1 00:00:00,650 --> 00:00:04,500 Er ... Let me talk today about projections. So the ... 2 00:00:05,400 --> 00:00:08,300 So our topic for today is projections. 3 00:00:10,500 --> 00:00:16,000 And the first appearance of projections is naturally in physics. 4 00:00:16,100 --> 00:00:17,950 When you look at ... 5 00:00:18,050 --> 00:00:22,450 When you are trying to solve a problem of, for example, pushing an object ... 6 00:00:22,550 --> 00:00:25,800 So if you want to push an object ... 7 00:00:26,900 --> 00:00:28,300 ... then you apply force. 8 00:00:28,400 --> 00:00:31,050 And we know force is a vector. 9 00:00:31,150 --> 00:00:33,450 And I'm applying that force ... 10 00:00:33,550 --> 00:00:39,300 ... and if I applied that force in one direction I'd fail to push the object. 11 00:00:39,400 --> 00:00:43,850 If I apply that force in another direction the object moves. 12 00:00:43,950 --> 00:00:44,900 And that's what I want. 13 00:00:45,000 --> 00:00:49,300 So the result depends on the direction of my vector of force. 14 00:00:49,400 --> 00:00:55,400 And from practice we know that the more my force is along the direction ... 15 00:00:55,500 --> 00:00:57,100 ... the better the effect is. 16 00:00:57,200 --> 00:00:59,400 And if it is perpendicular ... 17 00:00:59,500 --> 00:01:04,100 ... it's just a waste of my energy, waste of my effort. 18 00:01:05,200 --> 00:01:06,200 So er ... 19 00:01:08,250 --> 00:01:10,650 There are two things here to consider. 20 00:01:10,750 --> 00:01:18,550 One is a vector of the possible motion of the object. 21 00:01:18,650 --> 00:01:19,650 And ... 22 00:01:20,400 --> 00:01:28,700 Sometimes this vector is describing the only direction possible for the object to move. 23 00:01:28,800 --> 00:01:33,000 Like in the case of pushing a train. 24 00:01:33,100 --> 00:01:37,300 It is limited to the train tracks and never goes left or right. 25 00:01:37,400 --> 00:01:43,700 Er ... in case of pushing this eraser it will only go horizontally. 26 00:01:43,800 --> 00:01:47,200 Otherwise it will fall. So ... 27 00:01:49,300 --> 00:01:54,800 One vector describes the possible direction and the other vector ... 28 00:01:58,700 --> 00:02:00,600 Let's call it u. 29 00:02:02,250 --> 00:02:06,100 ... can be thought of as a force applied to the object. 30 00:02:06,200 --> 00:02:08,500 And then that force ... 31 00:02:09,250 --> 00:02:11,200 ... results in motion. 32 00:02:11,300 --> 00:02:14,300 And what you want to measure is ... 33 00:02:16,550 --> 00:02:19,850 ... a part of that force that ... 34 00:02:21,600 --> 00:02:25,600 ... acts on the object moving in the direction we want. 35 00:02:25,700 --> 00:02:28,400 So this is the ... 36 00:02:34,450 --> 00:02:39,400 ... part of the force that actually pushes the object as we want. 37 00:02:39,500 --> 00:02:40,500 And ... 38 00:02:43,250 --> 00:02:45,200 The rest of the force ... 39 00:02:48,350 --> 00:02:51,000 Well let me make it blue. Er ... 40 00:02:51,100 --> 00:02:55,300 ... is a waste of our effort. 41 00:03:03,000 --> 00:03:08,400 So our force is decomposed into the sum of two vectors. 42 00:03:08,500 --> 00:03:12,050 One is highly useful for the purpose. 43 00:03:12,150 --> 00:03:14,600 The other one is total waste. 44 00:03:14,700 --> 00:03:20,000 And what we want is to maximize the useful one, minimize the waste. 45 00:03:20,100 --> 00:03:24,350 Sometimes we cannot make the waste zero. 46 00:03:24,450 --> 00:03:28,850 Er ... but that's a question of technical limitations. 47 00:03:28,950 --> 00:03:34,900 So our purpose now is to compute both things. 48 00:03:35,000 --> 00:03:38,000 And first of all we will compute the useful part. 49 00:03:38,950 --> 00:03:43,750 So to compute that we will need of course that angle. 50 00:03:43,850 --> 00:03:45,850 Let's call it alpha. 51 00:03:45,950 --> 00:03:47,800 And then ... 52 00:03:49,250 --> 00:03:53,850 The magnitude of this useful part of the force ... 53 00:03:53,950 --> 00:03:58,000 ... is going to be called the component. 54 00:03:59,200 --> 00:04:02,700 This is the notation that we will use for the whole semester. 55 00:04:02,800 --> 00:04:11,000 Component of the vector u in the direction of the vector v. 56 00:04:13,700 --> 00:04:16,150 So that is going to be a number. 57 00:04:16,250 --> 00:04:17,250 And ... 58 00:04:17,550 --> 00:04:21,450 The value of that number is going to be ... 59 00:04:21,550 --> 00:04:23,500 ... from this triangle ... 60 00:04:23,600 --> 00:04:27,500 ... the magnitude of u times cosine of alpha. 61 00:04:36,950 --> 00:04:37,950 Now ... 62 00:04:39,250 --> 00:04:43,600 From computational point of view that formula is not highly useful. 63 00:04:43,700 --> 00:04:48,400 Because given two vectors u and v ... 64 00:04:49,000 --> 00:04:51,300 ... it is not easy to find the cosine ... 65 00:04:51,400 --> 00:04:56,300 ... unless you use the dot product and the formula we discussed last time. 66 00:04:56,400 --> 00:04:59,700 So the idea is to ... 67 00:05:00,650 --> 00:05:07,400 ... use ... well, to compute that quantity using the dot product of u and v. 68 00:05:07,500 --> 00:05:11,000 And to do that I would multiply and divide ... 69 00:05:11,500 --> 00:05:14,300 ... that quantity by the magnitude of v. 70 00:05:14,800 --> 00:05:19,550 Which doesn't change the quantity itself but the numerator ... 71 00:05:19,650 --> 00:05:22,000 ... now looks like the dot product. 72 00:05:23,000 --> 00:05:24,850 And actually it is the dot product. 73 00:05:24,950 --> 00:05:27,550 So it is dot product of u and v ... 74 00:05:27,650 --> 00:05:30,000 ... divided by magnitude of ... 75 00:05:31,000 --> 00:05:32,000 ... v. 76 00:05:32,450 --> 00:05:36,700 That's the simplest formula for computing ... 77 00:05:37,500 --> 00:05:42,800 ... the component of the force vector onto the given direction. 78 00:05:44,600 --> 00:05:53,750 Student: Can we [...]. 79 00:05:53,850 --> 00:05:57,200 Yes, you can do that. 80 00:05:57,300 --> 00:06:01,400 So if you just start with that formula ... 81 00:06:02,000 --> 00:06:06,100 ... u times cosine alpha and ... 82 00:06:09,250 --> 00:06:18,000 ... replace cosine with u dot v divided by magnitude u times magnitude v ... 83 00:06:19,600 --> 00:06:25,000 ... then u cancels and you arrive at the same formula. 84 00:06:27,800 --> 00:06:32,700 Well, so basically this is the formula that I suggest to memorize. 85 00:06:32,800 --> 00:06:36,750 And you will use it a lot this semester. 86 00:06:38,300 --> 00:06:41,000 Now let's talk a little about that formula. 87 00:06:41,100 --> 00:06:43,700 First of all that's a number. 88 00:06:43,800 --> 00:06:46,900 Although the whole thing contains a lot of vectors. 89 00:06:47,000 --> 00:06:51,450 The numerator is the dot product of two vectors resulting in a number. 90 00:06:51,550 --> 00:06:54,800 The denominator is the magnitude which is a number again. |

Video Lectures > Calculus III (2011 Summer) > Lecture 2011.06.15 Projection and Components of a Vector > Part 01 Formula for the Component of a Vector >