400","Text":"or the angle of vectors orthogonal projection on"},{"Start":"17:36.400 ","End":"17:40.510","Text":"the xy plane"},{"Start":"17:40.510 ","End":"17:47.440","Text":"relative to the x-axis."},{"Start":"17:47.440 ","End":"17:53.665","Text":"This is slightly more complicated to understand. What does this mean?"},{"Start":"17:53.665 ","End":"17:56.665","Text":"Imagine that here is my point,"},{"Start":"17:56.665 ","End":"18:03.200","Text":"my point D, and my vector D is from the origin until my point over here."},{"Start":"18:06.060 ","End":"18:10.120","Text":"What does the angle of the vectors orthogonal"},{"Start":"18:10.120 ","End":"18:14.469","Text":"projection on the xy plane relative to the x-axis mean?"},{"Start":"18:14.469 ","End":"18:22.015","Text":"If we draw my perpendicular line and we drop it from this point over here,"},{"Start":"18:22.015 ","End":"18:29.635","Text":"and we take it down perpendicularly until it reaches our x, y plane."},{"Start":"18:29.635 ","End":"18:35.124","Text":"The point at which I dotted line intercepts are xy plane."},{"Start":"18:35.124 ","End":"18:39.470","Text":"That point we draw over here."},{"Start":"18:40.860 ","End":"18:51.565","Text":"This point, if we are then to draw a line from the origin until this point,"},{"Start":"18:51.565 ","End":"18:54.805","Text":"imagine this as a straight line and of course,"},{"Start":"18:54.805 ","End":"18:57.370","Text":"this is a 90-degree angle,"},{"Start":"18:57.370 ","End":"19:02.964","Text":"so the line from the origin until this point on the x, y plane."},{"Start":"19:02.964 ","End":"19:09.259","Text":"This is called the orthogonal projection on the xy plane."},{"Start":"19:09.630 ","End":"19:14.229","Text":"Now a lot of times this orthogonal projection,"},{"Start":"19:14.229 ","End":"19:21.350","Text":"this blue line, is referred to as D. If we\u0027re speaking about vector D, D_xy."},{"Start":"19:21.960 ","End":"19:29.409","Text":"It\u0027s the projection of a vector D and what it would look like on the xy plane."},{"Start":"19:29.409 ","End":"19:33.250","Text":"Now another way of imagining this is if we have our vector D,"},{"Start":"19:33.250 ","End":"19:36.660","Text":"which is some arrow, or line,"},{"Start":"19:36.660 ","End":"19:43.050","Text":"or however you want to think of it in space and it\u0027s solid let\u0027s imagine it\u0027s some rod."},{"Start":"19:43.050 ","End":"19:46.364","Text":"If you imagine from up the top over here,"},{"Start":"19:46.364 ","End":"19:47.730","Text":"you have some lamp,"},{"Start":"19:47.730 ","End":"19:57.100","Text":"or the sun and it\u0027s shining a light on to your rod D. The shadow that"},{"Start":"19:57.100 ","End":"20:00.669","Text":"will be seen on the ground for imagining that our ground is"},{"Start":"20:00.669 ","End":"20:07.760","Text":"our xy plane will be this orthogonal projection, this shadow."},{"Start":"20:08.610 ","End":"20:13.765","Text":"That\u0027s a way to maybe imagine what that is in real life."},{"Start":"20:13.765 ","End":"20:15.850","Text":"It\u0027s the orthogonal projection,"},{"Start":"20:15.850 ","End":"20:19.610","Text":"which is sometimes also referred to as D_xy."},{"Start":"20:21.390 ","End":"20:30.955","Text":"We have the angle of this orthogonal projection on the xy plane relative to the x-axis."},{"Start":"20:30.955 ","End":"20:36.609","Text":"That means that it\u0027s going to be between this blue line,"},{"Start":"20:36.609 ","End":"20:40.404","Text":"the orthogonal projection, and the x-axis,"},{"Start":"20:40.404 ","End":"20:44.665","Text":"and this angle is drawn on the xy plane,"},{"Start":"20:44.665 ","End":"20:51.550","Text":"in 2-dimensions in the xy plane between this blue line,"},{"Start":"20:51.550 ","End":"20:55.014","Text":"this shadow of our vector and"},{"Start":"20:55.014 ","End":"21:01.280","Text":"the x-axis and this angle is denoted by the Greek letter Theta."},{"Start":"21:01.520 ","End":"21:05.025","Text":"Let\u0027s write this over here."},{"Start":"21:05.025 ","End":"21:07.845","Text":"It\u0027s denoted by the angle Theta."},{"Start":"21:07.845 ","End":"21:09.525","Text":"Now, if we were dealing with,"},{"Start":"21:09.525 ","End":"21:11.354","Text":"instead of the spherical format,"},{"Start":"21:11.354 ","End":"21:14.570","Text":"we were dealing with our polar format in 2D,"},{"Start":"21:14.570 ","End":"21:18.159","Text":"so you can imagine that it\u0027s going to be the exact same thing."},{"Start":"21:18.159 ","End":"21:20.740","Text":"It\u0027s the exact same angle Theta."},{"Start":"21:20.740 ","End":"21:26.680","Text":"Remember in 2D we have our x and y axes."},{"Start":"21:26.680 ","End":"21:29.200","Text":"This is x, this is y,"},{"Start":"21:29.200 ","End":"21:30.925","Text":"and then we have some vector,"},{"Start":"21:30.925 ","End":"21:32.260","Text":"so we have our r,"},{"Start":"21:32.260 ","End":"21:34.344","Text":"which is the size of this arrow,"},{"Start":"21:34.344 ","End":"21:36.865","Text":"and this angle is Theta."},{"Start":"21:36.865 ","End":"21:41.845","Text":"Remember, between r and x-axis."},{"Start":"21:41.845 ","End":"21:49.280","Text":"Here it\u0027s the exact same thing except we\u0027re dealing with 3 dimensions."},{"Start":"21:50.460 ","End":"21:55.000","Text":"If we were to ignore our 3rd dimension,"},{"Start":"21:55.000 ","End":"22:00.909","Text":"our z component or what have you if we\u0027re just dealing with 2 dimensions."},{"Start":"22:00.909 ","End":"22:04.719","Text":"This Theta is the exact same thing."},{"Start":"22:04.719 ","End":"22:10.029","Text":"Now the only thing is when we\u0027re working in 3 dimensions with our spherical format,"},{"Start":"22:10.029 ","End":"22:14.095","Text":"it\u0027s a little bit harder to find this angle just because first,"},{"Start":"22:14.095 ","End":"22:19.345","Text":"we have to get our orthogonal projection for our vector."},{"Start":"22:19.345 ","End":"22:23.034","Text":"We have to find this blue line over here."},{"Start":"22:23.034 ","End":"22:27.770","Text":"Then after that, we have to work out what our Theta is."},{"Start":"22:28.620 ","End":"22:35.665","Text":"Now let\u0027s show this also over here for our vector B."},{"Start":"22:35.665 ","End":"22:37.809","Text":"What I would do is,"},{"Start":"22:37.809 ","End":"22:42.189","Text":"I see my point over here at the edge of B. I\u0027m just going to draw"},{"Start":"22:42.189 ","End":"22:48.460","Text":"my straight line down until it intercepts with my x, y plane."},{"Start":"22:48.460 ","End":"22:51.520","Text":"Then I have this point and of course, this line,"},{"Start":"22:51.520 ","End":"22:56.079","Text":"this dotted line, is at 90 degrees to my x,y plane."},{"Start":"22:56.079 ","End":"22:58.209","Text":"Then from the origin,"},{"Start":"22:58.209 ","End":"23:02.035","Text":"I\u0027m going to draw a line to that point."},{"Start":"23:02.035 ","End":"23:05.110","Text":"That\u0027s called the orthogonal projection"},{"Start":"23:05.110 ","End":"23:13.060","Text":"or B_xy or D_xy."},{"Start":"23:13.060 ","End":"23:17.110","Text":"It\u0027s the orthogonal projection of our vector B."},{"Start":"23:17.110 ","End":"23:24.610","Text":"Then, between this orthogonal projection and our x-axis is going to be the angle Theta."},{"Start":"23:24.610 ","End":"23:27.230","Text":"That means over here."},{"Start":"23:28.140 ","End":"23:35.425","Text":"Now let\u0027s talk about what happens if our angle Theta is changed."},{"Start":"23:35.425 ","End":"23:39.115","Text":"If it\u0027s larger or smaller, what does that mean?"},{"Start":"23:39.115 ","End":"23:41.845","Text":"What will our vector look like?"},{"Start":"23:41.845 ","End":"23:44.665","Text":"So let\u0027s try and explain this."},{"Start":"23:44.665 ","End":"23:50.110","Text":"If I take my vector D and let\u0027s say that my angle Theta is 0."},{"Start":"23:50.110 ","End":"23:55.645","Text":"That means that there\u0027s an angle of 0 between my orthogonal projection,"},{"Start":"23:55.645 ","End":"24:00.804","Text":"my D_xy, the blue line, and my x-axis."},{"Start":"24:00.804 ","End":"24:08.274","Text":"That will mean that my orthogonal projection is on the x-axis."},{"Start":"24:08.274 ","End":"24:10.300","Text":"It\u0027s exactly on top."},{"Start":"24:10.300 ","End":"24:12.414","Text":"Then in 3D space,"},{"Start":"24:12.414 ","End":"24:18.890","Text":"considering I also have my angle Phi so my vector will be"},{"Start":"24:19.740 ","End":"24:29.960","Text":"somewhere over here where its orthogonal projection is exactly on my x-axis."},{"Start":"24:31.020 ","End":"24:37.315","Text":"Now let\u0027s imagine that my Theta angle is 0,"},{"Start":"24:37.315 ","End":"24:44.214","Text":"that means that my orthogonal projection for this vector D is going to be on the x-axis."},{"Start":"24:44.214 ","End":"24:49.525","Text":"Now imagine that my angle Theta becomes larger, it\u0027s growing."},{"Start":"24:49.525 ","End":"24:54.865","Text":"That means that my vector can spin around."},{"Start":"24:54.865 ","End":"25:00.980","Text":"My Theta angle can be between 0-360 degrees."},{"Start":"25:00.980 ","End":"25:05.862","Text":"When it\u0027s 0, it\u0027s on top of the x-axis,"},{"Start":"25:05.862 ","End":"25:07.764","Text":"and as it grows,"},{"Start":"25:07.764 ","End":"25:10.730","Text":"the vector will spin around."},{"Start":"25:10.920 ","End":"25:15.309","Text":"When for instance, my Theta\u0027s 90,"},{"Start":"25:15.309 ","End":"25:21.804","Text":"so that means that my orthogonal projection is going to be on the y-axis."},{"Start":"25:21.804 ","End":"25:25.269","Text":"My vector will look something like this."},{"Start":"25:25.269 ","End":"25:29.785","Text":"When my Theta grows to 180 so"},{"Start":"25:29.785 ","End":"25:35.289","Text":"my orthogonal projection is going to be in the negative x-direction."},{"Start":"25:35.289 ","End":"25:41.269","Text":"It\u0027s going to be somewhere here like so."},{"Start":"25:41.340 ","End":"25:47.830","Text":"Then my angle Theta can spend more so that"},{"Start":"25:47.830 ","End":"25:54.445","Text":"my vector can be somewhere around my z-axis."},{"Start":"25:54.445 ","End":"25:59.240","Text":"My vector is just rotating about the z-axis."},{"Start":"26:03.180 ","End":"26:06.145","Text":"Theta, unlike Phi,"},{"Start":"26:06.145 ","End":"26:15.980","Text":"can be anywhere between 0 degrees and 360 degrees."},{"Start":"26:16.680 ","End":"26:20.664","Text":"Of course, when Theta is equal to 360,"},{"Start":"26:20.664 ","End":"26:24.505","Text":"it\u0027s the exact same thing as saying that Theta is equal to 0."},{"Start":"26:24.505 ","End":"26:31.399","Text":"Because again, that means that the orthogonal projection will be on the x-axis."},{"Start":"26:32.310 ","End":"26:37.060","Text":"To conclude, when our Theta changes,"},{"Start":"26:37.060 ","End":"26:43.789","Text":"it rotates a vector about the z-axis."},{"Start":"26:44.900 ","End":"26:49.350","Text":"Now another way to think of this is that if we say that"},{"Start":"26:49.350 ","End":"26:53.769","Text":"our angle Phi is equal to 90 degrees,"},{"Start":"26:53.769 ","End":"26:59.950","Text":"that means that our vector is simply already on the x, y plane."},{"Start":"26:59.950 ","End":"27:03.775","Text":"The orthogonal projection will simply be the vector"},{"Start":"27:03.775 ","End":"27:08.589","Text":"d itself because it\u0027s lying on the x, y plane."},{"Start":"27:08.590 ","End":"27:14.280","Text":"Then, as our Theta changes,"},{"Start":"27:14.280 ","End":"27:19.524","Text":"our vector will just rotate around the x, y plane,"},{"Start":"27:19.524 ","End":"27:24.849","Text":"just as we\u0027re used to when we\u0027re dealing with 2 dimensions."},{"Start":"27:24.849 ","End":"27:28.405","Text":"That out Theta can change."},{"Start":"27:28.405 ","End":"27:35.030","Text":"That just means that our arrow or a vector is spinning around."},{"Start":"27:35.610 ","End":"27:40.840","Text":"Of course, if our Phi is bigger than 90 degrees,"},{"Start":"27:40.840 ","End":"27:47.290","Text":"that means that our vector D is pointing somewhere down in the negative z direction."},{"Start":"27:47.290 ","End":"27:49.060","Text":"It\u0027s a similar thing."},{"Start":"27:49.060 ","End":"27:52.450","Text":"Our Theta will spin a vector around,"},{"Start":"27:52.450 ","End":"28:00.379","Text":"but the top of the vector will be spinning below the x, y plane."},{"Start":"28:01.890 ","End":"28:06.670","Text":"The important thing to remember right now is that Phi,"},{"Start":"28:06.670 ","End":"28:08.500","Text":"the angle with the z-axis,"},{"Start":"28:08.500 ","End":"28:13.330","Text":"is always going to be between 0-180 degrees."},{"Start":"28:13.330 ","End":"28:16.509","Text":"Then our Theta, which is the angle of"},{"Start":"28:16.509 ","End":"28:21.966","Text":"the vector\u0027s orthogonal projection on the x y plane relative to the x-axis,"},{"Start":"28:21.966 ","End":"28:28.674","Text":"this is always going to be between 0-360 degrees."},{"Start":"28:28.674 ","End":"28:32.530","Text":"Now the reason that we do that, the reason that we say that our Phi has to be"},{"Start":"28:32.530 ","End":"28:38.214","Text":"between 0-180 degrees rather than up until 360,"},{"Start":"28:38.214 ","End":"28:39.625","Text":"is that this way,"},{"Start":"28:39.625 ","End":"28:45.339","Text":"in order to define any point in this 3-dimensional space,"},{"Start":"28:45.339 ","End":"28:48.669","Text":"we only have 1 way to do it."},{"Start":"28:48.669 ","End":"28:55.495","Text":"If our Phi was between 0-360 as well as our Theta being like so,"},{"Start":"28:55.495 ","End":"29:02.450","Text":"then we would have 2 ways to define where our vector is and we don\u0027t want that."},{"Start":"29:03.720 ","End":"29:07.044","Text":"Having it like this, there\u0027s 1 way to define,"},{"Start":"29:07.044 ","End":"29:11.169","Text":"which means that if we have our vector D pointing in this direction,"},{"Start":"29:11.169 ","End":"29:16.735","Text":"so if we want suddenly for it to be pointing in the opposite direction this way,"},{"Start":"29:16.735 ","End":"29:21.174","Text":"all we have to do is define a different Theta."},{"Start":"29:21.174 ","End":"29:26.275","Text":"Instead of our Theta being 45 degrees, if it\u0027s here,"},{"Start":"29:26.275 ","End":"29:32.569","Text":"it will be 275 or something."},{"Start":"29:33.210 ","End":"29:42.963","Text":"But the general idea is that in order to get our vector D to point over here, let\u0027s say,"},{"Start":"29:42.963 ","End":"29:48.235","Text":"it would just be done by rotating it via our Theta angle,"},{"Start":"29:48.235 ","End":"29:53.650","Text":"rather than saying that we can also rotate it through our Phi angle and then it would go"},{"Start":"29:53.650 ","End":"30:00.924","Text":"down under the x,y plane and then back off the x,y plane until this point."},{"Start":"30:00.924 ","End":"30:03.924","Text":"This means that there\u0027s 1 definition."},{"Start":"30:03.924 ","End":"30:05.965","Text":"We have our angle Phi,"},{"Start":"30:05.965 ","End":"30:17.089","Text":"and then its position around the origin and around the z-axis is related to our Theta."},{"Start":"30:17.310 ","End":"30:22.495","Text":"Now a tiny, tiny little notes, as we know,"},{"Start":"30:22.495 ","End":"30:30.339","Text":"our Phi and our Theta are simply Greek letters to denote which angle we\u0027re talking about."},{"Start":"30:30.339 ","End":"30:34.270","Text":"Specifically here, we said that our Phi was the angle with"},{"Start":"30:34.270 ","End":"30:39.289","Text":"the z-axis and our Theta was on the x,y plane."},{"Start":"30:39.300 ","End":"30:41.770","Text":"Now we\u0027re just denoting something,"},{"Start":"30:41.770 ","End":"30:43.719","Text":"we\u0027re just giving it a label."},{"Start":"30:43.719 ","End":"30:47.154","Text":"In some books and certain textbooks,"},{"Start":"30:47.154 ","End":"30:49.134","Text":"they sometimes switch this around."},{"Start":"30:49.134 ","End":"30:53.499","Text":"Some books will say that the angle with the z-axis is Theta and"},{"Start":"30:53.499 ","End":"30:58.719","Text":"that the angle on the x, y plane with the x-axis is Phi."},{"Start":"30:58.719 ","End":"31:06.295","Text":"What you have to do is you have to make sure that you understand their labeling."},{"Start":"31:06.295 ","End":"31:11.136","Text":"You have to see which angle they\u0027re talking about is in relation to the z-axis,"},{"Start":"31:11.136 ","End":"31:14.650","Text":"and then that\u0027s going to be between 0-180,"},{"Start":"31:14.650 ","End":"31:19.629","Text":"and the angle for the projection on the x, y plane relative to"},{"Start":"31:19.629 ","End":"31:25.055","Text":"the x-axis is always going to be between 0-360."},{"Start":"31:25.055 ","End":"31:30.864","Text":"Remember that these are simply symbols and they are not definitions."},{"Start":"31:30.864 ","End":"31:32.529","Text":"They\u0027re interchangeable."},{"Start":"31:32.529 ","End":"31:35.574","Text":"We could also say that this is Alpha and this is Beta."},{"Start":"31:35.574 ","End":"31:37.765","Text":"It makes no difference."},{"Start":"31:37.765 ","End":"31:40.730","Text":"That\u0027s the end of this lesson."}],"ID":9603},{"Watched":false,"Name":"Converting Between Cartesian And Spherical Coordinates","Duration":"19m 50s","ChapterTopicVideoID":9289,"CourseChapterTopicPlaylistID":85322,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9289.jpeg","UploadDate":"2017-11-08T11:29:04.6000000","DurationForVideoObject":"PT19M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.770","Text":"Hello. In this lesson,"},{"Start":"00:01.770 ","End":"00:04.290","Text":"we\u0027re going to be speaking about how to convert between"},{"Start":"00:04.290 ","End":"00:09.074","Text":"Cartesian coordinate systems and spherical coordinate systems."},{"Start":"00:09.074 ","End":"00:12.975","Text":"Now, the first thing that we\u0027re going to do is we\u0027re going to do"},{"Start":"00:12.975 ","End":"00:17.650","Text":"a little sum up of what we did in the last lesson."},{"Start":"00:17.650 ","End":"00:21.560","Text":"If we\u0027re imagining that this is our x-axis,"},{"Start":"00:21.560 ","End":"00:24.450","Text":"this is our y and this is our z,"},{"Start":"00:24.450 ","End":"00:29.915","Text":"then we\u0027re imagining that this orange line represents our vector A."},{"Start":"00:29.915 ","End":"00:37.380","Text":"Let\u0027s say that our vector a is 1, 2, 3."},{"Start":"00:37.880 ","End":"00:44.360","Text":"What that means, I\u0027m reminding you is that its x value is over here, 1,"},{"Start":"00:44.360 ","End":"00:48.120","Text":"its y value is over here, 2,"},{"Start":"00:48.120 ","End":"00:53.200","Text":"and its z value is somewhere over here, 3."},{"Start":"00:53.200 ","End":"00:56.300","Text":"As we said in the previous lesson,"},{"Start":"00:56.300 ","End":"00:59.870","Text":"in order to find this point"},{"Start":"00:59.870 ","End":"01:04.110","Text":"when we\u0027re given this information that our vector A is 1, 2, 3,"},{"Start":"01:04.110 ","End":"01:11.660","Text":"so we go across 1 on the x and we draw a dotted line which is"},{"Start":"01:11.660 ","End":"01:15.470","Text":"perpendicular to the x-axis and parallel to the y"},{"Start":"01:15.470 ","End":"01:20.390","Text":"up until 2 and then we go across until this intercept."},{"Start":"01:20.390 ","End":"01:23.285","Text":"Of course, this is 90 degrees,"},{"Start":"01:23.285 ","End":"01:25.354","Text":"this is 90 degrees."},{"Start":"01:25.354 ","End":"01:31.250","Text":"Then, the angle between these two dotted lines is also 90 degrees."},{"Start":"01:31.250 ","End":"01:34.809","Text":"Then we just go straight up along"},{"Start":"01:34.809 ","End":"01:39.540","Text":"the z-axis where this angle between the z-axis and the x,"},{"Start":"01:39.540 ","End":"01:42.980","Text":"y plane is also 90 degrees and we go up"},{"Start":"01:42.980 ","End":"01:48.660","Text":"3 corresponding to a z value and that\u0027s how we find this point."},{"Start":"01:49.030 ","End":"01:54.320","Text":"Then, what we spoke about was that the equivalent to"},{"Start":"01:54.320 ","End":"01:59.540","Text":"polar coordinates or spherical coordinates which is what we use in 3 dimensions."},{"Start":"01:59.540 ","End":"02:07.080","Text":"We said that this angle over here between the vector and the z-axis is our angle Phi."},{"Start":"02:07.080 ","End":"02:11.450","Text":"Then we said that if we take the orthogonal projection of"},{"Start":"02:11.450 ","End":"02:15.720","Text":"this vector A until this point over here."},{"Start":"02:15.720 ","End":"02:22.730","Text":"That\u0027s going to be this dotted line is the orthogonal projection on the x, y plane."},{"Start":"02:22.730 ","End":"02:31.130","Text":"We said that the angle between this orthogonal projection and our x-axis is called Theta."},{"Start":"02:31.130 ","End":"02:34.580","Text":"We also said that in certain textbooks,"},{"Start":"02:34.580 ","End":"02:37.840","Text":"they switch the Theta and the Phi."},{"Start":"02:37.840 ","End":"02:41.525","Text":"Just remember that this is simply a label,"},{"Start":"02:41.525 ","End":"02:44.180","Text":"it\u0027s not a definition and you could also label"},{"Start":"02:44.180 ","End":"02:50.660","Text":"these axes with any other Greek letter or any other symbol that you wish."},{"Start":"02:50.660 ","End":"02:53.070","Text":"It\u0027s just a symbol."},{"Start":"02:53.870 ","End":"02:57.739","Text":"Also that we said that the orthogonal projection,"},{"Start":"02:57.739 ","End":"03:01.400","Text":"this green dotted line of vector A on the x,"},{"Start":"03:01.400 ","End":"03:05.420","Text":"y plane was also called A_xy,"},{"Start":"03:05.420 ","End":"03:08.550","Text":"another way of writing that."},{"Start":"03:09.220 ","End":"03:12.830","Text":"I\u0027ve made this image slightly larger so"},{"Start":"03:12.830 ","End":"03:16.460","Text":"that it will be easier for us to see what\u0027s going on."},{"Start":"03:16.460 ","End":"03:19.700","Text":"The first thing that we\u0027re going to do now is we\u0027re going"},{"Start":"03:19.700 ","End":"03:22.595","Text":"to find the size of this vector."},{"Start":"03:22.595 ","End":"03:25.380","Text":"The magnitude of the vector."},{"Start":"03:25.720 ","End":"03:31.100","Text":"The magnitude of the vector or the size of the vector is simply going"},{"Start":"03:31.100 ","End":"03:35.685","Text":"to be this entire length over here."},{"Start":"03:35.685 ","End":"03:39.875","Text":"We\u0027re just trying to find what this length is and its symbol"},{"Start":"03:39.875 ","End":"03:44.680","Text":"is the absolute value of the vector A."},{"Start":"03:44.680 ","End":"03:49.280","Text":"Now I\u0027m going to write out the equation for finding this magnitude."},{"Start":"03:49.280 ","End":"03:57.005","Text":"We have that the magnitude of our vector A is simply equal to the square root of."},{"Start":"03:57.005 ","End":"03:59.630","Text":"Now, if we remember in 2 dimensions,"},{"Start":"03:59.630 ","End":"04:05.105","Text":"we would have had the square root of the x value of"},{"Start":"04:05.105 ","End":"04:12.050","Text":"our A vector squared plus the y value of our A vector squared."},{"Start":"04:12.050 ","End":"04:15.740","Text":"But now, because we have a third component, our z,"},{"Start":"04:15.740 ","End":"04:19.075","Text":"so we\u0027re also adding in"},{"Start":"04:19.075 ","End":"04:28.150","Text":"the square value of our z component of our A vector and the square root of all of that."},{"Start":"04:28.270 ","End":"04:32.435","Text":"This is the equation for finding the magnitude"},{"Start":"04:32.435 ","End":"04:36.515","Text":"of a vector when we\u0027re dealing with 3 dimensions."},{"Start":"04:36.515 ","End":"04:41.480","Text":"This is a really important equation so write this out in"},{"Start":"04:41.480 ","End":"04:43.880","Text":"your notebooks or in your equation sheets and it\u0027s"},{"Start":"04:43.880 ","End":"04:47.390","Text":"probably the most important equation of this lesson."},{"Start":"04:47.390 ","End":"04:52.880","Text":"Now what we\u0027re going to do is we\u0027re going to see why this is correct."},{"Start":"04:52.880 ","End":"04:57.320","Text":"We know that the length of the vector which is what we\u0027re trying to find,"},{"Start":"04:57.320 ","End":"05:02.615","Text":"the magnitude of it is equal to this equation over here."},{"Start":"05:02.615 ","End":"05:06.655","Text":"What does that mean? Our Ax is this."},{"Start":"05:06.655 ","End":"05:08.580","Text":"In this example, it\u0027s 1,"},{"Start":"05:08.580 ","End":"05:12.180","Text":"our A y in this example is 2,"},{"Start":"05:12.180 ","End":"05:16.930","Text":"and A z in this example is 3."},{"Start":"05:16.970 ","End":"05:21.395","Text":"A lot of the time we can write a vector"},{"Start":"05:21.395 ","End":"05:27.225","Text":"simply as its components so it would be the x component,"},{"Start":"05:27.225 ","End":"05:32.030","Text":"its y component, and its z component."},{"Start":"05:32.180 ","End":"05:36.020","Text":"Now let\u0027s take this example."},{"Start":"05:36.020 ","End":"05:37.685","Text":"Let\u0027s rub this out, we don\u0027t need this."},{"Start":"05:37.685 ","End":"05:43.665","Text":"Let\u0027s take this example and try and find the magnitude of our vector A over here."},{"Start":"05:43.665 ","End":"05:51.800","Text":"The magnitude of a vector A is simply equal to the square root of our Ax^2 so that\u0027s 1^2"},{"Start":"05:51.800 ","End":"06:00.990","Text":"plus our Ay^2 so that\u0027s 2^2 plus Az^2 which is 3^2."},{"Start":"06:01.130 ","End":"06:06.530","Text":"This is simply going to be equal to the square root of 1^2 is 1 plus"},{"Start":"06:06.530 ","End":"06:12.840","Text":"2^2 squared is 4 plus 3^2 is 9 so that\u0027s the square root of 14."},{"Start":"06:13.190 ","End":"06:18.305","Text":"That means that the magnitude of the vector or this length"},{"Start":"06:18.305 ","End":"06:24.810","Text":"in this vector space is equal to the square root of 14."},{"Start":"06:25.100 ","End":"06:31.610","Text":"Now what we want to do is we want to see why this makes sense and why it\u0027s correct."},{"Start":"06:31.610 ","End":"06:34.310","Text":"Let\u0027s begin."},{"Start":"06:34.310 ","End":"06:38.855","Text":"Now we know that this length is equal to"},{"Start":"06:38.855 ","End":"06:47.620","Text":"1 which means that this length is also equal to 1."},{"Start":"06:47.620 ","End":"06:51.950","Text":"Because this is meant to be at right angles and this"},{"Start":"06:51.950 ","End":"06:56.510","Text":"is also at right angles which means that these two lengths are exactly the same."},{"Start":"06:56.510 ","End":"07:01.235","Text":"Similarly, what color shall I use?"},{"Start":"07:01.235 ","End":"07:02.600","Text":"I\u0027ll just use gray again."},{"Start":"07:02.600 ","End":"07:11.455","Text":"Similarly, this length is equal to 2 along the y."},{"Start":"07:11.455 ","End":"07:17.420","Text":"Again, because everything is parallel or perpendicular however you want to look at it."},{"Start":"07:17.420 ","End":"07:22.770","Text":"That means that this length as well is also 2."},{"Start":"07:23.360 ","End":"07:27.200","Text":"Now what we want to do is we want to find"},{"Start":"07:27.200 ","End":"07:31.850","Text":"the magnitude or the length of this horizontal line."},{"Start":"07:31.850 ","End":"07:35.420","Text":"Now we know that this line is called"},{"Start":"07:35.420 ","End":"07:41.400","Text":"A_xy so let\u0027s write A_xy and we want to find its magnitude."},{"Start":"07:42.380 ","End":"07:46.520","Text":"How do we find this magnitude of this horizontal line?"},{"Start":"07:46.520 ","End":"07:50.090","Text":"Now we know that this is a right-angle triangle and we know that this is"},{"Start":"07:50.090 ","End":"07:56.260","Text":"a right-angle triangle so all we have to do is our usual Pythagoras."},{"Start":"07:56.260 ","End":"08:00.665","Text":"As we know our Pythagoras is going to be the square root"},{"Start":"08:00.665 ","End":"08:05.255","Text":"of this length squared plus this length squared,"},{"Start":"08:05.255 ","End":"08:09.920","Text":"so the square root of 2^2 plus 1^2."},{"Start":"08:09.920 ","End":"08:15.300","Text":"Let\u0027s write it. 1^2 plus 2^2."},{"Start":"08:15.300 ","End":"08:20.430","Text":"Normal Pythagoras to find the length of this line of our A_xy."},{"Start":"08:20.750 ","End":"08:27.545","Text":"Now what we want to do is we want to find the length of our actual vector."},{"Start":"08:27.545 ","End":"08:32.970","Text":"I\u0027m just going to rub out a few things to make this a bit more clear."},{"Start":"08:33.420 ","End":"08:40.010","Text":"Now, we want to bring down that line that we had before."},{"Start":"08:40.200 ","End":"08:43.960","Text":"We have it going from this point all the way"},{"Start":"08:43.960 ","End":"08:49.420","Text":"down until our intercept over here, and of course,"},{"Start":"08:49.420 ","End":"08:57.715","Text":"we know that this line coming down from this point is parallel to the z-axis,"},{"Start":"08:57.715 ","End":"09:02.245","Text":"which means that it\u0027s perpendicular to our xy plane."},{"Start":"09:02.245 ","End":"09:05.965","Text":"There is a right angle between our xy plane,"},{"Start":"09:05.965 ","End":"09:12.280","Text":"and this dotted line coming down from my point over here at the edge."},{"Start":"09:12.280 ","End":"09:18.280","Text":"That means that now in order to find the magnitude of our vector,"},{"Start":"09:18.280 ","End":"09:22.420","Text":"we\u0027re trying to find this horizontal,"},{"Start":"09:22.420 ","End":"09:25.690","Text":"and it\u0027s connected to this triangle,"},{"Start":"09:25.690 ","End":"09:28.150","Text":"which is standing in the middle."},{"Start":"09:28.150 ","End":"09:29.950","Text":"I\u0027ll highlight it."},{"Start":"09:29.950 ","End":"09:32.290","Text":"We have this triangle,"},{"Start":"09:32.290 ","End":"09:37.015","Text":"where over here we have a right angle."},{"Start":"09:37.015 ","End":"09:40.840","Text":"It\u0027s just like the right angle triangle that we had before on"},{"Start":"09:40.840 ","End":"09:45.830","Text":"our xy plane between the y-axis and our A_xy."},{"Start":"09:45.830 ","End":"09:50.955","Text":"But now, we\u0027re looking at it in 3-dimensions."},{"Start":"09:50.955 ","End":"09:54.790","Text":"But it\u0027s again a right angle triangle."},{"Start":"09:54.810 ","End":"09:58.750","Text":"Just like before, in order to find this length,"},{"Start":"09:58.750 ","End":"10:02.890","Text":"I\u0027m just going to use my simple Pythagoras."},{"Start":"10:02.890 ","End":"10:09.745","Text":"As we know that if this length over here is 3,"},{"Start":"10:09.745 ","End":"10:15.415","Text":"then this length over here is also 3."},{"Start":"10:15.415 ","End":"10:17.830","Text":"It\u0027s just our z value."},{"Start":"10:17.830 ","End":"10:20.260","Text":"Now let\u0027s use Pythagoras."},{"Start":"10:20.260 ","End":"10:26.965","Text":"We\u0027re trying to find the magnitude of our A vector which has this orange line over here."},{"Start":"10:26.965 ","End":"10:30.205","Text":"That\u0027s going to be equal to,"},{"Start":"10:30.205 ","End":"10:32.860","Text":"let\u0027s just write it as a squared and then"},{"Start":"10:32.860 ","End":"10:35.575","Text":"we don\u0027t have to write the square root over here."},{"Start":"10:35.575 ","End":"10:41.335","Text":"It\u0027s going to be equal to this line over here,"},{"Start":"10:41.335 ","End":"10:44.140","Text":"this side over here, squared."},{"Start":"10:44.140 ","End":"10:51.640","Text":"That\u0027s our A_xy^2 plus this line over here squared,"},{"Start":"10:51.640 ","End":"10:55.735","Text":"which is our 3^2, or our A_z^2."},{"Start":"10:55.735 ","End":"11:00.310","Text":"Let\u0027s write that as A_z^2,"},{"Start":"11:00.310 ","End":"11:04.045","Text":"and then we can put in our numbers."},{"Start":"11:04.045 ","End":"11:12.865","Text":"Our A_xy^2 is going to be equal to 1^2 plus 2^2,"},{"Start":"11:12.865 ","End":"11:14.950","Text":"the square root squared."},{"Start":"11:14.950 ","End":"11:21.460","Text":"The square root cancels out, plus our A_z^2."},{"Start":"11:21.460 ","End":"11:25.430","Text":"Our A_z is 3, so plus 3^2."},{"Start":"11:26.190 ","End":"11:31.960","Text":"Our A_xy was equal to the square root of 1^2 plus 2^2."},{"Start":"11:31.960 ","End":"11:39.190","Text":"That means that A_xy^2 is the square root of this expression squared,"},{"Start":"11:39.190 ","End":"11:42.040","Text":"which means that the square root is crossed out."},{"Start":"11:42.040 ","End":"11:47.260","Text":"Then plus our A_z^2, which is 3^2."},{"Start":"11:47.260 ","End":"11:49.060","Text":"Because we have our square here,"},{"Start":"11:49.060 ","End":"11:53.410","Text":"if we simply want to find our magnitude of our A,"},{"Start":"11:53.410 ","End":"11:55.390","Text":"we square root both sides."},{"Start":"11:55.390 ","End":"11:58.435","Text":"We\u0027ll have the square root of this,"},{"Start":"11:58.435 ","End":"12:01.960","Text":"which is 1^2 plus 2^2,"},{"Start":"12:01.960 ","End":"12:06.160","Text":"plus 3^2, and the square root of all of that,"},{"Start":"12:06.160 ","End":"12:08.035","Text":"and lo and behold,"},{"Start":"12:08.035 ","End":"12:10.490","Text":"look what we got."},{"Start":"12:10.620 ","End":"12:15.265","Text":"I hope now you understand how we got to this equation,"},{"Start":"12:15.265 ","End":"12:18.295","Text":"and then we can now see that it really is correct."},{"Start":"12:18.295 ","End":"12:22.765","Text":"All we did was we just use Pythagoras twice,"},{"Start":"12:22.765 ","End":"12:24.775","Text":"which you can do as well."},{"Start":"12:24.775 ","End":"12:25.900","Text":"But it\u0027s slightly longer."},{"Start":"12:25.900 ","End":"12:32.510","Text":"But just to remember this equation over here and you\u0027ll get to the answer immediately."},{"Start":"12:33.480 ","End":"12:38.485","Text":"Now we saw how to work out the magnitude of the vector,"},{"Start":"12:38.485 ","End":"12:40.360","Text":"and now what we\u0027re going to do is,"},{"Start":"12:40.360 ","End":"12:43.720","Text":"we\u0027re going to see how to find the direction."},{"Start":"12:43.720 ","End":"12:45.520","Text":"As we said last lesson,"},{"Start":"12:45.520 ","End":"12:50.485","Text":"the direction is defined by our Phi angle and our Theta angle."},{"Start":"12:50.485 ","End":"12:54.980","Text":"First we\u0027re going to discuss how to find our angle Phi."},{"Start":"12:56.190 ","End":"13:00.385","Text":"In order to find my angle Phi, as we know,"},{"Start":"13:00.385 ","End":"13:04.930","Text":"it\u0027s this angle between my vector A and the z-axis."},{"Start":"13:04.930 ","End":"13:07.435","Text":"The first thing that I\u0027m going to do is,"},{"Start":"13:07.435 ","End":"13:11.510","Text":"I\u0027m going to work with this triangle over here."},{"Start":"13:12.120 ","End":"13:19.990","Text":"I\u0027m going to draw a perpendicular line between this z-axis,"},{"Start":"13:19.990 ","End":"13:24.970","Text":"and this line over here"},{"Start":"13:24.970 ","End":"13:30.775","Text":"that connects the top point of my vector to its z value."},{"Start":"13:30.775 ","End":"13:36.575","Text":"Now we can see that I have a right-angled triangle."},{"Start":"13:36.575 ","End":"13:38.640","Text":"Just to be clear,"},{"Start":"13:38.640 ","End":"13:46.675","Text":"we\u0027re working with this triangle over here. Let\u0027s see."},{"Start":"13:46.675 ","End":"13:52.330","Text":"Now, we\u0027re just going to use our simple trigonometric identities."},{"Start":"13:52.330 ","End":"13:55.225","Text":"If we remember SOHCAHTOA,"},{"Start":"13:55.225 ","End":"13:57.460","Text":"we can use sine or cosine,"},{"Start":"13:57.460 ","End":"14:00.460","Text":"whichever one, I usually use cosine,"},{"Start":"14:00.460 ","End":"14:03.625","Text":"I just prefer it, but you can also use sine."},{"Start":"14:03.625 ","End":"14:05.845","Text":"Remembering from SOHCAHTOA,"},{"Start":"14:05.845 ","End":"14:08.185","Text":"if we\u0027re going to use cosine,"},{"Start":"14:08.185 ","End":"14:11.110","Text":"it\u0027s cosine of our angle."},{"Start":"14:11.110 ","End":"14:13.900","Text":"Here, it\u0027s specifically Phi,"},{"Start":"14:13.900 ","End":"14:18.940","Text":"is simply equal to our adjacent side."},{"Start":"14:18.940 ","End":"14:22.990","Text":"This is our adjacent side to our angle."},{"Start":"14:22.990 ","End":"14:26.740","Text":"Here, let\u0027s write a more general equation."},{"Start":"14:26.740 ","End":"14:31.330","Text":"It\u0027s our z components of our A vector,"},{"Start":"14:31.330 ","End":"14:34.135","Text":"3 as A_z,"},{"Start":"14:34.135 ","End":"14:38.305","Text":"divided by the hypotenuse,"},{"Start":"14:38.305 ","End":"14:43.285","Text":"the cosine of an angle as the adjacent divided by the hypotenuse."},{"Start":"14:43.285 ","End":"14:51.470","Text":"Over here, specifically, the hypotenuse is our magnitude of the vector."},{"Start":"14:52.170 ","End":"14:56.200","Text":"We have the magnitude of our vector."},{"Start":"14:56.200 ","End":"15:02.005","Text":"What we can write, is we can also just substitute in the longer equation."},{"Start":"15:02.005 ","End":"15:08.905","Text":"We\u0027ll have the cosine of our angle Phi is equal to A_z,"},{"Start":"15:08.905 ","End":"15:17.065","Text":"divided by the square root of A_x^2,"},{"Start":"15:17.065 ","End":"15:20.360","Text":"plus A_y ^2,"},{"Start":"15:20.360 ","End":"15:27.590","Text":"plus A_z^2, square root of all of that."},{"Start":"15:29.220 ","End":"15:33.610","Text":"This is the equation for finding your angle Phi,"},{"Start":"15:33.610 ","End":"15:38.005","Text":"and then obviously you just arccos both sides,"},{"Start":"15:38.005 ","End":"15:41.075","Text":"and you\u0027ll get your value for your angle."},{"Start":"15:41.075 ","End":"15:43.932","Text":"Now the last thing that we have to do is we"},{"Start":"15:43.932 ","End":"15:47.165","Text":"have to find an equation for our angle Theta,"},{"Start":"15:47.165 ","End":"15:49.366","Text":"which is this angle over here."},{"Start":"15:49.366 ","End":"15:51.582","Text":"Now in order to find our Theta,"},{"Start":"15:51.582 ","End":"15:56.963","Text":"we\u0027re going to do the exact same thing that we did in order to find our Phi."},{"Start":"15:56.963 ","End":"16:00.148","Text":"We\u0027re just going to draw this dotted line,"},{"Start":"16:00.148 ","End":"16:03.084","Text":"which is perpendicular to the x-axis,"},{"Start":"16:03.084 ","End":"16:08.190","Text":"parallel to the y-axis from our x value until this point over here."},{"Start":"16:08.190 ","End":"16:09.317","Text":"Now as we said,"},{"Start":"16:09.317 ","End":"16:10.638","Text":"this length is 1,"},{"Start":"16:10.638 ","End":"16:13.016","Text":"and just like this length is 2,"},{"Start":"16:13.016 ","End":"16:14.744","Text":"this length is also 2,"},{"Start":"16:14.744 ","End":"16:19.515","Text":"and of course, we know that these 2 lines are perpendicular,"},{"Start":"16:19.515 ","End":"16:21.489","Text":"so we have a right angle."},{"Start":"16:21.489 ","End":"16:25.783","Text":"Again, we\u0027re dealing with a right-angled triangle,"},{"Start":"16:25.783 ","End":"16:29.841","Text":"and we\u0027re dealing with this triangle over here."},{"Start":"16:29.841 ","End":"16:33.469","Text":"Now what we can do is we can also use cosine,"},{"Start":"16:33.469 ","End":"16:35.264","Text":"we can also use sine."},{"Start":"16:35.264 ","End":"16:38.474","Text":"However, if we\u0027re going to use that,"},{"Start":"16:38.474 ","End":"16:41.308","Text":"they deal with the hypotenuse,"},{"Start":"16:41.308 ","End":"16:45.100","Text":"and then we have to use our value for A_xy,"},{"Start":"16:45.100 ","End":"16:47.962","Text":"which means that we have to do,"},{"Start":"16:47.962 ","End":"16:52.331","Text":"even though it\u0027s very simple calculations,"},{"Start":"16:52.331 ","End":"16:55.580","Text":"slightly longer calculations,"},{"Start":"16:55.580 ","End":"16:58.420","Text":"and we\u0027re a little bit lazy."},{"Start":"16:58.420 ","End":"17:01.387","Text":"We want to get the shortcut."},{"Start":"17:01.387 ","End":"17:03.930","Text":"What we\u0027re going to do,"},{"Start":"17:03.930 ","End":"17:06.917","Text":"is we\u0027re going to use tan."},{"Start":"17:06.917 ","End":"17:08.284","Text":"As we know,"},{"Start":"17:08.284 ","End":"17:10.386","Text":"tan of an angle."},{"Start":"17:10.386 ","End":"17:18.500","Text":"Here specifically we\u0027re dealing with angle Theta,"},{"Start":"17:18.500 ","End":"17:23.050","Text":"is equal to opposite over adjacent."},{"Start":"17:23.050 ","End":"17:28.545","Text":"Here our opposite is this side over here."},{"Start":"17:28.545 ","End":"17:32.225","Text":"It\u0027s the side of the right-angle triangle,"},{"Start":"17:32.225 ","End":"17:35.355","Text":"which we know this is right angled because of here."},{"Start":"17:35.355 ","End":"17:39.430","Text":"Our side opposite to the angle is this,"},{"Start":"17:39.430 ","End":"17:42.425","Text":"and here we see specifically in our example,"},{"Start":"17:42.425 ","End":"17:47.465","Text":"it\u0027s 2, but we know that this is our A_y component."},{"Start":"17:47.465 ","End":"17:49.115","Text":"Let\u0027s write over here."},{"Start":"17:49.115 ","End":"17:55.505","Text":"It\u0027s the opposite, which here specifically is A_y divided by the adjacent."},{"Start":"17:55.505 ","End":"17:59.585","Text":"The adjacent side to our angle is this value over here,"},{"Start":"17:59.585 ","End":"18:04.950","Text":"which is our A_x component of our A vector,"},{"Start":"18:04.950 ","End":"18:07.805","Text":"and specifically, in our example, it\u0027s 1."},{"Start":"18:07.805 ","End":"18:12.100","Text":"But for a more general equation, it\u0027s A_x."},{"Start":"18:12.100 ","End":"18:14.260","Text":"We\u0027re simply going to have A_y,"},{"Start":"18:14.260 ","End":"18:18.070","Text":"divided by A_x,"},{"Start":"18:18.070 ","End":"18:22.250","Text":"and you\u0027ll notice that we\u0027re not even taking into account"},{"Start":"18:22.250 ","End":"18:26.330","Text":"our A_z components, our third dimension."},{"Start":"18:26.330 ","End":"18:29.855","Text":"We\u0027re simply using our 2-dimensions because we\u0027re dealing"},{"Start":"18:29.855 ","End":"18:33.935","Text":"with this orthogonal projection on our xy plane."},{"Start":"18:33.935 ","End":"18:37.280","Text":"We\u0027re dealing with something in 2D."},{"Start":"18:37.280 ","End":"18:44.925","Text":"Here\u0027s specifically our tan Theta is equal to our A_y,"},{"Start":"18:44.925 ","End":"18:46.815","Text":"which is 2,"},{"Start":"18:46.815 ","End":"18:48.660","Text":"divided by our A_x,"},{"Start":"18:48.660 ","End":"18:50.115","Text":"which is 1,"},{"Start":"18:50.115 ","End":"18:56.140","Text":"and then of course we arctan both sides in order to get our Theta."},{"Start":"18:56.670 ","End":"19:03.400","Text":"Here specifically we\u0027ll get in this example that our Theta is equal"},{"Start":"19:03.400 ","End":"19:09.820","Text":"to tan minus 1(2),"},{"Start":"19:09.820 ","End":"19:13.780","Text":"and then lets just sub in our values for our Phi."},{"Start":"19:13.780 ","End":"19:20.475","Text":"We\u0027ll get that our Phi is going to be equal to our cos to the minus 1(A_z),"},{"Start":"19:20.475 ","End":"19:27.200","Text":"which is 3 divided by the magnitude of our vector A,"},{"Start":"19:27.200 ","End":"19:28.565","Text":"which, as we saw before,"},{"Start":"19:28.565 ","End":"19:30.515","Text":"you can rewind the video,"},{"Start":"19:30.515 ","End":"19:33.765","Text":"was a square root of 14,"},{"Start":"19:33.765 ","End":"19:37.600","Text":"and this is what we will get."},{"Start":"19:37.600 ","End":"19:40.645","Text":"What\u0027s in the red boxes,"},{"Start":"19:40.645 ","End":"19:45.810","Text":"this is what you need to be writing down in your equation sheets."},{"Start":"19:47.100 ","End":"19:50.480","Text":"That\u0027s the end of the lesson."}],"ID":9601},{"Watched":false,"Name":"Converting Between Spherical And Cartesian Coordinates","Duration":"16m 48s","ChapterTopicVideoID":9290,"CourseChapterTopicPlaylistID":85322,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9290.jpeg","UploadDate":"2017-11-08T11:33:54.3600000","DurationForVideoObject":"PT16M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"Hello. In the last lesson,"},{"Start":"00:02.415 ","End":"00:05.520","Text":"we learned how to convert between Cartesian"},{"Start":"00:05.520 ","End":"00:09.405","Text":"coordinate system into a spherical coordinate system."},{"Start":"00:09.405 ","End":"00:12.555","Text":"In this lesson, we\u0027re going to be learning the opposite."},{"Start":"00:12.555 ","End":"00:15.600","Text":"We have a spherical coordinate system and we\u0027re going to be"},{"Start":"00:15.600 ","End":"00:20.320","Text":"converting these values back into Cartesian coordinates."},{"Start":"00:21.170 ","End":"00:28.110","Text":"Here what we\u0027ll have is our A vector which has magnitude and"},{"Start":"00:28.110 ","End":"00:36.930","Text":"direction and we\u0027re going to convert that into the Cartesian components for this vector."},{"Start":"00:37.270 ","End":"00:40.108","Text":"In our example over here,"},{"Start":"00:40.108 ","End":"00:44.920","Text":"our A vector is going to be given case,"},{"Start":"00:44.920 ","End":"00:47.990","Text":"so what do we have given for A vector?"},{"Start":"00:47.990 ","End":"00:51.140","Text":"Because we\u0027re working right now in spherical coordinates."},{"Start":"00:51.140 ","End":"00:56.925","Text":"We\u0027re going to have the magnitude of our A vector."},{"Start":"00:56.925 ","End":"01:01.280","Text":"What is that? Just a reminder from the previous lesson."},{"Start":"01:01.280 ","End":"01:07.260","Text":"Our magnitude is going to be this length over here."},{"Start":"01:07.850 ","End":"01:14.655","Text":"Then we\u0027re going to have our angle Theta,"},{"Start":"01:14.655 ","End":"01:18.675","Text":"which is going to be like this."},{"Start":"01:18.675 ","End":"01:23.720","Text":"If we say that this point is the tip of our vector,"},{"Start":"01:23.720 ","End":"01:28.115","Text":"we take a straight line down such that it\u0027s"},{"Start":"01:28.115 ","End":"01:35.625","Text":"perpendicular to the x-y plane and parallel to the z-axis."},{"Start":"01:35.625 ","End":"01:42.305","Text":"Then the line that joins this point over here to the origin"},{"Start":"01:42.305 ","End":"01:50.140","Text":"is called our orthogonal projection of our vector A on the x-y plane."},{"Start":"01:50.140 ","End":"01:57.965","Text":"Or its other name is also A_xy and we know that"},{"Start":"01:57.965 ","End":"02:06.150","Text":"our angle between our A_xy and the x-axis is called our Theta angle."},{"Start":"02:06.150 ","End":"02:11.825","Text":"Then the next value that we have is our Phi and as we know,"},{"Start":"02:11.825 ","End":"02:18.480","Text":"Phi is the angle between the z-axis and our vector A."},{"Start":"02:19.070 ","End":"02:23.115","Text":"Now let\u0027s give a simple example."},{"Start":"02:23.115 ","End":"02:32.295","Text":"Let\u0027s say that the magnitude of our vector A is equal to 6."},{"Start":"02:32.295 ","End":"02:38.760","Text":"Let\u0027s say that our Theta angle is equal to 60 degrees,"},{"Start":"02:38.760 ","End":"02:44.610","Text":"and let\u0027s say that our Phi angle is equal to 30 degrees."},{"Start":"02:44.610 ","End":"02:46.890","Text":"This is just an example."},{"Start":"02:46.890 ","End":"02:50.820","Text":"Now what do we want to do? Our question."},{"Start":"02:50.820 ","End":"02:55.650","Text":"Is we want to find what our A_x component is,"},{"Start":"02:55.650 ","End":"02:57.930","Text":"what our A_y component is,"},{"Start":"02:57.930 ","End":"03:01.215","Text":"and what our A_z component is."},{"Start":"03:01.215 ","End":"03:05.270","Text":"This is our question because we\u0027re trying to convert from"},{"Start":"03:05.270 ","End":"03:10.590","Text":"spherical coordinates into Cartesian coordinates."},{"Start":"03:11.030 ","End":"03:16.115","Text":"Just as a reminder our A_x is going to be,"},{"Start":"03:16.115 ","End":"03:21.140","Text":"if we draw a line that\u0027s perpendicular to the x-axis,"},{"Start":"03:21.140 ","End":"03:24.470","Text":"our A_x is going to be somewhere around here,"},{"Start":"03:24.470 ","End":"03:30.245","Text":"our A_y, if we draw a perpendicular line from this point to the y."},{"Start":"03:30.245 ","End":"03:35.210","Text":"Our A_y is going to be somewhere along here and a perpendicular line to"},{"Start":"03:35.210 ","End":"03:42.055","Text":"our z-axis and our A_z is going to be somewhere around here."},{"Start":"03:42.055 ","End":"03:46.290","Text":"Let\u0027s begin to try and find this."},{"Start":"03:46.290 ","End":"03:48.935","Text":"We\u0027re going to have our answer."},{"Start":"03:48.935 ","End":"03:52.520","Text":"Now the first thing that we\u0027re going to try and find is our A_z,"},{"Start":"03:52.520 ","End":"03:57.235","Text":"because that going to be our easiest thing to find at this stage."},{"Start":"03:57.235 ","End":"04:00.165","Text":"Let\u0027s find our A_z."},{"Start":"04:00.165 ","End":"04:01.815","Text":"Now, as we know,"},{"Start":"04:01.815 ","End":"04:05.025","Text":"our A_z is this value over here,"},{"Start":"04:05.025 ","End":"04:09.515","Text":"this value on the scale and also it\u0027s the same thing."},{"Start":"04:09.515 ","End":"04:12.460","Text":"This length over here is also A_z."},{"Start":"04:12.460 ","End":"04:16.400","Text":"What we can do is we can draw a perpendicular line to"},{"Start":"04:16.400 ","End":"04:21.280","Text":"our z-axis all the way until the tip of eye point."},{"Start":"04:21.280 ","End":"04:25.040","Text":"We know then, because we drew a perpendicular line,"},{"Start":"04:25.040 ","End":"04:26.855","Text":"that this is 90 degrees,"},{"Start":"04:26.855 ","End":"04:30.605","Text":"which means that we have a right angle triangle."},{"Start":"04:30.605 ","End":"04:36.830","Text":"Let\u0027s just highlight that so we\u0027re talking about this triangle over here."},{"Start":"04:36.830 ","End":"04:39.350","Text":"Now, in the previous lesson,"},{"Start":"04:39.350 ","End":"04:45.830","Text":"we learned that the equation to find out what our Phi is goes like this."},{"Start":"04:45.830 ","End":"04:56.770","Text":"Cosine of Phi is equal to A_z divided by the magnitude of our vector A."},{"Start":"04:56.860 ","End":"05:00.410","Text":"This is what we learned in the previous lesson."},{"Start":"05:00.410 ","End":"05:07.345","Text":"What we\u0027re trying to find is our A_z which means that we want to isolate out this."},{"Start":"05:07.345 ","End":"05:11.810","Text":"We\u0027ll get in the end that our A_z is equal to"},{"Start":"05:11.810 ","End":"05:19.655","Text":"the magnitude of our vector multiplied by cosine of the angle Phi."},{"Start":"05:19.655 ","End":"05:21.830","Text":"Then in our example,"},{"Start":"05:21.830 ","End":"05:28.475","Text":"the magnitude of the vector is 6 and cosine of Phi is cosine of 30,"},{"Start":"05:28.475 ","End":"05:36.670","Text":"which is equal to 5.2 to 1 decimal place."},{"Start":"05:37.310 ","End":"05:45.160","Text":"This is the general equation that you need in order to find your z component for"},{"Start":"05:45.160 ","End":"05:53.570","Text":"your A vector when you\u0027re converting from spherical coordinates into Cartesian."},{"Start":"05:53.960 ","End":"06:04.430","Text":"We can write over here that our A_z value is equal to 5.2 on the z-axis scale."},{"Start":"06:04.910 ","End":"06:11.875","Text":"Now we found A_z, and now we want to find out what our value for A-x is equal to."},{"Start":"06:11.875 ","End":"06:14.035","Text":"Now before we can do that,"},{"Start":"06:14.035 ","End":"06:19.555","Text":"the first thing that we have to do is we have to find out what this length is over here."},{"Start":"06:19.555 ","End":"06:26.675","Text":"What the length of this orthogonal projection of vector A on the x-y plane is equal to,"},{"Start":"06:26.675 ","End":"06:33.240","Text":"or what is the magnitude of our A_xy."},{"Start":"06:33.950 ","End":"06:36.935","Text":"This is super easy to do."},{"Start":"06:36.935 ","End":"06:42.760","Text":"As we know, this is a 90-degree angle because that\u0027s how we drew it."},{"Start":"06:42.760 ","End":"06:46.775","Text":"We also know that this is a 90-degree angle."},{"Start":"06:46.775 ","End":"06:51.980","Text":"It\u0027s parallel to the z-axis and it\u0027s perpendicular our x, y plane."},{"Start":"06:51.980 ","End":"06:55.850","Text":"We also know that this angle over here between,"},{"Start":"06:55.850 ","End":"06:57.110","Text":"again the z-axis,"},{"Start":"06:57.110 ","End":"07:01.500","Text":"and x-y plane is also at 90 degrees."},{"Start":"07:02.330 ","End":"07:07.175","Text":"Because these 2 lines the z-axis and this are also parallel."},{"Start":"07:07.175 ","End":"07:08.615","Text":"Then as we know,"},{"Start":"07:08.615 ","End":"07:14.980","Text":"this is also going to be at 90 degrees because we have some rectangle over here."},{"Start":"07:14.980 ","End":"07:19.489","Text":"In that case, if everything is equal to 90 degrees,"},{"Start":"07:19.489 ","End":"07:22.340","Text":"then we know that this length A_xy,"},{"Start":"07:22.340 ","End":"07:28.410","Text":"is simply equal to this length over here."},{"Start":"07:30.230 ","End":"07:37.485","Text":"Let\u0027s see. This is also equal to A_xy."},{"Start":"07:37.485 ","End":"07:40.390","Text":"These 2 lengths are equal."},{"Start":"07:40.820 ","End":"07:43.815","Text":"How are we going to find this length?"},{"Start":"07:43.815 ","End":"07:51.620","Text":"We know that this side of the triangle is the opposite side to our angle Phi."},{"Start":"07:51.620 ","End":"07:56.590","Text":"All we\u0027re going to use is we\u0027re going to use our sine function."},{"Start":"07:56.590 ","End":"08:02.540","Text":"All we have to do in order to find out what our A_xy is,"},{"Start":"08:03.180 ","End":"08:09.070","Text":"it\u0027s going to be equal to the magnitude of our vector A"},{"Start":"08:09.070 ","End":"08:15.590","Text":"multiplied by sine of our angle Phi."},{"Start":"08:16.460 ","End":"08:19.880","Text":"This is the equation that we need in order to find"},{"Start":"08:19.880 ","End":"08:23.255","Text":"our A_xy which is this length over here,"},{"Start":"08:23.255 ","End":"08:25.650","Text":"and this length over here."},{"Start":"08:25.650 ","End":"08:29.825","Text":"Now let\u0027s plug in our numbers from our specific example."},{"Start":"08:29.825 ","End":"08:37.415","Text":"We have that the magnitude of A is equal to 6 multiplied by sine of our angle Phi,"},{"Start":"08:37.415 ","End":"08:40.430","Text":"which is 30 degrees,"},{"Start":"08:40.430 ","End":"08:44.010","Text":"and that is equal to 3."},{"Start":"08:45.530 ","End":"08:49.155","Text":"We know that this is equal to 3,"},{"Start":"08:49.155 ","End":"08:56.608","Text":"which means that also this length over here is also equal to 3."},{"Start":"08:56.608 ","End":"09:03.505","Text":"Now we can move on to the next step,"},{"Start":"09:03.505 ","End":"09:10.160","Text":"which is to find what our value for A_x is equal to."},{"Start":"09:10.320 ","End":"09:14.740","Text":"Our value for A_x is this length over here,"},{"Start":"09:14.740 ","End":"09:19.195","Text":"which is also equal to the value at this point over here."},{"Start":"09:19.195 ","End":"09:21.310","Text":"How am I going to find this?"},{"Start":"09:21.310 ","End":"09:26.020","Text":"Once again, I\u0027m going to draw a perpendicular line that"},{"Start":"09:26.020 ","End":"09:31.150","Text":"goes from my value A_x until this point over here."},{"Start":"09:31.150 ","End":"09:35.170","Text":"This intercept on the x-y plane."},{"Start":"09:35.170 ","End":"09:39.595","Text":"I\u0027m drawing my perpendicular line,"},{"Start":"09:39.595 ","End":"09:41.710","Text":"imagine that this is straight,"},{"Start":"09:41.710 ","End":"09:46.330","Text":"and of course we have a 90-degree angle here."},{"Start":"09:46.330 ","End":"09:49.960","Text":"Now what I\u0027m going to do is I\u0027m going to use"},{"Start":"09:49.960 ","End":"09:55.120","Text":"the exact same geometry that I used in order to find my A_z,"},{"Start":"09:55.120 ","End":"09:58.915","Text":"except now I\u0027m working on my x-y plane,"},{"Start":"09:58.915 ","End":"10:01.990","Text":"and I\u0027m working instead of with my angle Phi,"},{"Start":"10:01.990 ","End":"10:04.930","Text":"I\u0027m working with my angle Theta."},{"Start":"10:04.930 ","End":"10:09.910","Text":"Let\u0027s just highlight which triangle we\u0027re looking at."},{"Start":"10:09.910 ","End":"10:12.895","Text":"It\u0027s this triangle right here,"},{"Start":"10:12.895 ","End":"10:17.980","Text":"and we have a right angle over here."},{"Start":"10:18.080 ","End":"10:23.895","Text":"We can use the exact same equation that we used up here."},{"Start":"10:23.895 ","End":"10:28.535","Text":"We\u0027ll have that our A_x divided by"},{"Start":"10:28.535 ","End":"10:37.090","Text":"our A_xy is going to be equal to cosine of Theta this time."},{"Start":"10:37.090 ","End":"10:39.775","Text":"Because we have our A_x,"},{"Start":"10:39.775 ","End":"10:40.990","Text":"which is our side,"},{"Start":"10:40.990 ","End":"10:44.680","Text":"which is adjacent to our angle Theta,"},{"Start":"10:44.680 ","End":"10:48.895","Text":"and then our A_xy is our hypotenuse."},{"Start":"10:48.895 ","End":"10:56.570","Text":"We have adjacent divided by hypotenuse is equal to cosine of the angle."},{"Start":"10:58.590 ","End":"11:02.080","Text":"Now all we have to do is we have to isolate"},{"Start":"11:02.080 ","End":"11:05.200","Text":"out our A_x because this is what we want to find out."},{"Start":"11:05.200 ","End":"11:11.145","Text":"That means that our Ax_ is simply equal to our value for A_xy,"},{"Start":"11:11.145 ","End":"11:18.700","Text":"that\u0027s this length over here multiplied by cosine of Theta."},{"Start":"11:19.160 ","End":"11:25.230","Text":"Now, in order to write this so that you can just have 1 expression,"},{"Start":"11:25.230 ","End":"11:29.165","Text":"and instead of also remembering what your A_xy expression is equal to,"},{"Start":"11:29.165 ","End":"11:31.810","Text":"we can substitute in our value."},{"Start":"11:31.810 ","End":"11:40.570","Text":"Our A_xy is the magnitude of our vector A multiplied by sine of Phi,"},{"Start":"11:40.570 ","End":"11:45.894","Text":"multiplied by cosine of Theta."},{"Start":"11:45.894 ","End":"11:49.760","Text":"This will be our value A_x."},{"Start":"11:50.640 ","End":"11:56.500","Text":"Now let\u0027s plug in our values from our specific example."},{"Start":"11:56.500 ","End":"12:01.180","Text":"We have that the magnitude of our A is given to us as"},{"Start":"12:01.180 ","End":"12:06.235","Text":"being equal to 6 multiplied by sine of Phi,"},{"Start":"12:06.235 ","End":"12:10.720","Text":"that\u0027s multiplied by sine of 30 degrees,"},{"Start":"12:10.720 ","End":"12:14.515","Text":"multiplied by cosine of Theta,"},{"Start":"12:14.515 ","End":"12:19.300","Text":"which is equal to 60 degrees,"},{"Start":"12:19.300 ","End":"12:26.900","Text":"and then that is simply equal to 3 divided by 2."},{"Start":"12:27.600 ","End":"12:37.315","Text":"Now finally, our fourth step in this conversion is to find our y component, A_y."},{"Start":"12:37.315 ","End":"12:44.905","Text":"Notice that our A_y is this length over here,"},{"Start":"12:44.905 ","End":"12:47.635","Text":"up until here, and of course,"},{"Start":"12:47.635 ","End":"12:50.920","Text":"it\u0027s the same value as this on the scale."},{"Start":"12:50.920 ","End":"12:53.890","Text":"Because we\u0027re dealing with 90-degree angles,"},{"Start":"12:53.890 ","End":"12:55.570","Text":"here we have a 90-degree angle,"},{"Start":"12:55.570 ","End":"12:57.505","Text":"here we have a 90-degree angle,"},{"Start":"12:57.505 ","End":"13:00.565","Text":"and here we have a 90-degree angle, and here."},{"Start":"13:00.565 ","End":"13:03.400","Text":"Everything is perpendicular to one another."},{"Start":"13:03.400 ","End":"13:08.065","Text":"We know that this length A_y is also"},{"Start":"13:08.065 ","End":"13:14.090","Text":"equal to this dotted line that we can see going along here."},{"Start":"13:14.370 ","End":"13:21.490","Text":"Our A_y is also equal to this length over here."},{"Start":"13:21.490 ","End":"13:24.130","Text":"Notice this isn\u0027t equal to the 3,"},{"Start":"13:24.130 ","End":"13:30.025","Text":"the 3 is this green horizontal line which is our A_xy."},{"Start":"13:30.025 ","End":"13:32.290","Text":"That\u0027s what the 3 is relating to,"},{"Start":"13:32.290 ","End":"13:36.850","Text":"but our A_y is this length over here."},{"Start":"13:36.850 ","End":"13:41.330","Text":"How are we going to find out what this value is equal to?"},{"Start":"13:41.520 ","End":"13:47.830","Text":"Now we can see that our A_y is the side of"},{"Start":"13:47.830 ","End":"13:54.860","Text":"our right angle triangle which is opposite to our angle Theta."},{"Start":"13:55.260 ","End":"13:59.500","Text":"We\u0027re going to use some trigonometry again,"},{"Start":"13:59.500 ","End":"14:04.655","Text":"and we\u0027re going to use the idea of opposite over our hypotenuse."},{"Start":"14:04.655 ","End":"14:06.795","Text":"Let\u0027s write that out."},{"Start":"14:06.795 ","End":"14:09.870","Text":"Our A_y is our opposite side because it\u0027s opposite"},{"Start":"14:09.870 ","End":"14:12.960","Text":"the angle Theta divided by our hypotenuse,"},{"Start":"14:12.960 ","End":"14:14.520","Text":"which is this green line,"},{"Start":"14:14.520 ","End":"14:16.610","Text":"which is our A_y,"},{"Start":"14:16.610 ","End":"14:19.450","Text":"which we figured out before."},{"Start":"14:19.450 ","End":"14:28.075","Text":"That from our SOHCAHTOA we know that opposite over hypotenuse is sine of the angle,"},{"Start":"14:28.075 ","End":"14:31.120","Text":"where here the angle is Theta."},{"Start":"14:31.120 ","End":"14:39.310","Text":"Remember, SOH is sine is opposite over hypotenuse."},{"Start":"14:39.310 ","End":"14:44.650","Text":"Now we want to isolate out our A_y because that\u0027s what we\u0027re looking for."},{"Start":"14:44.650 ","End":"14:53.065","Text":"We\u0027ll have that A_y is equal to A_xy multiplied by sine of Theta."},{"Start":"14:53.065 ","End":"14:56.800","Text":"Again, just like what we did over here,"},{"Start":"14:56.800 ","End":"15:00.820","Text":"we can say that our A_xy is equal to"},{"Start":"15:00.820 ","End":"15:07.885","Text":"the magnitude of our vector A multiplied by sine of Phi."},{"Start":"15:07.885 ","End":"15:10.750","Text":"That\u0027s our equation for A_xy,"},{"Start":"15:10.750 ","End":"15:17.360","Text":"and then multiplied by sine of Theta."},{"Start":"15:18.840 ","End":"15:23.155","Text":"This is the general equation for finding"},{"Start":"15:23.155 ","End":"15:29.035","Text":"our y component of A vector when we\u0027re dealing with Cartesian coordinates."},{"Start":"15:29.035 ","End":"15:32.995","Text":"Now let\u0027s see for our specific example over here."},{"Start":"15:32.995 ","End":"15:41.650","Text":"We have the magnitude of A is equal to 6 multiplied by sine of our angle Phi,"},{"Start":"15:41.650 ","End":"15:44.200","Text":"which is 30 degrees,"},{"Start":"15:44.200 ","End":"15:48.745","Text":"multiplied by sine of our angle Theta,"},{"Start":"15:48.745 ","End":"15:51.130","Text":"which is 60 degrees."},{"Start":"15:51.130 ","End":"15:53.620","Text":"When we plug this into the calculator,"},{"Start":"15:53.620 ","End":"16:01.100","Text":"we\u0027ll get that this is equal to 3 root 3 divided by 2."},{"Start":"16:02.640 ","End":"16:05.815","Text":"This is the end of the lesson."},{"Start":"16:05.815 ","End":"16:13.105","Text":"Now the most important things to remember is this equation over here for A_z,"},{"Start":"16:13.105 ","End":"16:15.775","Text":"this red block over here,"},{"Start":"16:15.775 ","End":"16:18.160","Text":"this equation for A_x,"},{"Start":"16:18.160 ","End":"16:20.020","Text":"this red block over here,"},{"Start":"16:20.020 ","End":"16:24.190","Text":"and this equation for A_y, this red block."},{"Start":"16:24.190 ","End":"16:28.285","Text":"This number 2 the magnitude of A_xy is"},{"Start":"16:28.285 ","End":"16:33.730","Text":"more just for understanding to understand where these equations come from,"},{"Start":"16:33.730 ","End":"16:39.985","Text":"but now that we\u0027ve substituted in the magnitude of A sine Phi into both,"},{"Start":"16:39.985 ","End":"16:42.040","Text":"you don\u0027t need to remember this."},{"Start":"16:42.040 ","End":"16:44.185","Text":"You have to remember this block,"},{"Start":"16:44.185 ","End":"16:46.075","Text":"this block, and this block."},{"Start":"16:46.075 ","End":"16:48.950","Text":"That\u0027s the end of this lesson."}],"ID":9602},{"Watched":false,"Name":"Example - Calculating Size and Angles of Vectors","Duration":"6m 5s","ChapterTopicVideoID":10449,"CourseChapterTopicPlaylistID":85322,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10449.jpeg","UploadDate":"2017-11-08T11:41:13.2100000","DurationForVideoObject":"PT6M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"Hello. In this lesson,"},{"Start":"00:01.830 ","End":"00:05.865","Text":"we\u0027re being given 2 vectors, A and B."},{"Start":"00:05.865 ","End":"00:09.360","Text":"We\u0027re being asked what is the size of each vector"},{"Start":"00:09.360 ","End":"00:13.410","Text":"and what is the angle between the 2 vectors?"},{"Start":"00:13.410 ","End":"00:20.130","Text":"Now, we\u0027ll remember that we define the dot product between 2 vectors."},{"Start":"00:20.130 ","End":"00:24.690","Text":"We have A.B and we know that that is equal to"},{"Start":"00:24.690 ","End":"00:29.955","Text":"taking the x-component of the A vector and the x-component of the B vector"},{"Start":"00:29.955 ","End":"00:33.000","Text":"and multiplying them together and then adding on to"},{"Start":"00:33.000 ","End":"00:38.210","Text":"that value the y component of A vector and the y component of"},{"Start":"00:38.210 ","End":"00:42.590","Text":"the B vector and multiplying them together and taking the z component"},{"Start":"00:42.590 ","End":"00:47.270","Text":"of the vector and the z component of the B vector and multiplying those together."},{"Start":"00:47.270 ","End":"00:53.620","Text":"Then we add up all of these terms and we\u0027ll come up with some kind of number quantity."},{"Start":"00:53.620 ","End":"00:57.290","Text":"The dot product gives us the number and not a vector"},{"Start":"00:57.290 ","End":"01:00.890","Text":"and we\u0027ll also remember that another definition for the dot product,"},{"Start":"01:00.890 ","End":"01:04.685","Text":"which will give us the same answer that we have here,"},{"Start":"01:04.685 ","End":"01:10.820","Text":"is taking the size of our A vector and multiplying it by the size of"},{"Start":"01:10.820 ","End":"01:19.510","Text":"our B vector and multiplying that by cosine of the angle between the 2 vectors."},{"Start":"01:20.270 ","End":"01:24.600","Text":"Let\u0027s tackle question Number 1 first."},{"Start":"01:24.600 ","End":"01:27.135","Text":"What is the size of each vector?"},{"Start":"01:27.135 ","End":"01:30.410","Text":"We say that the size of vector A,"},{"Start":"01:30.410 ","End":"01:34.780","Text":"so this is how we represent the size of vector A is simply"},{"Start":"01:34.780 ","End":"01:39.460","Text":"equal to the equation that we all know and love by Pythagoras."},{"Start":"01:39.460 ","End":"01:42.055","Text":"So it equals the x component squared."},{"Start":"01:42.055 ","End":"01:45.640","Text":"Here it\u0027s going to be 1^2 plus the y-component"},{"Start":"01:45.640 ","End":"01:51.150","Text":"squared plus 5^2 plus the z components squared."},{"Start":"01:51.150 ","End":"01:54.045","Text":"Here it\u0027s 10^2."},{"Start":"01:54.045 ","End":"02:04.335","Text":"That is going to be equal to the square root of 1 plus 25 plus 100,"},{"Start":"02:04.335 ","End":"02:10.550","Text":"which is equal to the square root of 126."},{"Start":"02:10.550 ","End":"02:13.235","Text":"This is the size of our vector A."},{"Start":"02:13.235 ","End":"02:17.245","Text":"Now let\u0027s work out the size of our vector B."},{"Start":"02:17.245 ","End":"02:19.765","Text":"Again, we\u0027re going to use Pythagoras."},{"Start":"02:19.765 ","End":"02:21.310","Text":"Our x component squared,"},{"Start":"02:21.310 ","End":"02:30.930","Text":"so 3^2 is 9 plus a y component squared so that\u0027s 4^2 is 16 plus our z components squared,"},{"Start":"02:30.930 ","End":"02:35.335","Text":"which is 5^2, which is 25,"},{"Start":"02:35.335 ","End":"02:38.690","Text":"which is equal to when we add all of that,"},{"Start":"02:38.690 ","End":"02:42.750","Text":"the square root of 50."},{"Start":"02:43.310 ","End":"02:51.030","Text":"These are answers to question Number 1 and that is great."},{"Start":"02:51.030 ","End":"02:55.595","Text":"Now what we\u0027re going to do is we\u0027re going to answer question Number 2."},{"Start":"02:55.595 ","End":"02:57.830","Text":"Question Number 1 was as easy as that."},{"Start":"02:57.830 ","End":"03:03.505","Text":"Now we\u0027re being asked what is the angle between the 2 vectors."},{"Start":"03:03.505 ","End":"03:08.915","Text":"What we\u0027re going to do now is we\u0027re going to use these 2 equations."},{"Start":"03:08.915 ","End":"03:16.140","Text":"We\u0027re going to use the dot-product to find it and we\u0027re going to say that A_x B_x"},{"Start":"03:16.140 ","End":"03:19.620","Text":"plus A_y B_y plus A_z B_z is equal to"},{"Start":"03:19.620 ","End":"03:23.570","Text":"the size of vector A multiplied by the size of vector B,"},{"Start":"03:23.570 ","End":"03:26.600","Text":"multiplied by cosine of the angle between the 2."},{"Start":"03:26.600 ","End":"03:31.535","Text":"Then what we\u0027re going to do is we\u0027re going to isolate out this angle Theta."},{"Start":"03:31.535 ","End":"03:33.945","Text":"Then we have our answer."},{"Start":"03:33.945 ","End":"03:36.350","Text":"What we\u0027re going to do is we\u0027re going to take"},{"Start":"03:36.350 ","End":"03:41.100","Text":"the x component of each vector and multiply them together."},{"Start":"03:41.200 ","End":"03:46.415","Text":"That\u0027s going to be 1 times 3 I\u0027ll really write this down."},{"Start":"03:46.415 ","End":"03:50.990","Text":"Plus the y component of each vector multiplied by each other."},{"Start":"03:50.990 ","End":"03:59.270","Text":"So that\u0027s going to be 5 times 4 plus the z component of each vector multiplied together."},{"Start":"03:59.270 ","End":"04:06.980","Text":"That\u0027s 10 times 5 and that is going to be equal to the size of our A vector,"},{"Start":"04:06.980 ","End":"04:14.210","Text":"which is equal to root of 126 multiplied by the size of our B vector,"},{"Start":"04:14.210 ","End":"04:22.640","Text":"which is roots 50 multiplied by the cosine of our angle Theta,"},{"Start":"04:22.640 ","End":"04:25.740","Text":"which is what we want to find out."},{"Start":"04:26.540 ","End":"04:36.605","Text":"So 1 times 3 is equal to 3 plus 5 times 4 is equal to 20 plus 10 times 5 is equal to 50."},{"Start":"04:36.605 ","End":"04:39.965","Text":"Then what we\u0027re going to do is we\u0027re going to divide"},{"Start":"04:39.965 ","End":"04:44.650","Text":"both sides by these square roots sines."},{"Start":"04:44.650 ","End":"04:48.945","Text":"By the sizes of vector A and sizes of vector B."},{"Start":"04:48.945 ","End":"04:55.610","Text":"All of this divided by the square root of 126 multiplied by the square root of"},{"Start":"04:55.610 ","End":"05:03.330","Text":"50 will give us cosine of our angle."},{"Start":"05:04.430 ","End":"05:10.280","Text":"Now we can plug all of this into our calculator and we\u0027ll get the cosine of"},{"Start":"05:10.280 ","End":"05:18.140","Text":"Theta is equal to 0.92 to 2 decimal places."},{"Start":"05:19.130 ","End":"05:28.805","Text":"Now all I have to do in order to find out this angle is I have to OK, cos both sides."},{"Start":"05:28.805 ","End":"05:34.100","Text":"I\u0027ll do cos to the minus 1 over here,"},{"Start":"05:34.100 ","End":"05:41.250","Text":"and then that will equal approximately 23 degrees."},{"Start":"05:41.800 ","End":"05:44.735","Text":"Now we have it, it\u0027s as easy as that."},{"Start":"05:44.735 ","End":"05:47.600","Text":"This is the angle between 2 vectors."},{"Start":"05:47.600 ","End":"05:52.685","Text":"What\u0027s important is when we\u0027re working out the size of each vector,"},{"Start":"05:52.685 ","End":"05:57.440","Text":"we use Pythagoras and when we\u0027re working out the angle between the 2 vectors,"},{"Start":"05:57.440 ","End":"06:02.750","Text":"we have to use both of these definitions for the dot-product."},{"Start":"06:02.750 ","End":"06:05.610","Text":"That\u0027s the end of this lesson."}],"ID":10797},{"Watched":false,"Name":"Example - Sum is Perpendicular to Difference","Duration":"8m 56s","ChapterTopicVideoID":10450,"CourseChapterTopicPlaylistID":85322,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10450.jpeg","UploadDate":"2017-11-08T11:44:56.2600000","DurationForVideoObject":"PT8M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"Hello. In this lesson,"},{"Start":"00:01.830 ","End":"00:08.370","Text":"we have to prove that if the sum of 2 vectors is perpendicular to their difference,"},{"Start":"00:08.370 ","End":"00:11.890","Text":"then their lengths are equal."},{"Start":"00:12.200 ","End":"00:14.730","Text":"How are we going to prove this?"},{"Start":"00:14.730 ","End":"00:19.005","Text":"What we\u0027re going to do is we\u0027re going to do exactly what they said."},{"Start":"00:19.005 ","End":"00:22.950","Text":"We\u0027re going to give some arbitrary vectors."},{"Start":"00:22.950 ","End":"00:25.785","Text":"We\u0027ll have some vector A,"},{"Start":"00:25.785 ","End":"00:30.120","Text":"which has an x component,"},{"Start":"00:30.120 ","End":"00:32.250","Text":"a y component,"},{"Start":"00:32.250 ","End":"00:34.920","Text":"and a z component."},{"Start":"00:34.920 ","End":"00:38.730","Text":"Then also some arbitrary vector B,"},{"Start":"00:38.730 ","End":"00:42.235","Text":"which has also an x component,"},{"Start":"00:42.235 ","End":"00:44.280","Text":"a y component,"},{"Start":"00:44.280 ","End":"00:46.885","Text":"and a z component."},{"Start":"00:46.885 ","End":"00:50.860","Text":"Then what we\u0027re going to do is we\u0027re going to see what happens when"},{"Start":"00:50.860 ","End":"00:55.475","Text":"we add them and what happens when we subtract them."},{"Start":"00:55.475 ","End":"01:01.885","Text":"Let\u0027s define vector C as the vector that we achieve"},{"Start":"01:01.885 ","End":"01:08.355","Text":"when we add our vector A and our vector B to one another."},{"Start":"01:08.355 ","End":"01:11.920","Text":"As we know, when we\u0027re adding vectors."},{"Start":"01:11.920 ","End":"01:14.920","Text":"In the x component of this new vector,"},{"Start":"01:14.920 ","End":"01:20.920","Text":"we add in the 2x components of A and B,"},{"Start":"01:20.920 ","End":"01:24.875","Text":"so we\u0027ll have A_x plus B_x."},{"Start":"01:24.875 ","End":"01:33.980","Text":"Then the y component of our new vector C will be equal to A_y plus B_y."},{"Start":"01:33.980 ","End":"01:44.365","Text":"Then the z component of our new vector will be A_z plus B_z. Pretty standard."},{"Start":"01:44.365 ","End":"01:49.690","Text":"Now let\u0027s define our vector D as the vector that"},{"Start":"01:49.690 ","End":"01:55.910","Text":"we get when we subtract these 2 vectors, A and B."},{"Start":"01:56.900 ","End":"02:00.915","Text":"When we subtract B from A,"},{"Start":"02:00.915 ","End":"02:11.250","Text":"so what we\u0027ll get is in the x component of a new vector D we\u0027ll have A_x minus B_x,"},{"Start":"02:11.250 ","End":"02:15.855","Text":"and the y component will be A_y minus B_y,"},{"Start":"02:15.855 ","End":"02:21.645","Text":"and the z component will be A_z minus B_z."},{"Start":"02:21.645 ","End":"02:24.970","Text":"The exact same thing, just with a minus."},{"Start":"02:24.970 ","End":"02:27.920","Text":"These are vectors C and D representing"},{"Start":"02:27.920 ","End":"02:31.145","Text":"the addition of our vectors and the subtraction of our vectors."},{"Start":"02:31.145 ","End":"02:32.885","Text":"Now what we want to do,"},{"Start":"02:32.885 ","End":"02:36.205","Text":"so we have the sum of the 2 vectors, which is C,"},{"Start":"02:36.205 ","End":"02:40.130","Text":"and what we want to know is if it\u0027s perpendicular to their difference,"},{"Start":"02:40.130 ","End":"02:41.840","Text":"so that\u0027s our vector D,"},{"Start":"02:41.840 ","End":"02:44.255","Text":"then their lengths are equal."},{"Start":"02:44.255 ","End":"02:45.935","Text":"That\u0027s what we\u0027re trying to prove."},{"Start":"02:45.935 ","End":"02:49.880","Text":"Now we want to prove and we want to see what"},{"Start":"02:49.880 ","End":"02:54.510","Text":"happens when vectors C and D are perpendicular."},{"Start":"02:54.520 ","End":"02:59.900","Text":"The sum of the 2 vectors is perpendicular to their difference."},{"Start":"02:59.900 ","End":"03:03.330","Text":"Vector C is perpendicular,"},{"Start":"03:03.330 ","End":"03:06.540","Text":"this is how we denote the perpendicular,"},{"Start":"03:06.540 ","End":"03:09.655","Text":"to vector D. If so,"},{"Start":"03:09.655 ","End":"03:16.680","Text":"that would mean that the dot-product C.D,"},{"Start":"03:16.680 ","End":"03:19.655","Text":"which as we know is according to the equation of"},{"Start":"03:19.655 ","End":"03:23.660","Text":"the size of our vector C multiplied by the size of"},{"Start":"03:23.660 ","End":"03:32.390","Text":"our vector D multiplied by cosine of the angle between them must be equal to 0."},{"Start":"03:32.390 ","End":"03:34.145","Text":"Why so?"},{"Start":"03:34.145 ","End":"03:39.800","Text":"Because if our vector C is perpendicular to our vector D,"},{"Start":"03:39.800 ","End":"03:44.410","Text":"that means that the angle between the 2 vectors is 90."},{"Start":"03:44.410 ","End":"03:51.670","Text":"As we know, cosine of 90 degrees is equal to 0,"},{"Start":"03:51.670 ","End":"03:54.090","Text":"if Theta\u0027s equal to 90."},{"Start":"03:54.090 ","End":"03:58.820","Text":"That happens every time we have 2 vectors which are perpendicular."},{"Start":"03:58.820 ","End":"04:04.950","Text":"We\u0027ll know that they\u0027re perpendicular if their dot product is equal to 0."},{"Start":"04:05.210 ","End":"04:09.880","Text":"Now let\u0027s solve this equation."},{"Start":"04:10.550 ","End":"04:17.376","Text":"What we\u0027re going to do now is we\u0027re going to work out what C.D is equal to."},{"Start":"04:17.376 ","End":"04:21.380","Text":"Let\u0027s write this here. We have C.D."},{"Start":"04:21.590 ","End":"04:29.485","Text":"As we know, that means that we take each component of the vectors,"},{"Start":"04:29.485 ","End":"04:31.780","Text":"multiply them together, and then add on."},{"Start":"04:31.780 ","End":"04:36.355","Text":"We\u0027ll take the x component multiplied by this x component plus"},{"Start":"04:36.355 ","End":"04:39.190","Text":"this y component multiplied by this y component"},{"Start":"04:39.190 ","End":"04:43.125","Text":"plus this z component multiplied by this z component."},{"Start":"04:43.125 ","End":"04:45.420","Text":"Let\u0027s see. First,"},{"Start":"04:45.420 ","End":"04:51.555","Text":"we\u0027ll have A_x plus B_x multiplied by A_x minus B_x."},{"Start":"04:51.555 ","End":"04:54.470","Text":"As we know from simple mathematics,"},{"Start":"04:54.470 ","End":"05:04.460","Text":"that that will be equal to A_x^2 minus B_x^2."},{"Start":"05:04.460 ","End":"05:05.840","Text":"Then for the y components,"},{"Start":"05:05.840 ","End":"05:08.840","Text":"so we add on A_y plus B_y,"},{"Start":"05:08.840 ","End":"05:12.125","Text":"multiplied by A_y minus B_y."},{"Start":"05:12.125 ","End":"05:19.150","Text":"That will be equal to A_y^2 minus B_y^2."},{"Start":"05:19.150 ","End":"05:21.915","Text":"For the z components, so we add on."},{"Start":"05:21.915 ","End":"05:27.165","Text":"That will be A_z plus B_z multiplied by A_z minus B_z."},{"Start":"05:27.165 ","End":"05:35.520","Text":"That will be A_z^2 minus B_z^2."},{"Start":"05:35.520 ","End":"05:37.815","Text":"Now let\u0027s rewrite this out."},{"Start":"05:37.815 ","End":"05:42.715","Text":"Here we can take all the components of our A vector."},{"Start":"05:42.715 ","End":"05:49.735","Text":"We have A_x^2 plus A_y^2,"},{"Start":"05:49.735 ","End":"05:56.150","Text":"I\u0027m just writing this out in a different order, plus A_z^2."},{"Start":"05:57.050 ","End":"06:01.300","Text":"Then we have negative."},{"Start":"06:01.700 ","End":"06:06.630","Text":"The coefficient of all of the B terms have a minus,"},{"Start":"06:06.630 ","End":"06:09.535","Text":"so I can just take the minus outside of the brackets."},{"Start":"06:09.535 ","End":"06:15.100","Text":"Then I have B_x^2"},{"Start":"06:15.100 ","End":"06:21.590","Text":"plus B_y^2 plus B_z^2."},{"Start":"06:22.400 ","End":"06:24.550","Text":"If I open out the brackets,"},{"Start":"06:24.550 ","End":"06:28.420","Text":"they all become minus signs, all these positives."},{"Start":"06:28.420 ","End":"06:33.680","Text":"I\u0027ve just rewritten this section of the equation out again."},{"Start":"06:34.070 ","End":"06:39.315","Text":"What is A_x^2 plus A_y^2 plus A_z^2?"},{"Start":"06:39.315 ","End":"06:42.120","Text":"Let\u0027s show it like this."},{"Start":"06:42.120 ","End":"06:48.000","Text":"We know that the size of our vector A"},{"Start":"06:48.000 ","End":"06:54.195","Text":"is simply going to be A_x^2 plus A_y^2 plus A_z^2."},{"Start":"06:54.195 ","End":"06:56.395","Text":"Then we take the square root of all of that."},{"Start":"06:56.395 ","End":"07:02.890","Text":"Remember the size of a vector is simply applying Pythagoras\u0027 theorem to each component."},{"Start":"07:02.890 ","End":"07:07.405","Text":"It will be the square root of A_x^2 plus A_y^2 plus A_z^2."},{"Start":"07:07.405 ","End":"07:09.415","Text":"That\u0027s exactly what this is here."},{"Start":"07:09.415 ","End":"07:15.290","Text":"This is the size of vector A^2."},{"Start":"07:15.290 ","End":"07:17.640","Text":"That\u0027s what this is over here."},{"Start":"07:17.640 ","End":"07:19.025","Text":"If we took the square root,"},{"Start":"07:19.025 ","End":"07:20.885","Text":"it would just be the size of our vector A."},{"Start":"07:20.885 ","End":"07:30.900","Text":"Then what we have here inside the bracket is again B_x^2 plus B_y^2 plus B_z^2."},{"Start":"07:30.900 ","End":"07:36.830","Text":"That\u0027s again simply equal to the size of our vector B^2."},{"Start":"07:36.830 ","End":"07:38.855","Text":"If we would take the square root of this,"},{"Start":"07:38.855 ","End":"07:41.850","Text":"we would have the size of our vector B."},{"Start":"07:42.620 ","End":"07:47.590","Text":"As we know that if C is perpendicular to D,"},{"Start":"07:47.590 ","End":"07:52.570","Text":"so that means that the dot product between vector C and D is equal to 0."},{"Start":"07:52.570 ","End":"07:58.755","Text":"Here we have that all of what we just wrote out over here is equal to 0."},{"Start":"07:58.755 ","End":"08:02.340","Text":"Let\u0027s scroll down a touch."},{"Start":"08:02.340 ","End":"08:08.070","Text":"Then we can move over this term to the other side of the equal side,"},{"Start":"08:08.070 ","End":"08:13.390","Text":"and what we\u0027ll get is that the size of A vector squared is"},{"Start":"08:13.390 ","End":"08:18.780","Text":"equal to the size of our B vector squared,"},{"Start":"08:18.780 ","End":"08:21.535","Text":"which if we take the square root of both sides,"},{"Start":"08:21.535 ","End":"08:29.105","Text":"we\u0027ll get that the size of A vector is equal to the size of our B vector."},{"Start":"08:29.105 ","End":"08:31.640","Text":"Now what is the size of these vectors?"},{"Start":"08:31.640 ","End":"08:34.585","Text":"It means their length, their magnitude."},{"Start":"08:34.585 ","End":"08:36.935","Text":"That is exactly what we wanted to prove."},{"Start":"08:36.935 ","End":"08:43.880","Text":"We wanted to prove that if the sum of the 2 vectors is perpendicular to that difference,"},{"Start":"08:43.880 ","End":"08:45.590","Text":"so that\u0027s what we did over here,"},{"Start":"08:45.590 ","End":"08:49.070","Text":"then their lengths or their magnitudes are equal,"},{"Start":"08:49.070 ","End":"08:52.430","Text":"which is what we got right over here."},{"Start":"08:52.430 ","End":"08:56.820","Text":"We proved that, and that is the end of the lesson."}],"ID":10798},{"Watched":false,"Name":"Example - Perpendicular Vector","Duration":"6m 57s","ChapterTopicVideoID":10451,"CourseChapterTopicPlaylistID":85322,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10451.jpeg","UploadDate":"2017-11-08T11:48:40.6800000","DurationForVideoObject":"PT6M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"Hello. In this question,"},{"Start":"00:02.070 ","End":"00:03.569","Text":"we are given two vectors,"},{"Start":"00:03.569 ","End":"00:05.675","Text":"vector A and vector B,"},{"Start":"00:05.675 ","End":"00:08.510","Text":"where vector B has 3 components,"},{"Start":"00:08.510 ","End":"00:14.595","Text":"where one component is a known value and the other two components are unknown."},{"Start":"00:14.595 ","End":"00:19.565","Text":"The question is to find the components of vector B, the unknown components."},{"Start":"00:19.565 ","End":"00:24.425","Text":"If we\u0027re told that vector B is perpendicular to vector A,"},{"Start":"00:24.425 ","End":"00:26.980","Text":"and then its length is 10."},{"Start":"00:26.980 ","End":"00:29.495","Text":"As we can see, we have two unknowns,"},{"Start":"00:29.495 ","End":"00:34.895","Text":"B_x and B_y that we\u0027re trying to solve to find out what they are."},{"Start":"00:34.895 ","End":"00:38.050","Text":"Which means that we need 2 equations to find that out."},{"Start":"00:38.050 ","End":"00:44.000","Text":"We\u0027re being told that the 2 equations that we\u0027re going to use is the fact that B is"},{"Start":"00:44.000 ","End":"00:51.090","Text":"perpendicular to A and that the size of B is equal to 10."},{"Start":"00:51.680 ","End":"00:56.490","Text":"Let\u0027s go with the first equation."},{"Start":"00:56.490 ","End":"01:01.200","Text":"We\u0027re being told that the 2 vectors are perpendicular."},{"Start":"01:01.200 ","End":"01:04.785","Text":"This is how we denote 2 perpendicular vectors."},{"Start":"01:04.785 ","End":"01:10.710","Text":"If they\u0027re perpendicular, we know that 2 perpendicular vectors,"},{"Start":"01:10.850 ","End":"01:16.725","Text":"their dot-product is always equal to 0."},{"Start":"01:16.725 ","End":"01:22.190","Text":"Because that\u0027s because we know that the dot product between these 2 vectors is equal"},{"Start":"01:22.190 ","End":"01:27.350","Text":"to the size of the 2 vectors multiplied by cosine of the angle between the 2."},{"Start":"01:27.350 ","End":"01:29.495","Text":"If we\u0027re being told that they\u0027re perpendicular,"},{"Start":"01:29.495 ","End":"01:36.300","Text":"that means that the angle between the 2 is 90 degrees and cosine of 90 is equal to 0."},{"Start":"01:36.400 ","End":"01:41.875","Text":"Let\u0027s work out what A.B is equal 2."},{"Start":"01:41.875 ","End":"01:49.210","Text":"We know that we multiply the x component of A multiplied by the x component of B."},{"Start":"01:49.210 ","End":"01:53.585","Text":"Then we add on the y component of A multiplied by the y component of B."},{"Start":"01:53.585 ","End":"01:59.570","Text":"Then we add on the z component of A multiplied by the z component of B."},{"Start":"01:59.570 ","End":"02:07.580","Text":"Here we\u0027ll have 1 multiplied by B_x plus 4 multiplied by"},{"Start":"02:07.580 ","End":"02:16.700","Text":"B_y plus h multiplied by 0."},{"Start":"02:16.700 ","End":"02:19.100","Text":"Eight multiplied by 0 is equal to 0."},{"Start":"02:19.100 ","End":"02:24.140","Text":"Then we\u0027ll have y multiplied by B_x is simply B_x."},{"Start":"02:24.140 ","End":"02:32.720","Text":"Then 4 multiplied by B_y will just simply equal plus 4B_y."},{"Start":"02:32.720 ","End":"02:37.800","Text":"Then we know that this is equal to 0."},{"Start":"02:38.990 ","End":"02:43.955","Text":"Now if we move the components over to that other side,"},{"Start":"02:43.955 ","End":"02:51.390","Text":"will get therefore that B_x is equal to negative 4B_y."},{"Start":"02:53.360 ","End":"02:57.380","Text":"This is part of the answer."},{"Start":"02:57.380 ","End":"03:00.140","Text":"Now, we\u0027re going to use the second piece of"},{"Start":"03:00.140 ","End":"03:03.500","Text":"information that we know for the second equation."},{"Start":"03:03.500 ","End":"03:08.095","Text":"That\u0027s that the length of our vector B is equal to 10."},{"Start":"03:08.095 ","End":"03:10.760","Text":"As we know, the length or the size,"},{"Start":"03:10.760 ","End":"03:14.285","Text":"or the magnitude of a vector is denoted by this."},{"Start":"03:14.285 ","End":"03:18.415","Text":"This we were given in the question that it\u0027s equal to 10."},{"Start":"03:18.415 ","End":"03:23.255","Text":"That means that when we use Pythagoras\u0027s theorem,"},{"Start":"03:23.255 ","End":"03:28.115","Text":"so that is that the square root of B_x squared"},{"Start":"03:28.115 ","End":"03:33.560","Text":"plus B_y squared plus the z component,"},{"Start":"03:33.560 ","End":"03:36.185","Text":"which here is 0 squared."},{"Start":"03:36.185 ","End":"03:40.550","Text":"This will give us the magnitude of our vector B,"},{"Start":"03:40.550 ","End":"03:42.650","Text":"which as we were told in the question,"},{"Start":"03:42.650 ","End":"03:45.900","Text":"is going to be equal to 10."},{"Start":"03:46.670 ","End":"03:54.170","Text":"Now we can substitute n. We know that our B_x is equal to negative 4 B_y."},{"Start":"03:54.170 ","End":"03:58.430","Text":"Let\u0027s substitute that n. We\u0027ll have the square root of B_x squared."},{"Start":"03:58.430 ","End":"04:03.900","Text":"That\u0027s negative for what we worked out here for B_y"},{"Start":"04:03.900 ","End":"04:11.505","Text":"squared plus B_y squared plus 0 squared is 0."},{"Start":"04:11.505 ","End":"04:14.650","Text":"All of this is equal to 10."},{"Start":"04:14.650 ","End":"04:17.300","Text":"Now what we\u0027re going to do is we\u0027re going to square"},{"Start":"04:17.300 ","End":"04:20.570","Text":"both sides to get rid of the square root sign over here."},{"Start":"04:20.570 ","End":"04:25.460","Text":"Then we know that a negative number squared is a positive number."},{"Start":"04:25.460 ","End":"04:32.735","Text":"What we\u0027ll have is we\u0027ll have 16 B_y squared."},{"Start":"04:32.735 ","End":"04:34.760","Text":"I\u0027m just doing two steps in one."},{"Start":"04:34.760 ","End":"04:36.260","Text":"I\u0027m squaring both sides,"},{"Start":"04:36.260 ","End":"04:40.640","Text":"and also I\u0027m squaring this over here."},{"Start":"04:40.640 ","End":"04:50.730","Text":"The inner quantities and then plus B_y squared is equal to 10 squared, which is 100."},{"Start":"04:50.730 ","End":"04:52.845","Text":"Now I can add that both up,"},{"Start":"04:52.845 ","End":"04:58.785","Text":"and then we\u0027ll get that 17 B_y squared is equal to 100."},{"Start":"04:58.785 ","End":"05:00.770","Text":"Let\u0027s write that here,"},{"Start":"05:00.770 ","End":"05:07.130","Text":"so 17 B_y squared is equal to 100."},{"Start":"05:07.130 ","End":"05:11.220","Text":"Now what I want to do is I want to isolate out my B_y,"},{"Start":"05:11.440 ","End":"05:17.905","Text":"so I\u0027m going to divide both sides by 17 and then take the square root."},{"Start":"05:17.905 ","End":"05:21.315","Text":"I\u0027m isolating out my B_y because that\u0027s what I want to know."},{"Start":"05:21.315 ","End":"05:26.540","Text":"I\u0027ll have 100 divided by 17,"},{"Start":"05:26.540 ","End":"05:31.730","Text":"and then I\u0027ll take the square root of all of that."},{"Start":"05:31.730 ","End":"05:40.650","Text":"Then we know that B_x is equal to negative 4 B_y."},{"Start":"05:40.760 ","End":"05:43.790","Text":"If B_y is equal to this,"},{"Start":"05:43.790 ","End":"05:46.175","Text":"so then I know that B_x,"},{"Start":"05:46.175 ","End":"05:48.155","Text":"which is equal to negative 4 B_y,"},{"Start":"05:48.155 ","End":"05:58.600","Text":"is going to be equal to negative 4 multiplied by the square root of 100 divided by 17."},{"Start":"06:00.980 ","End":"06:04.985","Text":"What we were meant to do is to find the components of"},{"Start":"06:04.985 ","End":"06:09.210","Text":"our vector B given these two pieces of information."},{"Start":"06:09.210 ","End":"06:12.360","Text":"Now we\u0027ve done that, and now we have a y component"},{"Start":"06:12.360 ","End":"06:15.660","Text":"of our B vector and our x component of our B vector."},{"Start":"06:15.660 ","End":"06:17.480","Text":"Now we can simply write,"},{"Start":"06:17.480 ","End":"06:18.950","Text":"let\u0027s write it over here,"},{"Start":"06:18.950 ","End":"06:25.620","Text":"that our B vector is going to be given as our x component, which was B_x,"},{"Start":"06:25.620 ","End":"06:27.805","Text":"which is negative 4,"},{"Start":"06:27.805 ","End":"06:37.800","Text":"multiplied by the square root of 100 divided by 17 then the y component of our B vector,"},{"Start":"06:37.800 ","End":"06:44.685","Text":"which was B_y, is simply just 100 divided by 17."},{"Start":"06:44.685 ","End":"06:50.110","Text":"We were already given a z component, which is 0."},{"Start":"06:51.080 ","End":"06:54.335","Text":"Here we have our answer to the question,"},{"Start":"06:54.335 ","End":"06:57.240","Text":"and that is the end of the lesson."}],"ID":10799},{"Watched":false,"Name":"Example- Net Force and Angles","Duration":"8m 41s","ChapterTopicVideoID":10452,"CourseChapterTopicPlaylistID":85322,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10452.jpeg","UploadDate":"2017-11-08T11:52:48.9270000","DurationForVideoObject":"PT8M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.995","Text":"Hello. In this question,"},{"Start":"00:01.995 ","End":"00:06.165","Text":"we\u0027re being given 2 forces which are acting on a body."},{"Start":"00:06.165 ","End":"00:10.725","Text":"The 2 forces are denoted by these vectors A and B."},{"Start":"00:10.725 ","End":"00:13.125","Text":"Question number 1 is asking us,"},{"Start":"00:13.125 ","End":"00:15.555","Text":"what is the net force?"},{"Start":"00:15.555 ","End":"00:17.460","Text":"What is the net force?"},{"Start":"00:17.460 ","End":"00:21.060","Text":"The net force is the total force acting on the body."},{"Start":"00:21.060 ","End":"00:24.435","Text":"It takes into account all of the forces acting on the body."},{"Start":"00:24.435 ","End":"00:27.300","Text":"Here, we have 2 forces acting, A and B,"},{"Start":"00:27.300 ","End":"00:32.895","Text":"so the net force is going to be A plus B."},{"Start":"00:32.895 ","End":"00:35.775","Text":"Let\u0027s define a new vector C,"},{"Start":"00:35.775 ","End":"00:40.005","Text":"which is equal to the addition of A plus B,"},{"Start":"00:40.005 ","End":"00:43.005","Text":"so C is our net force."},{"Start":"00:43.005 ","End":"00:46.775","Text":"That is going to be equal to. How do we do this?"},{"Start":"00:46.775 ","End":"00:49.370","Text":"We add up all of the x components,"},{"Start":"00:49.370 ","End":"00:53.660","Text":"so that\u0027s 1+3 in the x section."},{"Start":"00:53.660 ","End":"00:55.580","Text":"Then for the y component of C,"},{"Start":"00:55.580 ","End":"00:57.695","Text":"we add up the y components."},{"Start":"00:57.695 ","End":"01:01.300","Text":"That\u0027s 4+6."},{"Start":"01:01.300 ","End":"01:03.270","Text":"For the z components,"},{"Start":"01:03.270 ","End":"01:09.690","Text":"we add the z components of A and B, which is 5+7."},{"Start":"01:09.690 ","End":"01:11.705","Text":"Let\u0027s see what that is equal to."},{"Start":"01:11.705 ","End":"01:18.050","Text":"That will simply be equal to 4, 10, 12."},{"Start":"01:18.050 ","End":"01:20.550","Text":"This is the net force."},{"Start":"01:20.630 ","End":"01:23.805","Text":"That\u0027s the answer to question number 1."},{"Start":"01:23.805 ","End":"01:26.320","Text":"Now, let\u0027s look at question number 2."},{"Start":"01:26.320 ","End":"01:30.680","Text":"We\u0027re being asked what is the size of the net force?"},{"Start":"01:30.680 ","End":"01:38.030","Text":"The size of a vector is denoted like so and the size of it,"},{"Start":"01:38.030 ","End":"01:39.915","Text":"we use simply Pythagoras."},{"Start":"01:39.915 ","End":"01:43.190","Text":"It\u0027s the square root of the x component squared,"},{"Start":"01:43.190 ","End":"01:44.570","Text":"so that\u0027s 4^2,"},{"Start":"01:44.570 ","End":"01:46.639","Text":"plus the y component squared,"},{"Start":"01:46.639 ","End":"01:48.470","Text":"so that\u0027s 10^2,"},{"Start":"01:48.470 ","End":"01:53.250","Text":"plus the z component squared, so that\u0027s 12^2."},{"Start":"01:53.900 ","End":"01:59.510","Text":"Then once we work that out or plug it into a calculator,"},{"Start":"01:59.510 ","End":"02:04.380","Text":"we\u0027ll get that this is equal to the square root of 260."},{"Start":"02:05.900 ","End":"02:09.360","Text":"That\u0027s the answer to question number 2."},{"Start":"02:09.360 ","End":"02:13.015","Text":"Now, let\u0027s do question number 3."},{"Start":"02:13.015 ","End":"02:15.920","Text":"What is the angle between the net force,"},{"Start":"02:15.920 ","End":"02:17.650","Text":"so that\u0027s our vector C,"},{"Start":"02:17.650 ","End":"02:20.140","Text":"and each of the axes?"},{"Start":"02:20.140 ","End":"02:25.700","Text":"What we\u0027re actually asking is what is the angle between vector C and the x-axis,"},{"Start":"02:25.700 ","End":"02:27.335","Text":"vector C and the y-axis,"},{"Start":"02:27.335 ","End":"02:31.415","Text":"and the angle between vector C and the z-axis?"},{"Start":"02:31.415 ","End":"02:37.280","Text":"First of all, what\u0027s good to remember is in order to find the angle,"},{"Start":"02:37.280 ","End":"02:39.740","Text":"our equation is cosine of Alpha,"},{"Start":"02:39.740 ","End":"02:42.905","Text":"where Alpha is the angle between 2 vectors,"},{"Start":"02:42.905 ","End":"02:50.325","Text":"is equal to our A vector multiplied by our B vector"},{"Start":"02:50.325 ","End":"02:58.845","Text":"divided by the size of our A vector multiplied by the size of our B vector."},{"Start":"02:58.845 ","End":"03:02.660","Text":"Now, these are just arbitrary vectors and this would give us"},{"Start":"03:02.660 ","End":"03:07.680","Text":"the angle between our A vector and our B vector."},{"Start":"03:08.540 ","End":"03:12.650","Text":"If we did this equation for these 2,"},{"Start":"03:12.650 ","End":"03:16.025","Text":"we would get the angle between vector A and vector B."},{"Start":"03:16.025 ","End":"03:21.350","Text":"But what we want is the angle between vector C and the x,"},{"Start":"03:21.350 ","End":"03:22.775","Text":"y, and z axes."},{"Start":"03:22.775 ","End":"03:26.500","Text":"Let\u0027s remind ourselves what the x-axis is."},{"Start":"03:26.500 ","End":"03:30.170","Text":"The x-axis can be denoted by an x-hat."},{"Start":"03:30.170 ","End":"03:32.990","Text":"The unit vector in the x direction,"},{"Start":"03:32.990 ","End":"03:36.380","Text":"which is given by the coordinates 1, 0, 0."},{"Start":"03:36.380 ","End":"03:42.185","Text":"Then the y-axis can be denoted by the unit vector y,"},{"Start":"03:42.185 ","End":"03:46.430","Text":"which is denoted by 0, 1, 0."},{"Start":"03:46.430 ","End":"03:51.650","Text":"Then the z-axis can be denoted by the unit vectors Z,"},{"Start":"03:51.650 ","End":"03:55.920","Text":"which is given by 0, 0, 1."},{"Start":"03:55.920 ","End":"03:58.395","Text":"Now, what we want to do is we want to find"},{"Start":"03:58.395 ","End":"04:05.725","Text":"the angle between C and all of these unit vectors."},{"Start":"04:05.725 ","End":"04:08.460","Text":"Let\u0027s start working this out."},{"Start":"04:08.460 ","End":"04:14.230","Text":"Let\u0027s work it out between C and the x-axis."},{"Start":"04:14.990 ","End":"04:18.045","Text":"Then we\u0027ll just use this equation."},{"Start":"04:18.045 ","End":"04:23.000","Text":"We\u0027ll get that cosine of Alpha is going to be equal to"},{"Start":"04:23.000 ","End":"04:29.055","Text":"our vector C dot-product with our other vectors."},{"Start":"04:29.055 ","End":"04:34.340","Text":"That will be our x-hat vector divided by the size of"},{"Start":"04:34.340 ","End":"04:41.580","Text":"our vector C multiplied by the size of our vector x."},{"Start":"04:42.820 ","End":"04:46.310","Text":"Then let\u0027s do this."},{"Start":"04:46.310 ","End":"04:49.790","Text":"We\u0027ll have our C is 4,10,12."},{"Start":"04:49.790 ","End":"04:52.804","Text":"The dot-product, if we remember correctly,"},{"Start":"04:52.804 ","End":"04:57.830","Text":"it\u0027s going to be equal to our x component multiplied by our x component plus"},{"Start":"04:57.830 ","End":"05:00.380","Text":"our y component multiplied by our y component"},{"Start":"05:00.380 ","End":"05:04.095","Text":"plus our z component multiplied by our z component."},{"Start":"05:04.095 ","End":"05:06.320","Text":"We\u0027ll have x times x,"},{"Start":"05:06.320 ","End":"05:11.200","Text":"so that\u0027s 4 times 1 plus 10 times 0,"},{"Start":"05:11.200 ","End":"05:14.020","Text":"which is 0 plus 12 times 0 which is 0."},{"Start":"05:14.020 ","End":"05:17.980","Text":"I\u0027ll simply just have 4 because 4 times 1 is equal to"},{"Start":"05:17.980 ","End":"05:22.930","Text":"4 divided by the size of our C vector,"},{"Start":"05:22.930 ","End":"05:27.805","Text":"which we got was root 260 multiplied by the size of our x vector,"},{"Start":"05:27.805 ","End":"05:29.290","Text":"which if we can see,"},{"Start":"05:29.290 ","End":"05:31.340","Text":"we can see that it\u0027s the unit vector."},{"Start":"05:31.340 ","End":"05:35.335","Text":"That should give you a clue that the size of this vector is 1."},{"Start":"05:35.335 ","End":"05:37.810","Text":"But otherwise, we can use Pythagoras."},{"Start":"05:37.810 ","End":"05:44.275","Text":"The size of our x vector is going to be the square root of 1^2 plus 0^2 plus 0^2,"},{"Start":"05:44.275 ","End":"05:47.725","Text":"which is simply going to be the square root of 1^2,"},{"Start":"05:47.725 ","End":"05:49.495","Text":"which is just 1."},{"Start":"05:49.495 ","End":"05:54.430","Text":"That\u0027s just going to be the size of vector C,"},{"Start":"05:54.430 ","End":"05:59.400","Text":"so that\u0027s root 260 multiplied by 1,"},{"Start":"05:59.400 ","End":"06:01.350","Text":"which is just this."},{"Start":"06:01.350 ","End":"06:05.635","Text":"Now, once we work this out,"},{"Start":"06:05.635 ","End":"06:08.335","Text":"and then arc cos both sides,"},{"Start":"06:08.335 ","End":"06:11.410","Text":"cos to the minus 1 over here and on this side,"},{"Start":"06:11.410 ","End":"06:20.240","Text":"we\u0027ll get that Alpha=75.63 degrees."},{"Start":"06:21.110 ","End":"06:24.010","Text":"That was pretty straightforward."},{"Start":"06:24.010 ","End":"06:29.750","Text":"This is the angle between our net force and the x-axis."},{"Start":"06:32.030 ","End":"06:35.040","Text":"Let\u0027s write here x."},{"Start":"06:35.040 ","End":"06:41.330","Text":"Now, let\u0027s work out the angle between our C vector and the y-axis."},{"Start":"06:41.330 ","End":"06:47.805","Text":"Again, we\u0027ll take our vector dot product with our y-axis."},{"Start":"06:47.805 ","End":"06:55.680","Text":"Here, we\u0027ll have 4 times 0 is 0 plus 10 times 1 is 10 plus 12 times 0 is 0."},{"Start":"06:55.680 ","End":"07:02.675","Text":"We\u0027ll have cosine of Alpha with the y-axis is going to be our C.y,"},{"Start":"07:02.675 ","End":"07:05.360","Text":"which is simply equal to 10,"},{"Start":"07:05.360 ","End":"07:11.485","Text":"divided by, again, the size of our C vector is root 260,"},{"Start":"07:11.485 ","End":"07:17.870","Text":"and multiplied by the size of our unit vector in the y direction, which is 1."},{"Start":"07:17.870 ","End":"07:21.480","Text":"That\u0027s simply equal to that."},{"Start":"07:21.830 ","End":"07:24.920","Text":"Then once we arc cos both sides,"},{"Start":"07:24.920 ","End":"07:35.130","Text":"we\u0027ll get that our angle between our C vector and our y-axis will be equal to 51.67."},{"Start":"07:36.170 ","End":"07:42.695","Text":"Now, let\u0027s work out the angle between our C vector and the z-axis."},{"Start":"07:42.695 ","End":"07:46.475","Text":"Let\u0027s call this cosine of Alpha z."},{"Start":"07:46.475 ","End":"07:51.270","Text":"Again, we\u0027ll do our C.z."},{"Start":"07:51.270 ","End":"07:54.165","Text":"Here, we\u0027ll have 4 times 0 is 0,"},{"Start":"07:54.165 ","End":"07:55.845","Text":"plus 10 times 0,"},{"Start":"07:55.845 ","End":"08:00.180","Text":"which is 0, plus 12 times 1, which is 12."},{"Start":"08:00.180 ","End":"08:04.780","Text":"So 12 divided by and then the size of our C vector is root"},{"Start":"08:04.780 ","End":"08:11.170","Text":"260 multiplied by the size of our unit vector in the z direction which is 1,"},{"Start":"08:11.170 ","End":"08:13.015","Text":"which is just going to be equal to this."},{"Start":"08:13.015 ","End":"08:23.660","Text":"Then we\u0027ll get that the angle between our C vector and our z-axis will be equal to 41.9."},{"Start":"08:25.040 ","End":"08:27.265","Text":"That\u0027s the end of the lesson."},{"Start":"08:27.265 ","End":"08:31.250","Text":"I suggest writing out this equation, which is what?"},{"Start":"08:31.250 ","End":"08:34.580","Text":"Every time you\u0027re being asked to find the angle between 2 vectors,"},{"Start":"08:34.580 ","End":"08:39.340","Text":"you use this equation where obviously A and B are arbitrary vectors."},{"Start":"08:39.340 ","End":"08:42.040","Text":"That\u0027s the end of this lesson."}],"ID":10800}],"Thumbnail":null,"ID":85322},{"Name":"Curl (Also Called Rotor)","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Curl Operator","Duration":"3m 14s","ChapterTopicVideoID":9458,"CourseChapterTopicPlaylistID":85323,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/9458.jpeg","UploadDate":"2021-11-18T09:06:23.1130000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.920","Text":"Hello. We already talked about the gradient and divergence."},{"Start":"00:04.920 ","End":"00:07.785","Text":"That leaves us one more tool we need to talk about,"},{"Start":"00:07.785 ","End":"00:09.150","Text":"which is the rotor."},{"Start":"00:09.150 ","End":"00:15.135","Text":"A rotor is a multiplication between the nabla and a vector function."},{"Start":"00:15.135 ","End":"00:18.450","Text":"If I have a general function f,"},{"Start":"00:18.450 ","End":"00:20.340","Text":"then I\u0027ll multiply that,"},{"Start":"00:20.340 ","End":"00:23.849","Text":"do a vector multiplication with the nabla using a determinant."},{"Start":"00:23.849 ","End":"00:25.170","Text":"On the top row,"},{"Start":"00:25.170 ","End":"00:26.295","Text":"I\u0027ll put x hat,"},{"Start":"00:26.295 ","End":"00:28.125","Text":"y hat, and z hat."},{"Start":"00:28.125 ","End":"00:29.550","Text":"In the middle row."},{"Start":"00:29.550 ","End":"00:33.045","Text":"I\u0027ll put d over d_x,"},{"Start":"00:33.045 ","End":"00:35.685","Text":"d over d_y, and d over d_z."},{"Start":"00:35.685 ","End":"00:40.500","Text":"In the bottom row, I\u0027ll put F_x, F_y, and F_z."},{"Start":"00:40.500 ","End":"00:43.115","Text":"As you can see the result here,"},{"Start":"00:43.115 ","End":"00:46.580","Text":"this is just like with any other determinant,"},{"Start":"00:46.580 ","End":"00:48.400","Text":"the solution that you\u0027ll get."},{"Start":"00:48.400 ","End":"00:51.515","Text":"This is the formula, of course, for Cartesian coordinates."},{"Start":"00:51.515 ","End":"00:55.100","Text":"You can see below for cylindrical or spherical coordinates."},{"Start":"00:55.100 ","End":"00:57.125","Text":"What\u0027s important is to remember this formula."},{"Start":"00:57.125 ","End":"00:58.550","Text":"This is the mathematical side."},{"Start":"00:58.550 ","End":"01:00.439","Text":"You should know how to perform this operation."},{"Start":"01:00.439 ","End":"01:02.735","Text":"We\u0027ll talk about the meaning in a minute."},{"Start":"01:02.735 ","End":"01:06.725","Text":"If you\u0027d like, you can write this on your formula sheet if there\u0027s room."},{"Start":"01:06.725 ","End":"01:09.290","Text":"But it\u0027s very similar to any other determinant and you"},{"Start":"01:09.290 ","End":"01:11.570","Text":"just need to know how to perform this."},{"Start":"01:11.570 ","End":"01:15.125","Text":"For example, if I\u0027m looking for the x element of a rotor,"},{"Start":"01:15.125 ","End":"01:21.925","Text":"I would write df_z over df_y minus df_y over d_z."},{"Start":"01:21.925 ","End":"01:24.230","Text":"Notice that this gives me a vector."},{"Start":"01:24.230 ","End":"01:27.575","Text":"You should remember that the gradient will give me a vector,"},{"Start":"01:27.575 ","End":"01:30.095","Text":"the divergence will give me a scalar,"},{"Start":"01:30.095 ","End":"01:33.605","Text":"and the rotor will give me a vector again."},{"Start":"01:33.605 ","End":"01:38.420","Text":"Notice below we have the formulas to do this in cylindrical and spherical coordinates."},{"Start":"01:38.420 ","End":"01:41.390","Text":"You don\u0027t have to remember this off the top of your head."},{"Start":"01:41.390 ","End":"01:44.510","Text":"You should write it in your formula sheet and understand what\u0027s going on."},{"Start":"01:44.510 ","End":"01:48.050","Text":"Again, this is like a normal determinant and these formulas"},{"Start":"01:48.050 ","End":"01:51.875","Text":"shouldn\u0027t be particularly unfamiliar but there are nonetheless important."},{"Start":"01:51.875 ","End":"01:55.355","Text":"Now let\u0027s talk about the meaning behind this whole thing."},{"Start":"01:55.355 ","End":"02:01.745","Text":"What the rotor is telling me is how much a given function turns around its origin."},{"Start":"02:01.745 ","End":"02:07.150","Text":"You can see here I\u0027ve written out two examples of formulas with a strong rotor."},{"Start":"02:07.150 ","End":"02:11.740","Text":"These are two formulas with none at all or very weak one in the very least."},{"Start":"02:11.740 ","End":"02:14.305","Text":"You can see here that if that\u0027s my center point,"},{"Start":"02:14.305 ","End":"02:15.805","Text":"if I perform the rotor,"},{"Start":"02:15.805 ","End":"02:19.120","Text":"I\u0027ll see that the function spins around very strongly,"},{"Start":"02:19.120 ","End":"02:22.840","Text":"very quickly, perhaps, or at least a lot around my origin."},{"Start":"02:22.840 ","End":"02:25.420","Text":"Then this function as well, if I graph it out,"},{"Start":"02:25.420 ","End":"02:30.805","Text":"there\u0027s some rotation happening that you can see around that central point."},{"Start":"02:30.805 ","End":"02:33.580","Text":"Of course, on the opposite side in this function,"},{"Start":"02:33.580 ","End":"02:35.200","Text":"nothing is spinning around the center."},{"Start":"02:35.200 ","End":"02:36.985","Text":"In fact, things are only going outwards."},{"Start":"02:36.985 ","End":"02:38.740","Text":"There\u0027s no rotation at all."},{"Start":"02:38.740 ","End":"02:40.615","Text":"Similarly, with this formula,"},{"Start":"02:40.615 ","End":"02:42.010","Text":"there\u0027s no rotation at all."},{"Start":"02:42.010 ","End":"02:44.495","Text":"Everything is going in a straight line to the right."},{"Start":"02:44.495 ","End":"02:48.920","Text":"What the rotor is really revealing to us is how strong the curvature is,"},{"Start":"02:48.920 ","End":"02:52.940","Text":"how strong the rotation is in a given function and how the field lines will show that."},{"Start":"02:52.940 ","End":"02:57.470","Text":"A good real-life example of this is a vortex, some sort of cyclone,"},{"Start":"02:57.470 ","End":"03:00.470","Text":"or something like that where you can clearly see strong rotation in"},{"Start":"03:00.470 ","End":"03:05.225","Text":"the middle of the function for this phenomenon."},{"Start":"03:05.225 ","End":"03:07.910","Text":"I\u0027ll just reiterate, it\u0027s important for you to understand"},{"Start":"03:07.910 ","End":"03:13.140","Text":"the mathematical operation here because we will use it quite a bit in this course."}],"ID":9619}],"Thumbnail":null,"ID":85323}]

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