WEBVTT 00:00:01.000 --> 00:00:03.000 The following content is provided under a Creative 00:00:03.000 --> 00:00:05.000 Commons license. Your support will help MIT 00:00:05.000 --> 00:00:08.000 OpenCourseWare continue to offer high quality educational 00:00:08.000 --> 00:00:13.000 resources for free. To make a donation or to view 00:00:13.000 --> 00:00:18.000 additional materials from hundreds of MIT courses, 00:00:18.000 --> 00:00:23.000 visit MIT OpenCourseWare at ocw.mit.edu. 00:00:23.000 --> 00:00:25.000 So let's start right away with stuff that we will need to see 00:00:25.000 --> 00:00:28.000 before we can go on to more advanced things. 00:00:28.000 --> 00:00:31.000 So, hopefully yesterday in recitation, you heard a bit 00:00:31.000 --> 00:00:34.000 about vectors. How many of you actually knew 00:00:34.000 --> 00:00:39.000 about vectors before that? OK, that's the vast majority. 00:00:39.000 --> 00:00:42.000 If you are not one of those people, well, 00:00:42.000 --> 00:00:45.000 hopefully you'll learn about vectors right now. 00:00:45.000 --> 00:00:48.000 I'm sorry that the learning curve will be a bit steeper for 00:00:48.000 --> 00:00:50.000 the first week. But hopefully, 00:00:50.000 --> 00:00:55.000 you'll adjust fine. If you have trouble with 00:00:55.000 --> 00:00:59.000 vectors, do go to your recitation instructor's office 00:00:59.000 --> 00:01:03.000 hours for extra practice if you feel the need to. 00:01:03.000 --> 00:01:09.000 You will see it's pretty easy. So, just to remind you, 00:01:09.000 --> 00:01:18.000 a vector is a quantity that has both a direction and a magnitude 00:01:18.000 --> 00:01:20.000 of length. 00:01:33.000 --> 00:01:38.000 So -- So, concretely the way you draw a vector is by some 00:01:38.000 --> 00:01:40.000 arrow, like that, OK? 00:01:40.000 --> 00:01:43.000 And so, it has a length, and it's pointing in some 00:01:43.000 --> 00:01:45.000 direction. And, so, now, 00:01:45.000 --> 00:01:49.000 the way that we compute things with vectors, 00:01:49.000 --> 00:01:53.000 typically, as we introduce a coordinate system. 00:01:53.000 --> 00:01:57.000 So, if we are in the plane, x-y-axis, if we are in space, 00:01:57.000 --> 00:02:00.000 x-y-z axis. So, usually I will try to draw 00:02:00.000 --> 00:02:04.000 my x-y-z axis consistently to look like this. 00:02:04.000 --> 00:02:07.000 And then, I can represent my vector in terms of its 00:02:07.000 --> 00:02:10.000 components along the coordinate axis. 00:02:10.000 --> 00:02:13.000 So, that means when I have this row, I can ask, 00:02:13.000 --> 00:02:15.000 how much does it go in the x direction? 00:02:15.000 --> 00:02:17.000 How much does it go in the y direction? 00:02:17.000 --> 00:02:20.000 How much does it go in the z direction? 00:02:20.000 --> 00:02:25.000 And, so, let's call this a vector A. 00:02:25.000 --> 00:02:29.000 So, it's more convention. When we have a vector quantity, 00:02:29.000 --> 00:02:32.000 we put an arrow on top to remind us that it's a vector. 00:02:32.000 --> 00:02:35.000 If it's in the textbook, then sometimes it's in bold 00:02:35.000 --> 00:02:39.000 because it's easier to typeset. If you've tried in your 00:02:39.000 --> 00:02:44.000 favorite word processor, bold is easy and vectors are 00:02:44.000 --> 00:02:49.000 not easy. So, the vector you can try to 00:02:49.000 --> 00:02:56.000 decompose terms of unit vectors directed along the coordinate 00:02:56.000 --> 00:02:59.000 axis. So, the convention is there is 00:02:59.000 --> 00:03:03.000 a vector that we call ***amp***lt;i***amp***gt; 00:03:03.000 --> 00:03:08.000 hat that points along the x axis and has length one. 00:03:08.000 --> 00:03:10.000 There's a vector called ***amp***lt;j***amp***gt; 00:03:10.000 --> 00:03:12.000 hat that does the same along the y axis, 00:03:12.000 --> 00:03:14.000 and the ***amp***lt;k***amp***gt; 00:03:14.000 --> 00:03:16.000 hat that does the same along the z axis. 00:03:16.000 --> 00:03:20.000 And, so, we can express any vector in terms of its 00:03:20.000 --> 00:03:24.000 components. So, the other notation is 00:03:24.000 --> 00:03:29.000 ***amp***lt;a1, a2, a3 ***amp***gt; 00:03:29.000 --> 00:03:37.000 between these square brackets. Well, in angular brackets. 00:03:37.000 --> 00:03:42.000 So, the length of a vector we denote by, if you want, 00:03:42.000 --> 00:03:47.000 it's the same notation as the absolute value. 00:03:47.000 --> 00:03:50.000 So, that's going to be a number, as we say, 00:03:50.000 --> 00:03:54.000 now, a scalar quantity. OK, so, a scalar quantity is a 00:03:54.000 --> 00:03:58.000 usual numerical quantity as opposed to a vector quantity. 00:03:58.000 --> 00:04:08.000 And, its direction is sometimes called dir A, 00:04:08.000 --> 00:04:13.000 and that can be obtained just by scaling the vector down to 00:04:13.000 --> 00:04:17.000 unit length, for example, 00:04:17.000 --> 00:04:26.000 by dividing it by its length. So -- Well, there's a lot of 00:04:26.000 --> 00:04:32.000 notation to be learned. So, for example, 00:04:32.000 --> 00:04:37.000 if I have two points, P and Q, then I can draw a 00:04:37.000 --> 00:04:42.000 vector from P to Q. And, that vector is called 00:04:42.000 --> 00:04:46.000 vector PQ, OK? So, maybe we'll call it A. 00:04:46.000 --> 00:04:48.000 But, a vector doesn't really have, necessarily, 00:04:48.000 --> 00:04:50.000 a starting point and an ending point. 00:04:50.000 --> 00:04:54.000 OK, so if I decide to start here and I go by the same 00:04:54.000 --> 00:04:57.000 distance in the same direction, this is also vector A. 00:04:57.000 --> 00:05:04.000 It's the same thing. So, a lot of vectors we'll draw 00:05:04.000 --> 00:05:08.000 starting at the origin, but we don't have to. 00:05:08.000 --> 00:05:19.000 So, let's just check and see how things went in recitation. 00:05:19.000 --> 00:05:23.000 So, let's say that I give you the vector 00:05:23.000 --> 00:05:34.000 ***amp***lt;3,2,1***amp***gt;. And so, what do you think about 00:05:34.000 --> 00:05:46.000 the length of this vector? OK, I see an answer forming. 00:05:46.000 --> 00:05:49.000 So, a lot of you are answering the same thing. 00:05:49.000 --> 00:05:54.000 Maybe it shouldn't spoil it for those who haven't given it yet. 00:05:54.000 --> 00:05:59.000 OK, I think the overwhelming vote is in favor of answer 00:05:59.000 --> 00:06:02.000 number two. I see some sixes, I don't know. 00:06:02.000 --> 00:06:06.000 That's a perfectly good answer, too, but hopefully in a few 00:06:06.000 --> 00:06:10.000 minutes it won't be I don't know anymore. 00:06:10.000 --> 00:06:17.000 So, let's see. How do we find -- -- the length 00:06:17.000 --> 00:06:24.000 of a vector three, two, one? 00:06:24.000 --> 00:06:30.000 Well, so, this vector, A, it comes towards us along 00:06:30.000 --> 00:06:37.000 the x axis by three units. It goes to the right along the 00:06:37.000 --> 00:06:42.000 y axis by two units, and then it goes up by one unit 00:06:42.000 --> 00:06:46.000 along the z axis. OK, so, it's pointing towards 00:06:46.000 --> 00:06:51.000 here. That's pretty hard to draw. 00:06:51.000 --> 00:06:55.000 So, how do we get its length? Well, maybe we can start with 00:06:55.000 --> 00:06:58.000 something easier, the length of the vector in the 00:06:58.000 --> 00:07:01.000 plane. So, observe that A is obtained 00:07:01.000 --> 00:07:04.000 from a vector, B, in the plane. 00:07:04.000 --> 00:07:09.000 Say, B equals three (i) hat plus two (j) hat. 00:07:09.000 --> 00:07:15.000 And then, we just have to, still, go up by one unit, 00:07:15.000 --> 00:07:17.000 OK? So, let me try to draw a 00:07:17.000 --> 00:07:20.000 picture in this vertical plane that contains A and B. 00:07:20.000 --> 00:07:23.000 If I draw it in the vertical plane, 00:07:23.000 --> 00:07:27.000 so, that's the Z axis, that's not any particular axis, 00:07:27.000 --> 00:07:38.000 then my vector B will go here, and my vector A will go above 00:07:38.000 --> 00:07:43.000 it. And here, that's one unit. 00:07:43.000 --> 00:07:49.000 And, here I have a right angle. So, I can use the Pythagorean 00:07:49.000 --> 00:07:57.000 theorem to find that length A^2 equals length B^2 plus one. 00:07:57.000 --> 00:08:00.000 Now, we are reduced to finding the length of B. 00:08:00.000 --> 00:08:02.000 The length of B, we can again find using the 00:08:02.000 --> 00:08:06.000 Pythagorean theorem in the XY plane because here we have the 00:08:06.000 --> 00:08:09.000 right angle. Here we have three units, 00:08:09.000 --> 00:08:12.000 and here we have two units. OK, so, if you do the 00:08:12.000 --> 00:08:15.000 calculations, you will see that, 00:08:15.000 --> 00:08:18.000 well, length of B is square root of (3^2 2^2), 00:08:18.000 --> 00:08:23.000 that's 13. So, the square root of 13 -- -- 00:08:23.000 --> 00:08:32.000 and length of A is square root of length B^2 plus one (square 00:08:32.000 --> 00:08:41.000 it if you want) which is going to be square root of 13 plus one 00:08:41.000 --> 00:08:49.000 is the square root of 14, hence, answer number two which 00:08:49.000 --> 00:08:54.000 almost all of you gave. OK, so the general formula, 00:08:54.000 --> 00:09:02.000 if you follow it with it, in general if we have a vector 00:09:02.000 --> 00:09:07.000 with components a1, a2, a3, 00:09:07.000 --> 00:09:16.000 then the length of A is the square root of a1^2 plus a2^2 00:09:16.000 --> 00:09:23.000 plus a3^2. OK, any questions about that? 00:09:23.000 --> 00:09:29.000 Yes? Yes. 00:09:29.000 --> 00:09:32.000 So, in general, we indeed can consider vectors 00:09:32.000 --> 00:09:36.000 in abstract spaces that have any number of coordinates. 00:09:36.000 --> 00:09:38.000 And that you have more components. 00:09:38.000 --> 00:09:40.000 In this class, we'll mostly see vectors with 00:09:40.000 --> 00:09:44.000 two or three components because they are easier to draw, 00:09:44.000 --> 00:09:47.000 and because a lot of the math that we'll see works exactly the 00:09:47.000 --> 00:09:50.000 same way whether you have three variables or a million 00:09:50.000 --> 00:09:52.000 variables. If we had a factor with more 00:09:52.000 --> 00:09:55.000 components, then we would have a lot of trouble drawing it. 00:09:55.000 --> 00:09:58.000 But we could still define its length in the same way, 00:09:58.000 --> 00:10:01.000 by summing the squares of the components. 00:10:01.000 --> 00:10:04.000 So, I'm sorry to say that here, multi-variable, 00:10:04.000 --> 00:10:07.000 multi will mean mostly two or three. 00:10:07.000 --> 00:10:13.000 But, be assured that it works just the same way if you have 00:10:13.000 --> 00:10:20.000 10,000 variables. Just, calculations are longer. 00:10:20.000 --> 00:10:28.000 OK, more questions? So, what else can we do with 00:10:28.000 --> 00:10:31.000 vectors? Well, another thing that I'm 00:10:31.000 --> 00:10:35.000 sure you know how to do with vectors is to add them to scale 00:10:35.000 --> 00:10:39.000 them. So, vector addition, 00:10:39.000 --> 00:10:48.000 so, if you have two vectors, A and B, then you can form, 00:10:48.000 --> 00:10:52.000 their sum, A plus B. How do we do that? 00:10:52.000 --> 00:10:54.000 Well, first, I should tell you, 00:10:54.000 --> 00:10:56.000 vectors, they have this double life. 00:10:56.000 --> 00:10:59.000 They are, at the same time, geometric objects that we can 00:10:59.000 --> 00:11:02.000 draw like this in pictures, and there are also 00:11:02.000 --> 00:11:06.000 computational objects that we can represent by numbers. 00:11:06.000 --> 00:11:09.000 So, every question about vectors will have two answers, 00:11:09.000 --> 00:11:11.000 one geometric, and one numerical. 00:11:11.000 --> 00:11:14.000 OK, so let's start with the geometric. 00:11:14.000 --> 00:11:17.000 So, let's say that I have two vectors, A and B, 00:11:17.000 --> 00:11:21.000 given to me. And, let's say that I thought 00:11:21.000 --> 00:11:24.000 of drawing them at the same place to start with. 00:11:24.000 --> 00:11:28.000 Well, to take the sum, what I should do is actually 00:11:28.000 --> 00:11:33.000 move B so that it starts at the end of A, at the head of A. 00:11:33.000 --> 00:11:38.000 OK, so this is, again, vector B. So, observe, 00:11:38.000 --> 00:11:41.000 this actually forms, now, a parallelogram, 00:11:41.000 --> 00:11:43.000 right? So, this side is, 00:11:43.000 --> 00:11:48.000 again, vector A. And now, if we take the 00:11:48.000 --> 00:11:57.000 diagonal of that parallelogram, this is what we call A plus B, 00:11:57.000 --> 00:12:00.000 OK, so, the idea being that to move along A plus B, 00:12:00.000 --> 00:12:03.000 it's the same as to move first along A and then along B, 00:12:03.000 --> 00:12:09.000 or, along B, then along A. A plus B equals B plus A. 00:12:09.000 --> 00:12:13.000 OK, now, if we do it numerically, 00:12:13.000 --> 00:12:19.000 then all you do is you just add the first component of A with 00:12:19.000 --> 00:12:23.000 the first component of B, the second with the second, 00:12:23.000 --> 00:12:28.000 and the third with the third. OK, say that A was 00:12:28.000 --> 00:12:31.000 ***amp***lt;a1, a2, a3***amp***gt; 00:12:31.000 --> 00:12:35.000 B was ***amp***lt;b1, b2, b3***amp***gt;, 00:12:35.000 --> 00:12:40.000 then you just add this way. OK, so it's pretty 00:12:40.000 --> 00:12:44.000 straightforward. So, for example, 00:12:44.000 --> 00:12:48.000 I said that my vector over there, its components are three, 00:12:48.000 --> 00:12:54.000 two, one. But, I also wrote it as 3i 2j k. 00:12:54.000 --> 00:12:57.000 What does that mean? OK, so I need to tell you first 00:12:57.000 --> 00:13:06.000 about multiplying by a scalar. So, this is about addition. 00:13:06.000 --> 00:13:11.000 So, multiplication by a scalar, it's very easy. 00:13:11.000 --> 00:13:15.000 If you have a vector, A, then you can form a vector 00:13:15.000 --> 00:13:20.000 2A just by making it go twice as far in the same direction. 00:13:20.000 --> 00:13:24.000 Or, we can make half A more modestly. 00:13:24.000 --> 00:13:31.000 We can even make minus A, and so on. 00:13:31.000 --> 00:13:35.000 So now, you see, if I do the calculation, 00:13:35.000 --> 00:13:38.000 3i 2j k, well, what does it mean? 00:13:38.000 --> 00:13:43.000 3i is just going to go along the x axis, but by distance of 00:13:43.000 --> 00:13:47.000 three instead of one. And then, 2j goes two units 00:13:47.000 --> 00:13:51.000 along the y axis, and k goes up by one unit. 00:13:51.000 --> 00:13:54.000 Well, if you add these together, you will go from the 00:13:54.000 --> 00:13:58.000 origin, then along the x axis, then parallel to the y axis, 00:13:58.000 --> 00:14:02.000 and then up. And, you will end up, 00:14:02.000 --> 00:14:05.000 indeed, at the endpoint of a vector. 00:14:05.000 --> 00:14:19.000 OK, any questions at this point? Yes? 00:14:19.000 --> 00:14:21.000 Exactly. To add vectors geometrically, 00:14:21.000 --> 00:14:25.000 you just put the head of the first vector and the tail of the 00:14:25.000 --> 00:14:30.000 second vector in the same place. And then, it's head to tail 00:14:30.000 --> 00:14:35.000 addition. Any other questions? 00:14:35.000 --> 00:14:41.000 Yes? That's correct. 00:14:41.000 --> 00:14:43.000 If you subtract two vectors, that just means you add the 00:14:43.000 --> 00:14:45.000 opposite of a vector. So, for example, 00:14:45.000 --> 00:14:49.000 if I wanted to do A minus B, I would first go along A and 00:14:49.000 --> 00:14:52.000 then along minus B, which would take me somewhere 00:14:52.000 --> 00:14:55.000 over there, OK? So, A minus B, 00:14:55.000 --> 00:15:01.000 if you want, would go from here to here. 00:15:01.000 --> 00:15:08.000 OK, so hopefully you've kind of seen that stuff either before in 00:15:08.000 --> 00:15:13.000 your lives, or at least yesterday. 00:15:13.000 --> 00:15:23.000 So, I'm going to use that as an excuse to move quickly forward. 00:15:23.000 --> 00:15:28.000 So, now we are going to learn a few more operations about 00:15:28.000 --> 00:15:31.000 vectors. And, these operations will be 00:15:31.000 --> 00:15:34.000 useful to us when we start trying to do a bit of geometry. 00:15:34.000 --> 00:15:37.000 So, of course, you've all done some geometry. 00:15:37.000 --> 00:15:40.000 But, we are going to see that geometry can be done using 00:15:40.000 --> 00:15:42.000 vectors. And, in many ways, 00:15:42.000 --> 00:15:44.000 it's the right language for that, 00:15:44.000 --> 00:15:47.000 and in particular when we learn about functions we really will 00:15:47.000 --> 00:15:51.000 want to use vectors more than, maybe, the other kind of 00:15:51.000 --> 00:15:54.000 geometry that you've seen before. 00:15:54.000 --> 00:15:56.000 I mean, of course, it's just a language in a way. 00:15:56.000 --> 00:15:59.000 I mean, we are just reformulating things that you 00:15:59.000 --> 00:16:02.000 have seen, you already know since childhood. 00:16:02.000 --> 00:16:07.000 But, you will see that notation somehow helps to make it more 00:16:07.000 --> 00:16:10.000 straightforward. So, what is dot product? 00:16:10.000 --> 00:16:16.000 Well, dot product as a way of multiplying two vectors to get a 00:16:16.000 --> 00:16:21.000 number, a scalar. And, well, let me start by 00:16:21.000 --> 00:16:25.000 giving you a definition in terms of components. 00:16:25.000 --> 00:16:29.000 What we do, let's say that we have a vector, 00:16:29.000 --> 00:16:32.000 A, with components a1, a2, a3, vector B with 00:16:32.000 --> 00:16:34.000 components b1, b2, b3. 00:16:34.000 --> 00:16:38.000 Well, we multiply the first components by the first 00:16:38.000 --> 00:16:43.000 components, the second by the second, the third by the third. 00:16:43.000 --> 00:16:46.000 If you have N components, you keep going. 00:16:46.000 --> 00:16:49.000 And, you sum all of these together. 00:16:49.000 --> 00:16:55.000 OK, and important: this is a scalar. 00:16:55.000 --> 00:16:59.000 OK, you do not get a vector. You get a number. 00:16:59.000 --> 00:17:01.000 I know it sounds completely obvious from the definition 00:17:01.000 --> 00:17:03.000 here, but in the middle of the action 00:17:03.000 --> 00:17:07.000 when you're going to do complicated problems, 00:17:07.000 --> 00:17:14.000 it's sometimes easy to forget. So, that's the definition. 00:17:14.000 --> 00:17:17.000 What is it good for? Why would we ever want to do 00:17:17.000 --> 00:17:20.000 that? That's kind of a strange 00:17:20.000 --> 00:17:23.000 operation. So, probably to see what it's 00:17:23.000 --> 00:17:27.000 good for, I should first tell you what it is geometrically. 00:17:27.000 --> 00:17:29.000 OK, so what does it do geometrically? 00:17:38.000 --> 00:17:42.000 Well, what you do when you multiply two vectors in this 00:17:42.000 --> 00:17:45.000 way, I claim the answer is equal to 00:17:45.000 --> 00:17:51.000 the length of A times the length of B times the cosine of the 00:17:51.000 --> 00:17:59.000 angle between them. So, I have my vector, A, 00:17:59.000 --> 00:18:04.000 and if I have my vector, B, and I have some angle between 00:18:04.000 --> 00:18:06.000 them, I multiply the length of A 00:18:06.000 --> 00:18:10.000 times the length of B times the cosine of that angle. 00:18:10.000 --> 00:18:13.000 So, that looks like a very artificial operation. 00:18:13.000 --> 00:18:16.000 I mean, why would want to do that complicated multiplication? 00:18:16.000 --> 00:18:21.000 Well, the basic answer is it tells us at the same time about 00:18:21.000 --> 00:18:25.000 lengths and about angles. And, the extra bonus thing is 00:18:25.000 --> 00:18:29.000 that it's very easy to compute if you have components, 00:18:29.000 --> 00:18:32.000 see, that formula is actually pretty easy. 00:18:32.000 --> 00:18:39.000 So, OK, maybe I should first tell you, how do we get this 00:18:39.000 --> 00:18:41.000 from that? Because, you know, 00:18:41.000 --> 00:18:44.000 in math, one tries to justify everything to prove theorems. 00:18:44.000 --> 00:18:45.000 So, if you want, that's the theorem. 00:18:45.000 --> 00:18:47.000 That's the first theorem in 18.02. 00:18:47.000 --> 00:18:52.000 So, how do we prove the theorem? How do we check that this is, 00:18:52.000 --> 00:18:55.000 indeed, correct using this definition? 00:18:55.000 --> 00:19:06.000 So, in more common language, what does this geometric 00:19:06.000 --> 00:19:11.000 definition mean? Well, the first thing it means, 00:19:11.000 --> 00:19:14.000 before we multiply two vectors, let's start multiplying a 00:19:14.000 --> 00:19:17.000 vector with itself. That's probably easier. 00:19:17.000 --> 00:19:19.000 So, if we multiply a vector, A, with itself, 00:19:19.000 --> 00:19:22.000 using this dot product, so, by the way, 00:19:22.000 --> 00:19:24.000 I should point out, we put this dot here. 00:19:24.000 --> 00:19:28.000 That's why it's called dot product. 00:19:28.000 --> 00:19:33.000 So, what this tells us is we should get the same thing as 00:19:33.000 --> 00:19:38.000 multiplying the length of A with itself, so, squared, 00:19:38.000 --> 00:19:43.000 times the cosine of the angle. But now, the cosine of an 00:19:43.000 --> 00:19:49.000 angle, of zero, cosine of zero you all know is 00:19:49.000 --> 00:19:52.000 one. OK, so that's going to be 00:19:52.000 --> 00:19:56.000 length A^2. Well, doesn't stand a chance of 00:19:56.000 --> 00:19:57.000 being true? Well, let's see. 00:19:57.000 --> 00:20:03.000 If we do AdotA using this formula, we will get a1^2 a2^2 00:20:03.000 --> 00:20:07.000 a3^2. That is, indeed, 00:20:07.000 --> 00:20:14.000 the square of the length. So, check. 00:20:14.000 --> 00:20:18.000 That works. OK, now, what about two 00:20:18.000 --> 00:20:23.000 different vectors? Can we understand what this 00:20:23.000 --> 00:20:27.000 says, and how it relates to that? 00:20:27.000 --> 00:20:33.000 So, let's say that I have two different vectors, 00:20:33.000 --> 00:20:40.000 A and B, and I want to try to understand what's going on. 00:20:40.000 --> 00:20:45.000 So, my claim is that we are going to be able to understand 00:20:45.000 --> 00:20:49.000 the relation between this and that in terms of the law of 00:20:49.000 --> 00:20:52.000 cosines. So, the law of cosines is 00:20:52.000 --> 00:20:56.000 something that tells you about the length of the third side in 00:20:56.000 --> 00:21:00.000 the triangle like this in terms of these two sides, 00:21:00.000 --> 00:21:07.000 and the angle here. OK, so the law of cosines, 00:21:07.000 --> 00:21:11.000 which hopefully you have seen before, says that, 00:21:11.000 --> 00:21:14.000 so let me give a name to this side. 00:21:14.000 --> 00:21:19.000 Let's call this side C, and as a vector, 00:21:19.000 --> 00:21:29.000 C is A minus B. It's minus B plus A. 00:21:29.000 --> 00:21:37.000 So, it's getting a bit cluttered here. 00:21:37.000 --> 00:21:45.000 So, the law of cosines says that the length of the third 00:21:45.000 --> 00:21:53.000 side in this triangle is equal to length A2 plus length B2. 00:21:53.000 --> 00:21:56.000 Well, if I stopped here, that would be Pythagoras, 00:21:56.000 --> 00:22:01.000 but I don't have a right angle. So, I have a third term which 00:22:01.000 --> 00:22:07.000 is twice length A, length B, cosine theta, 00:22:07.000 --> 00:22:10.000 OK? Has everyone seen this formula 00:22:10.000 --> 00:22:13.000 sometime? I hear some yeah's. 00:22:13.000 --> 00:22:16.000 I hear some no's. Well, it's a fact about, 00:22:16.000 --> 00:22:19.000 I mean, you probably haven't seen it with vectors, 00:22:19.000 --> 00:22:22.000 but it's a fact about the side lengths in a triangle. 00:22:22.000 --> 00:22:27.000 And, well, let's say, if you haven't seen it before, 00:22:27.000 --> 00:22:32.000 then this is going to be a proof of the law of cosines if 00:22:32.000 --> 00:22:39.000 you believe this. Otherwise, it's the other way 00:22:39.000 --> 00:22:43.000 around. So, let's try to see how this 00:22:43.000 --> 00:22:47.000 relates to what I'm saying about the dot product. 00:22:47.000 --> 00:22:54.000 So, I've been saying that length C^2, that's the same 00:22:54.000 --> 00:22:56.000 thing as CdotC, OK? 00:22:56.000 --> 00:23:01.000 That, we have checked. Now, CdotC, well, 00:23:01.000 --> 00:23:06.000 C is A minus B. So, it's A minus B, 00:23:06.000 --> 00:23:09.000 dot product, A minus B. 00:23:09.000 --> 00:23:11.000 Now, what do we want to do in a situation like that? 00:23:11.000 --> 00:23:16.000 Well, we want to expand this into a sum of four terms. 00:23:16.000 --> 00:23:19.000 Are we allowed to do that? Well, we have this dot product 00:23:19.000 --> 00:23:22.000 that's a mysterious new operation. 00:23:22.000 --> 00:23:24.000 We don't really know. Well, the answer is yes, 00:23:24.000 --> 00:23:27.000 we can do it. You can check from this 00:23:27.000 --> 00:23:31.000 definition that it behaves in the usual way in terms of 00:23:31.000 --> 00:23:34.000 expanding, vectoring, and so on. 00:23:34.000 --> 00:23:49.000 So, I can write that as AdotA minus AdotB minus BdotA plus 00:23:49.000 --> 00:23:55.000 BdotB. So, AdotA is length A^2. 00:23:55.000 --> 00:23:56.000 Let me jump ahead to the last term. 00:23:56.000 --> 00:24:01.000 BdotB is length B^2, and then these two terms, 00:24:01.000 --> 00:24:04.000 well, they're the same. You can check from the 00:24:04.000 --> 00:24:07.000 definition that AdotB and BdotA are the same thing. 00:24:20.000 --> 00:24:24.000 Well, you see that this term, I mean, this is the only 00:24:24.000 --> 00:24:30.000 difference between these two formulas for the length of C. 00:24:30.000 --> 00:24:34.000 So, if you believe in the law of cosines, then it tells you 00:24:34.000 --> 00:24:39.000 that, yes, this a proof that AdotB equals length A length B 00:24:39.000 --> 00:24:41.000 cosine theta. Or, vice versa, 00:24:41.000 --> 00:24:45.000 if you've never seen the law of cosines, you are willing to 00:24:45.000 --> 00:24:49.000 believe this. Then, this is the proof of the 00:24:49.000 --> 00:24:53.000 law of cosines. So, the law of cosines, 00:24:53.000 --> 00:24:59.000 or this interpretation, are equivalent to each other. 00:24:59.000 --> 00:25:07.000 OK, any questions? Yes? 00:25:07.000 --> 00:25:12.000 So, in the second one there isn't a cosine theta because I'm 00:25:12.000 --> 00:25:16.000 just expanding a dot product. OK, so I'm just writing C 00:25:16.000 --> 00:25:19.000 equals A minus B, and then I'm expanding this 00:25:19.000 --> 00:25:22.000 algebraically. And then, I get to an answer 00:25:22.000 --> 00:25:24.000 that has an A.B. So then, if I wanted to express 00:25:24.000 --> 00:25:27.000 that without a dot product, then I would have to introduce 00:25:27.000 --> 00:25:31.000 a cosine. And, I would get the same as 00:25:31.000 --> 00:25:34.000 that, OK? So, yeah, if you want, 00:25:34.000 --> 00:25:38.000 the next step to recall the law of cosines would be plug in this 00:25:38.000 --> 00:25:43.000 formula for AdotB. And then you would have a 00:25:43.000 --> 00:25:58.000 cosine. OK, let's keep going. 00:25:58.000 --> 00:26:03.000 OK, so what is this good for? Now that we have a definition, 00:26:03.000 --> 00:26:06.000 we should figure out what we can do with it. 00:26:06.000 --> 00:26:11.000 So, what are the applications of dot product? 00:26:11.000 --> 00:26:14.000 Well, will this discover new applications of dot product 00:26:14.000 --> 00:26:17.000 throughout the entire semester,but let me tell you at 00:26:17.000 --> 00:26:20.000 least about those that are readily visible. 00:26:20.000 --> 00:26:33.000 So, one is to compute lengths and angles, especially angles. 00:26:33.000 --> 00:26:39.000 So, let's do an example. Let's say that, 00:26:39.000 --> 00:26:44.000 for example, I have in space, 00:26:44.000 --> 00:26:51.000 I have a point, P, which is at (1,0,0). 00:26:51.000 --> 00:26:55.000 I have a point, Q, which is at (0,1,0). 00:26:55.000 --> 00:26:58.000 So, it's at distance one here, one here. 00:26:58.000 --> 00:27:03.000 And, I have a third point, R at (0,0,2), 00:27:03.000 --> 00:27:07.000 so it's at height two. And, let's say that I'm 00:27:07.000 --> 00:27:11.000 curious, and I'm wondering what is the angle here? 00:27:11.000 --> 00:27:15.000 So, here I have a triangle in space connect P, 00:27:15.000 --> 00:27:20.000 Q, and R, and I'm wondering, what is this angle here? 00:27:20.000 --> 00:27:23.000 OK, so, of course, one solution is to build a 00:27:23.000 --> 00:27:25.000 model and then go and measure the angle. 00:27:25.000 --> 00:27:28.000 But, we can do better than that. We can just find the angle 00:27:28.000 --> 00:27:32.000 using dot product. So, how would we do that? 00:27:32.000 --> 00:27:38.000 Well, so, if we look at this formula, we see, 00:27:38.000 --> 00:27:44.000 so, let's say that we want to find the angle here. 00:27:44.000 --> 00:27:50.000 Well, let's look at the formula for PQdotPR. 00:27:50.000 --> 00:27:56.000 Well, we said it should be length PQ times length PR times 00:27:56.000 --> 00:27:59.000 the cosine of the angle, OK? 00:27:59.000 --> 00:28:01.000 Now, what do we know, and what do we not know? 00:28:01.000 --> 00:28:04.000 Well, certainly at this point we don't know the cosine of the 00:28:04.000 --> 00:28:06.000 angle. That's what we would like to 00:28:06.000 --> 00:28:08.000 find. The lengths, 00:28:08.000 --> 00:28:11.000 certainly we can compute. We know how to find these 00:28:11.000 --> 00:28:14.000 lengths. And, this dot product we know 00:28:14.000 --> 00:28:17.000 how to compute because we have an easy formula here. 00:28:17.000 --> 00:28:20.000 OK, so we can compute everything else and then find 00:28:20.000 --> 00:28:25.000 theta. So, I'll tell you what we will 00:28:25.000 --> 00:28:31.000 do is we will find theta -- -- in this way. 00:28:31.000 --> 00:28:34.000 We'll take the dot product of PQ with PR, and then we'll 00:28:34.000 --> 00:28:36.000 divide by the lengths. 00:29:14.000 --> 00:29:27.000 OK, so let's see. So, we said cosine theta is 00:29:27.000 --> 00:29:33.000 PQdotPR over length PQ length PR. 00:29:33.000 --> 00:29:36.000 So, let's try to figure out what this vector, 00:29:36.000 --> 00:29:39.000 PQ, well, to go from P to Q, 00:29:39.000 --> 00:29:43.000 I should go minus one unit along the x direction plus one 00:29:43.000 --> 00:29:46.000 unit along the y direction. And, I'm not moving in the z 00:29:46.000 --> 00:29:49.000 direction. So, to go from P to Q, 00:29:49.000 --> 00:29:54.000 I have to move by ***amp***lt;-1,1,0***amp***gt;. 00:29:54.000 --> 00:29:59.000 To go from P to R, I go -1 along the x axis and 2 00:29:59.000 --> 00:30:04.000 along the z axis. So, PR, I claim, is this. 00:30:04.000 --> 00:30:12.000 OK, then, the lengths of these vectors, well,(-1)^2 (1)^2 00:30:12.000 --> 00:30:19.000 (0)^2, square root, and then same thing with the 00:30:19.000 --> 00:30:24.000 other one. OK, so, the denominator will 00:30:24.000 --> 00:30:30.000 become the square root of 2, and there's a square root of 5. 00:30:30.000 --> 00:30:34.000 What about the numerator? Well, so, remember, 00:30:34.000 --> 00:30:37.000 to do the dot product, we multiply this by this, 00:30:37.000 --> 00:30:40.000 and that by that, that by that. 00:30:40.000 --> 00:30:45.000 And, we add. Minus 1 times minus 1 makes 1 00:30:45.000 --> 00:30:49.000 plus 1 times 0, that's 0. 00:30:49.000 --> 00:30:55.000 Zero times 2 is 0 again. So, we will get 1 over square 00:30:55.000 --> 00:30:59.000 root of 10. That's the cosine of the angle. 00:30:59.000 --> 00:31:03.000 And, of course if we want the actual angle, 00:31:03.000 --> 00:31:08.000 well, we have to take a calculator, find the inverse 00:31:08.000 --> 00:31:12.000 cosine, and you'll find it's about 71.5°. 00:31:12.000 --> 00:31:18.000 Actually, we'll be using mostly radians, but for today, 00:31:18.000 --> 00:31:26.000 that's certainly more speaking. OK, any questions about that? 00:31:26.000 --> 00:31:29.000 No? OK, so in particular, 00:31:29.000 --> 00:31:32.000 I should point out one thing that's really neat about the 00:31:32.000 --> 00:31:34.000 answer. I mean, we got this number. 00:31:34.000 --> 00:31:37.000 We don't really know what it means exactly because it mixes 00:31:37.000 --> 00:31:39.000 together the lengths and the angle. 00:31:39.000 --> 00:31:41.000 But, one thing that's interesting here, 00:31:41.000 --> 00:31:45.000 it's the sign of the answer, the fact that we got a positive 00:31:45.000 --> 00:31:48.000 number. So, if you think about it, 00:31:48.000 --> 00:31:50.000 the lengths are always positive. 00:31:50.000 --> 00:31:56.000 So, the sign of a dot product is the same as a sign of cosine 00:31:56.000 --> 00:32:00.000 theta. So, in fact, 00:32:00.000 --> 00:32:13.000 the sign of AdotB is going to be positive if the angle is less 00:32:13.000 --> 00:32:17.000 than 90°. So, that means geometrically, 00:32:17.000 --> 00:32:21.000 my two vectors are going more or less in the same direction. 00:32:21.000 --> 00:32:27.000 They make an acute angle. It's going to be zero if the 00:32:27.000 --> 00:32:33.000 angle is exactly 90°, OK, because that's when the 00:32:33.000 --> 00:32:39.000 cosine will be zero. And, it will be negative if the 00:32:39.000 --> 00:32:43.000 angle is more than 90°. So, that means they go, 00:32:43.000 --> 00:32:46.000 however, in opposite directions. 00:32:46.000 --> 00:32:50.000 So, that's basically one way to think about what dot product 00:32:50.000 --> 00:32:54.000 measures. It measures how much the two 00:32:54.000 --> 00:32:58.000 vectors are going along each other. 00:32:58.000 --> 00:33:02.000 OK, and that actually leads us to the next application. 00:33:02.000 --> 00:33:05.000 So, let's see, did I have a number one there? 00:33:05.000 --> 00:33:07.000 Yes. So, if I had a number one, 00:33:07.000 --> 00:33:12.000 I must have number two. The second application is to 00:33:12.000 --> 00:33:16.000 detect orthogonality. It's to figure out when two 00:33:16.000 --> 00:33:21.000 things are perpendicular. OK, so orthogonality is just a 00:33:21.000 --> 00:33:26.000 complicated word from Greek to say things are perpendicular. 00:33:26.000 --> 00:33:34.000 So, let's just take an example. Let's say I give you the 00:33:34.000 --> 00:33:41.000 equation x 2y 3z = 0. OK, so that defines a certain 00:33:41.000 --> 00:33:46.000 set of points in space, and what do you think the set 00:33:46.000 --> 00:33:52.000 of solutions look like if I give you this equation? 00:33:52.000 --> 00:34:01.000 So far I see one, two, three answers, 00:34:01.000 --> 00:34:06.000 OK. So, I see various competing 00:34:06.000 --> 00:34:11.000 answers, but, yeah, I see a lot of people 00:34:11.000 --> 00:34:18.000 voting for answer number four. I see also some I don't knows, 00:34:18.000 --> 00:34:22.000 and some other things. But, the majority vote seems to 00:34:22.000 --> 00:34:26.000 be a plane. And, indeed that's the correct 00:34:26.000 --> 00:34:28.000 answer. So, how do we see that it's a 00:34:28.000 --> 00:34:28.000 plane? 00:34:43.000 --> 00:34:49.000 So, I should say, this is the equation of a 00:34:49.000 --> 00:34:52.000 plane. So, there's many ways to see 00:34:52.000 --> 00:34:55.000 that, and I'm not going to give you all of them. 00:34:55.000 --> 00:34:58.000 But, here's one way to think about it. 00:34:58.000 --> 00:35:03.000 So, let's think geometrically about how to express this 00:35:03.000 --> 00:35:09.000 condition in terms of vectors. So, let's take the origin O, 00:35:09.000 --> 00:35:13.000 by convention is the point (0,0,0). 00:35:13.000 --> 00:35:18.000 And, let's take a point, P, that will satisfy this 00:35:18.000 --> 00:35:21.000 equation on it, so, at coordinates x, 00:35:21.000 --> 00:35:24.000 y, z. So, what does this condition 00:35:24.000 --> 00:35:28.000 here mean? Well, it means the following 00:35:28.000 --> 00:35:32.000 thing. So, let's take the vector, OP. 00:35:32.000 --> 00:35:37.000 OK, so vector OP, of course, has components x, 00:35:37.000 --> 00:35:40.000 y, z. Now, we can think of this as 00:35:40.000 --> 00:35:44.000 actually a dot product between OP and a mysterious vector that 00:35:44.000 --> 00:35:47.000 won't remain mysterious for very long, 00:35:47.000 --> 00:35:50.000 namely, the vector one, two, three. 00:35:50.000 --> 00:35:59.000 OK, so, this condition is the same as OP.A equals zero, 00:35:59.000 --> 00:36:03.000 right? If I take the dot product 00:36:03.000 --> 00:36:09.000 OPdotA I get x times one plus y times two plus z times three. 00:36:09.000 --> 00:36:14.000 But now, what does it mean that the dot product between OP and A 00:36:14.000 --> 00:36:19.000 is zero? Well, it means that OP and A 00:36:19.000 --> 00:36:25.000 are perpendicular. OK, so I have this vector, A. 00:36:25.000 --> 00:36:28.000 I'm not going to be able to draw it realistically. 00:36:28.000 --> 00:36:32.000 Let's say it goes this way. Then, a point, 00:36:32.000 --> 00:36:37.000 P, solves this equation exactly when the vector from O to P is 00:36:37.000 --> 00:36:40.000 perpendicular to A. And, I claim that defines a 00:36:40.000 --> 00:36:41.000 plane. For example, 00:36:41.000 --> 00:36:45.000 if it helps you to see it, take a vertical vector. 00:36:45.000 --> 00:36:47.000 What does it mean to be perpendicular to the vertical 00:36:47.000 --> 00:36:49.000 vector? It means you are horizontal. 00:36:49.000 --> 00:36:56.000 It's the horizontal plane. Here, it's a plane that passes 00:36:56.000 --> 00:37:05.000 through the origin and is perpendicular to this vector, 00:37:05.000 --> 00:37:14.000 A. OK, so what we get is a plane 00:37:14.000 --> 00:37:25.000 through the origin perpendicular to A. 00:37:25.000 --> 00:37:29.000 And, in general, what you should remember is 00:37:29.000 --> 00:37:35.000 that two vectors have a dot product equal to zero if and 00:37:35.000 --> 00:37:41.000 only if that's equivalent to the cosine of the angle between them 00:37:41.000 --> 00:37:46.000 is zero. That means the angle is 90°. 00:37:46.000 --> 00:37:51.000 That means A and B are perpendicular. 00:37:51.000 --> 00:37:57.000 So, we have a very fast way of checking whether two vectors are 00:37:57.000 --> 00:38:01.000 perpendicular. So, one additional application 00:38:01.000 --> 00:38:05.000 I think we'll see actually tomorrow is to find the 00:38:05.000 --> 00:38:10.000 components of a vector along a certain direction. 00:38:10.000 --> 00:38:13.000 So, I claim we can use this intuition I gave about dot 00:38:13.000 --> 00:38:16.000 product telling us how much to vectors go in the same direction 00:38:16.000 --> 00:38:19.000 to actually give a precise meaning to the notion of 00:38:19.000 --> 00:38:22.000 component for vector, not just along the x, 00:38:22.000 --> 00:38:27.000 y, or z axis, but along any direction in 00:38:27.000 --> 00:38:31.000 space. So, I think I should probably 00:38:31.000 --> 00:38:34.000 stop here. But, I will see you tomorrow at 00:38:34.000 --> 00:38:38.000 2:00 here, and we'll learn more about that and about cross 00:38:38.000 --> 00:38:44.000 products.