1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high-quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,210 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,210 --> 00:00:17,862 at ocw.mit.edu. 8 00:00:34,880 --> 00:00:35,690 PROFESSOR: Hi. 9 00:00:35,690 --> 00:00:39,080 In the last unit, we introduced the notion 10 00:00:39,080 --> 00:00:44,210 of the structure of vectors in terms of arrows in the plane. 11 00:00:44,210 --> 00:00:49,740 And today, we want to talk about the added geometric property 12 00:00:49,740 --> 00:00:52,800 of what I could call three-dimensional vectors 13 00:00:52,800 --> 00:00:55,040 or three-dimensional arrows. 14 00:00:55,040 --> 00:00:57,750 Now before I say that-- I guess I've already said it-- 15 00:00:57,750 --> 00:00:59,470 but before I say any more about it, 16 00:00:59,470 --> 00:01:01,220 let me point out, of course, that this 17 00:01:01,220 --> 00:01:03,250 is a matter of semantics. 18 00:01:03,250 --> 00:01:08,010 Obviously an arrow, being a line, has only one dimension, 19 00:01:08,010 --> 00:01:12,050 be it in the direction of the x-axis, be it in the x-y plane, 20 00:01:12,050 --> 00:01:14,530 or be it in three-dimensional space. 21 00:01:14,530 --> 00:01:17,010 When we say "three-dimensional vectors," 22 00:01:17,010 --> 00:01:21,270 we do not negate the fact that we're talking about arrows, 23 00:01:21,270 --> 00:01:24,870 but rather that we are dealing with a coordinate system which 24 00:01:24,870 --> 00:01:28,480 takes into consideration the fact that the line is drawn 25 00:01:28,480 --> 00:01:30,640 through three-dimensional space. 26 00:01:30,640 --> 00:01:32,620 In other words, where as I call the lecture 27 00:01:32,620 --> 00:01:36,230 three-dimensional vectors or arrows-- and by the way, 28 00:01:36,230 --> 00:01:39,080 both in the notes and as I'm lecturing, 29 00:01:39,080 --> 00:01:42,920 I will very often, whenever I write the word "vector," 30 00:01:42,920 --> 00:01:45,120 put "arrow" in parentheses, whenever 31 00:01:45,120 --> 00:01:47,660 I write the word "arrow," put "vector" in parentheses 32 00:01:47,660 --> 00:01:50,190 so that you can see the juxtaposition 33 00:01:50,190 --> 00:01:54,330 between these two, the geometry versus the physical or 34 00:01:54,330 --> 00:01:55,850 mathematical concept. 35 00:01:55,850 --> 00:01:57,770 But the idea is something like this. 36 00:01:57,770 --> 00:02:01,320 In three-dimensional space, we have a natural extension 37 00:02:01,320 --> 00:02:02,980 of Cartesian coordinates. 38 00:02:02,980 --> 00:02:05,450 Rather than talk about the x-y plane, 39 00:02:05,450 --> 00:02:09,699 we pick a third axis, a third number line, which 40 00:02:09,699 --> 00:02:13,530 goes through the origin, perpendicular to the x-y plane, 41 00:02:13,530 --> 00:02:17,010 and which has the sense that if the x-axis is 42 00:02:17,010 --> 00:02:22,350 rotated into the y-axis through the positive 90-degree angle 43 00:02:22,350 --> 00:02:25,420 here, that the z-axis is in the direction 44 00:02:25,420 --> 00:02:28,950 in which a right-handed screw would turn. 45 00:02:28,950 --> 00:02:30,410 I won't belabor this. 46 00:02:30,410 --> 00:02:32,370 It's done very, very nicely in the textbook. 47 00:02:32,370 --> 00:02:35,710 And I'm sure that you have seen this type of coordinate system 48 00:02:35,710 --> 00:02:36,360 before. 49 00:02:36,360 --> 00:02:37,818 If you haven't, it's something that 50 00:02:37,818 --> 00:02:40,910 takes a matter of some 20 or 30 seconds to pick up. 51 00:02:40,910 --> 00:02:42,850 But at any rate, what we're saying 52 00:02:42,850 --> 00:02:45,820 is, let's imagine this three-dimensional coordinate 53 00:02:45,820 --> 00:02:48,930 system, three-dimensional Cartesian coordinates. 54 00:02:48,930 --> 00:02:52,800 The convention is that just as, in the plane, 55 00:02:52,800 --> 00:02:57,480 we label the point by its x and y components, in three space, 56 00:02:57,480 --> 00:03:02,070 a point is labeled by its x, y, and z components. 57 00:03:02,070 --> 00:03:06,070 So for example, if I take a vector in three-space-- meaning 58 00:03:06,070 --> 00:03:06,570 what? 59 00:03:06,570 --> 00:03:09,580 It's a line that goes through three-dimensional space-- I 60 00:03:09,580 --> 00:03:11,310 again shift it parallel to itself, 61 00:03:11,310 --> 00:03:13,350 so it begins at my origin. 62 00:03:13,350 --> 00:03:16,340 And at the risk of causing some confusion here, 63 00:03:16,340 --> 00:03:18,150 I still think it's worth doing. 64 00:03:18,150 --> 00:03:20,660 Let me call the vector A, and let 65 00:03:20,660 --> 00:03:24,680 me call the point at which the vector terminates the point A. 66 00:03:24,680 --> 00:03:29,290 And let's suppose that the point A has coordinates A_1, A_2, 67 00:03:29,290 --> 00:03:30,130 A_3. 68 00:03:30,130 --> 00:03:32,790 The reason that this doesn't make much difference 69 00:03:32,790 --> 00:03:36,390 if you get confused is that if the vector, which 70 00:03:36,390 --> 00:03:39,150 originates at the origin, terminates 71 00:03:39,150 --> 00:03:41,850 at the point whose coordinates a_1, a_2, 72 00:03:41,850 --> 00:03:46,810 and a_3, then the components of that vector will have what? 73 00:03:46,810 --> 00:03:49,070 The i component will be a_1. 74 00:03:49,070 --> 00:03:51,480 The j component will be a_2. 75 00:03:51,480 --> 00:03:54,940 And by the way, we let k be the unit vector 76 00:03:54,940 --> 00:03:56,910 in the positive z direction. 77 00:03:56,910 --> 00:04:00,100 And the k component will be a_3. 78 00:04:00,100 --> 00:04:03,260 To see this thing pictorially, notice 79 00:04:03,260 --> 00:04:05,440 the three-dimensional effect of the vector A. 80 00:04:05,440 --> 00:04:07,830 It means that if we drop a perpendicular from the head 81 00:04:07,830 --> 00:04:15,300 of A to the x-y plane, then, you see, this distance is a_1. 82 00:04:15,300 --> 00:04:17,300 This distance is a_2. 83 00:04:17,300 --> 00:04:19,540 And this height is a_3. 84 00:04:19,540 --> 00:04:23,810 And notice, again, in terms of adding vectors head to tail et 85 00:04:23,810 --> 00:04:29,200 cetera, notice that as a vector, this would be the vector a_1*i. 86 00:04:29,200 --> 00:04:33,460 This would be the vector a_2*j. 87 00:04:33,460 --> 00:04:37,420 And this would be the vector a_3*k. 88 00:04:37,420 --> 00:04:41,350 In other words, the vector A is the sum of the three vectors 89 00:04:41,350 --> 00:04:45,560 a_1*i, a_2*j, and a_3*k. 90 00:04:45,560 --> 00:04:47,430 In other words, to summarize this, 91 00:04:47,430 --> 00:04:49,260 the vector A is simply what? 92 00:04:49,260 --> 00:04:52,660 a_1*i, plus a_2*j, plus a_3*k. 93 00:04:52,660 --> 00:04:55,630 And again, notice the juxtapositioning, if you wish, 94 00:04:55,630 --> 00:04:58,910 between the components of the vector 95 00:04:58,910 --> 00:05:02,810 and the coordinates of the point. 96 00:05:02,810 --> 00:05:06,640 Also, we can get ahold of the magnitude of this vector. 97 00:05:06,640 --> 00:05:09,300 Remember, the magnitude is the length. 98 00:05:09,300 --> 00:05:12,270 And how can we figure out the length of this vector? 99 00:05:12,270 --> 00:05:13,930 Notice that, by the way, the geometry 100 00:05:13,930 --> 00:05:16,275 gets much tougher in three-dimensional space. 101 00:05:16,275 --> 00:05:18,650 But I want to show you something interesting in a minute. 102 00:05:18,650 --> 00:05:21,190 Let's suffer through the geometry for a minute. 103 00:05:21,190 --> 00:05:23,690 Let's see how we can find the length of the vector A. 104 00:05:23,690 --> 00:05:27,670 Well, notice, by the way, that A happens to be-- 105 00:05:27,670 --> 00:05:30,660 or the magnitude of A happens to be the hypotenuse 106 00:05:30,660 --> 00:05:32,160 of a right triangle. 107 00:05:32,160 --> 00:05:34,190 What right triangle is it? 108 00:05:34,190 --> 00:05:37,110 It's the right triangle that joins 109 00:05:37,110 --> 00:05:42,010 the origin-- let me call this point here B-- and A. 110 00:05:42,010 --> 00:05:45,270 In other words, triangle OBA is a right triangle 111 00:05:45,270 --> 00:05:48,840 because OB is in the x-y plane and AB is 112 00:05:48,840 --> 00:05:51,300 perpendicular to the x-y plane. 113 00:05:51,300 --> 00:05:55,320 Therefore, by the Pythagorean theorem, the magnitude of A 114 00:05:55,320 --> 00:05:59,950 will be the square root of the square of the magnitude of OB 115 00:05:59,950 --> 00:06:02,730 plus the square of the magnitude of AB. 116 00:06:02,730 --> 00:06:10,750 On the other hand, notice that OCB is also a right triangle. 117 00:06:10,750 --> 00:06:14,480 So by the Pythagorean theorem, the length of OB 118 00:06:14,480 --> 00:06:20,510 is the square root of a_1 squared plus a_2 squared. 119 00:06:20,510 --> 00:06:23,800 And therefore, you see, by the Pythagorean theorem, 120 00:06:23,800 --> 00:06:26,250 the square of the magnitude of A is 121 00:06:26,250 --> 00:06:28,310 this squared plus this squared. 122 00:06:28,310 --> 00:06:30,820 In other words, the magnitude of A 123 00:06:30,820 --> 00:06:34,550 is the square root of a_1 squared plus a_2 squared 124 00:06:34,550 --> 00:06:35,990 plus a_3 squared. 125 00:06:35,990 --> 00:06:38,150 In other words, in Cartesian coordinates, 126 00:06:38,150 --> 00:06:40,250 to find the magnitude of a vector, 127 00:06:40,250 --> 00:06:43,780 you need only take the positive square root 128 00:06:43,780 --> 00:06:46,070 of the sum of the squares of the components. 129 00:06:46,070 --> 00:06:48,750 It's a regular distance formula, the Pythagorean theorem. 130 00:06:48,750 --> 00:06:50,930 By the way, just a very, very quick review 131 00:06:50,930 --> 00:06:52,970 from part one of our course-- remember 132 00:06:52,970 --> 00:06:55,160 that technically speaking, the square root 133 00:06:55,160 --> 00:06:57,130 is a double-valued function. 134 00:06:57,130 --> 00:07:00,820 Namely, any positive number has two square roots. 135 00:07:00,820 --> 00:07:01,320 All right? 136 00:07:01,320 --> 00:07:04,200 In other words, the square root of 16 is 4. 137 00:07:04,200 --> 00:07:06,150 It's also negative 4. 138 00:07:06,150 --> 00:07:09,470 But by convention, when we don't indicate a sign, 139 00:07:09,470 --> 00:07:12,180 we're always referring to the positive square root. 140 00:07:12,180 --> 00:07:13,280 And that's fine. 141 00:07:13,280 --> 00:07:15,220 Because after all, the magnitude-- 142 00:07:15,220 --> 00:07:17,010 the length of an arrow-- is always 143 00:07:17,010 --> 00:07:18,540 looked upon as being what? 144 00:07:18,540 --> 00:07:20,550 Some non-negative number. 145 00:07:20,550 --> 00:07:24,390 Notice that the negativeness is taken care of by the sense. 146 00:07:24,390 --> 00:07:25,130 OK? 147 00:07:25,130 --> 00:07:26,900 And this is what's rather interesting. 148 00:07:26,900 --> 00:07:28,380 I think it's worth noting. 149 00:07:28,380 --> 00:07:31,260 If we use Cartesian coordinates in one dimension, 150 00:07:31,260 --> 00:07:34,480 all you have is the x-axis, in which case 151 00:07:34,480 --> 00:07:38,720 any vector is some scale or multiple of the unit vector i. 152 00:07:38,720 --> 00:07:42,190 In other words, A is equal to a_1*i. 153 00:07:42,190 --> 00:07:45,750 In this case, the magnitude of A is just what? 154 00:07:45,750 --> 00:07:48,470 It's the square root of a_1 squared. 155 00:07:48,470 --> 00:07:50,890 And notice that means the positive square root-- 156 00:07:50,890 --> 00:07:53,330 notice, by the way, that that's the usual meaning 157 00:07:53,330 --> 00:07:54,570 of absolute value. 158 00:07:54,570 --> 00:07:57,380 After all, if a_1 were already positive, 159 00:07:57,380 --> 00:08:00,640 taking the positive square root of its square root-- 160 00:08:00,640 --> 00:08:01,564 of its square. 161 00:08:01,564 --> 00:08:02,480 These tongue twisters. 162 00:08:02,480 --> 00:08:04,250 The positive square root of the square 163 00:08:04,250 --> 00:08:05,920 would give you the number back again. 164 00:08:05,920 --> 00:08:08,080 On the other hand, if a_1 were negative, 165 00:08:08,080 --> 00:08:11,620 and then you square it, and then take the positive square root, 166 00:08:11,620 --> 00:08:14,780 all you do is change the sign of a_1. 167 00:08:14,780 --> 00:08:16,940 And since it was negative, changing its sign 168 00:08:16,940 --> 00:08:18,420 makes it positive. 169 00:08:18,420 --> 00:08:21,020 So this is the usual absolute value, all right? 170 00:08:21,020 --> 00:08:22,940 Now in terms of the previous unit, 171 00:08:22,940 --> 00:08:25,240 we saw that in two-dimensional space, 172 00:08:25,240 --> 00:08:27,620 using i and j as our basic vectors, 173 00:08:27,620 --> 00:08:31,520 that if A was the vector a_1*i plus a_2*j, 174 00:08:31,520 --> 00:08:34,970 then the magnitude of A, again, by the Pythagorean theorem, 175 00:08:34,970 --> 00:08:40,130 was just a square root of a_1 squared plus a_2 squared. 176 00:08:40,130 --> 00:08:42,330 And now we've seen that structurally-- 177 00:08:42,330 --> 00:08:46,010 even though geometrically it was harder to show-- 178 00:08:46,010 --> 00:08:50,230 that if A in three-dimensional space is a_1*i plus a_2*j plus 179 00:08:50,230 --> 00:08:53,250 a_3*k, then A is what? 180 00:08:53,250 --> 00:08:55,190 The positive square root of a_1 squared 181 00:08:55,190 --> 00:08:57,410 plus a_2 squared plus a_3 squared. 182 00:08:57,410 --> 00:09:01,030 And notice you see this structural resemblance. 183 00:09:01,030 --> 00:09:03,230 This is a beautiful structural resemblance. 184 00:09:03,230 --> 00:09:05,610 Why is it a beautiful structural resemblance? 185 00:09:05,610 --> 00:09:07,950 Because now, maybe we're getting the feeling 186 00:09:07,950 --> 00:09:10,510 that once our rules and recipes are made up, 187 00:09:10,510 --> 00:09:13,610 we don't have to worry about how difficult the geometry is. 188 00:09:13,610 --> 00:09:15,510 In other words, except for the fact 189 00:09:15,510 --> 00:09:17,910 that you have an extra component to worry about here, 190 00:09:17,910 --> 00:09:20,320 there is no basic difference structurally 191 00:09:20,320 --> 00:09:23,500 between finding the length of a vector in one-dimensional space 192 00:09:23,500 --> 00:09:25,500 or in three-dimensional space. 193 00:09:25,500 --> 00:09:28,610 And in fact, another reason that I'm harping on this is the fact 194 00:09:28,610 --> 00:09:31,740 that in a little while-- meaning within the next few lessons-- 195 00:09:31,740 --> 00:09:34,730 we are going to be talking about some horrible thing called 196 00:09:34,730 --> 00:09:38,620 n-dimensional space, which geometrically cannot be drawn, 197 00:09:38,620 --> 00:09:41,940 but which analytically preserves the same structure that 198 00:09:41,940 --> 00:09:43,310 we're talking about here. 199 00:09:43,310 --> 00:09:46,520 You see, the whole bag is still this idea of structure. 200 00:09:46,520 --> 00:09:49,580 We're trying to emphasize that the recipes remain 201 00:09:49,580 --> 00:09:52,650 the same independently of the dimension. 202 00:09:52,650 --> 00:09:56,970 And just to carry this idea a few steps further, what we're 203 00:09:56,970 --> 00:10:00,620 saying is, remember in the last unit, we talked about the fact 204 00:10:00,620 --> 00:10:02,780 that the nice thing about i and j components 205 00:10:02,780 --> 00:10:05,070 was that the definition for adding two vectors 206 00:10:05,070 --> 00:10:08,680 was that if they were in terms of i and j components, 207 00:10:08,680 --> 00:10:10,860 you just add it component by component? 208 00:10:10,860 --> 00:10:13,500 Well again, without belaboring the point 209 00:10:13,500 --> 00:10:18,900 and leaving the details to the text and to the exercises, 210 00:10:18,900 --> 00:10:21,650 let's keep in mind that it's relatively easy to show 211 00:10:21,650 --> 00:10:24,780 that the same formal definition of addition-- namely, 212 00:10:24,780 --> 00:10:27,680 placing the arrows head to tail, et cetera-- 213 00:10:27,680 --> 00:10:29,764 forgetting about Cartesian coordinates, but notice 214 00:10:29,764 --> 00:10:31,180 the definition of addition doesn't 215 00:10:31,180 --> 00:10:32,530 mention the coordinate system. 216 00:10:32,530 --> 00:10:35,300 It just says put the vectors head to tail, et cetera. 217 00:10:35,300 --> 00:10:37,730 And what I'm saying is it's easy to show, 218 00:10:37,730 --> 00:10:41,530 under those conditions, that if A is the vector a_1*i plus 219 00:10:41,530 --> 00:10:46,640 a_2*j plus a_3*k, and B is the vector b_1*i plus b_2*j plus 220 00:10:46,640 --> 00:10:49,710 b_3*k, then you still add vectors the same way 221 00:10:49,710 --> 00:10:52,300 in three-space as you did in two-space. 222 00:10:52,300 --> 00:10:55,220 Again, the geometric arguments are a little bit more 223 00:10:55,220 --> 00:10:57,570 sophisticated, if only because it's harder 224 00:10:57,570 --> 00:10:59,620 to draw three dimensions to scale 225 00:10:59,620 --> 00:11:01,210 than it is two dimensions. 226 00:11:01,210 --> 00:11:02,350 But the idea is what? 227 00:11:02,350 --> 00:11:05,620 A plus B is just obtained by adding what? 228 00:11:05,620 --> 00:11:08,740 The two i components, the two j components, 229 00:11:08,740 --> 00:11:11,010 the two k components. 230 00:11:11,010 --> 00:11:13,740 And similarly, for scalar multiplication, 231 00:11:13,740 --> 00:11:16,490 in the same way in terms of i and j components, 232 00:11:16,490 --> 00:11:19,740 you simply multiplied each complement by the scalar c. 233 00:11:19,740 --> 00:11:23,940 If c is any number and A is still the vector a_1*i plus 234 00:11:23,940 --> 00:11:28,330 a_2*j plus a_3k, the scalar multiple c times A turns out 235 00:11:28,330 --> 00:11:32,270 to be c*a_1*i plus c*a_3*j plus c*a_3*k. 236 00:11:32,270 --> 00:11:35,470 That you just multiply component by component. 237 00:11:35,470 --> 00:11:38,990 And by the way, don't lose track of this important point, 238 00:11:38,990 --> 00:11:43,820 that the scalar multiple c*A means the same thing in two 239 00:11:43,820 --> 00:11:45,770 dimensions as it did three. 240 00:11:45,770 --> 00:11:49,480 Because after all, when you draw that vector, once that arrow 241 00:11:49,480 --> 00:11:54,100 is drawn, it's a one-dimensional thing if you pick as your axis 242 00:11:54,100 --> 00:11:56,919 the line of action of the particular arrow. 243 00:11:56,919 --> 00:11:58,460 In other words, scalar multiplication 244 00:11:58,460 --> 00:11:59,450 means the same thing. 245 00:11:59,450 --> 00:12:02,280 What's happening is that you're looking at it in terms 246 00:12:02,280 --> 00:12:03,820 of different components. 247 00:12:03,820 --> 00:12:06,780 You're looking at it in terms of i, j, and k, rather than just i 248 00:12:06,780 --> 00:12:09,740 and j, or in terms of any other coordinate system. 249 00:12:09,740 --> 00:12:11,930 But carrying this out in particular, 250 00:12:11,930 --> 00:12:15,180 notice then, the connection between the vector minus B-- 251 00:12:15,180 --> 00:12:19,360 negative B-- and the vector negative 1 times 252 00:12:19,360 --> 00:12:23,790 B. Namely, the vector negative B, which has the same magnitude 253 00:12:23,790 --> 00:12:27,690 and direction as B, but the opposite sense, 254 00:12:27,690 --> 00:12:29,560 is the same way as saying what? 255 00:12:29,560 --> 00:12:31,337 Multiply B by minus 1. 256 00:12:31,337 --> 00:12:33,920 Because that gives you a vector which has the same magnitude-- 257 00:12:33,920 --> 00:12:37,780 namely 1 times as much as B-- the same direction, 258 00:12:37,780 --> 00:12:38,810 and the opposite sense. 259 00:12:38,810 --> 00:12:40,720 In other words, the vector negative B, 260 00:12:40,720 --> 00:12:45,610 in three-dimensional space, is minus b_1 i, minus b_2 j, 261 00:12:45,610 --> 00:12:47,440 minus b_3 k. 262 00:12:47,440 --> 00:12:52,490 And therefore, since A minus B still means A plus negative B, 263 00:12:52,490 --> 00:12:56,370 the vector A minus B is simply obtained how? 264 00:12:56,370 --> 00:12:58,680 You subtract the same way as we did 265 00:12:58,680 --> 00:13:02,390 in ordinary one-dimensional analytic geometry, 266 00:13:02,390 --> 00:13:03,610 in terms of a number line. 267 00:13:03,610 --> 00:13:06,390 In other words, you subtract component 268 00:13:06,390 --> 00:13:08,800 by component-- subtracting what? 269 00:13:14,570 --> 00:13:18,430 The first vector minus the corresponding component 270 00:13:18,430 --> 00:13:19,660 of the second vector. 271 00:13:19,660 --> 00:13:22,470 In other words, in Cartesian coordinates, the vector A 272 00:13:22,470 --> 00:13:27,910 minus B-- once we know a_1, a_2, and a_3 as the components of A, 273 00:13:27,910 --> 00:13:30,740 b_1, b_2, and b_3 as the components of B, 274 00:13:30,740 --> 00:13:33,530 we just subtract component by component 275 00:13:33,530 --> 00:13:35,770 to get this particular result. And this is, I 276 00:13:35,770 --> 00:13:38,010 think, very important to understand. 277 00:13:38,010 --> 00:13:40,900 By the same token, as simple as this is, 278 00:13:40,900 --> 00:13:42,320 it's going to cause-- if we're not 279 00:13:42,320 --> 00:13:45,230 careful-- great misinterpretation. 280 00:13:45,230 --> 00:13:50,170 Namely, the vector arithmetic that we're talking about 281 00:13:50,170 --> 00:13:53,420 does not depend on Cartesian coordinates. 282 00:13:53,420 --> 00:13:56,580 Vectors or arrows, as we're looking at them, 283 00:13:56,580 --> 00:13:58,970 and the arrows exist no matter what coordinate system 284 00:13:58,970 --> 00:13:59,710 we're using. 285 00:13:59,710 --> 00:14:02,470 What is particularly important, however, 286 00:14:02,470 --> 00:14:06,410 is that many of the luxuries of using vector arithmetic 287 00:14:06,410 --> 00:14:08,890 hinge on using Cartesian coordinates. 288 00:14:08,890 --> 00:14:10,590 Now what do I mean by that? 289 00:14:10,590 --> 00:14:14,340 Well, let me just emphasize: Note the need 290 00:14:14,340 --> 00:14:16,950 for Cartesian coordinates. 291 00:14:16,950 --> 00:14:19,420 Let me talk instead, you see, about something 292 00:14:19,420 --> 00:14:21,110 called polar coordinates. 293 00:14:21,110 --> 00:14:23,060 Now later on, we're going to talk 294 00:14:23,060 --> 00:14:26,560 about polar coordinates in more detail-- in fact, 295 00:14:26,560 --> 00:14:28,540 in very, very much more detail. 296 00:14:28,540 --> 00:14:30,930 But for the time being, let's view polar 297 00:14:30,930 --> 00:14:34,270 coordinates simply as a radar-type thing, 298 00:14:34,270 --> 00:14:37,590 as a range-and-bearing type navigation. 299 00:14:37,590 --> 00:14:39,870 Namely, in polar coordinates, the way 300 00:14:39,870 --> 00:14:42,600 you specify a vector in the plane, say, 301 00:14:42,600 --> 00:14:49,020 is you specify its length, which we'll call r, and the angle 302 00:14:49,020 --> 00:14:52,980 that it makes with the positive x-axis, that we'll call theta. 303 00:14:52,980 --> 00:14:57,800 In other words, in polar coordinates, the vector A, 304 00:14:57,800 --> 00:15:00,420 which I've labeled r_2 comma theta_2 , 305 00:15:00,420 --> 00:15:06,030 that would indicate that the magnitude of A is r_2. 306 00:15:06,030 --> 00:15:09,340 And that the angle that A makes with the positive x-axis 307 00:15:09,340 --> 00:15:13,050 is theta_2. 308 00:15:13,050 --> 00:15:16,510 And when I label the vector B as r_1 comma theta_1, 309 00:15:16,510 --> 00:15:19,060 it's simply my way of saying that the magnitude 310 00:15:19,060 --> 00:15:22,500 of the vector B is some length, which we'll call r_1. 311 00:15:22,500 --> 00:15:26,790 And the angle that B makes with the positive x-axis 312 00:15:26,790 --> 00:15:29,110 is some angle which we'll call theta_1. 313 00:15:29,110 --> 00:15:30,160 OK? 314 00:15:30,160 --> 00:15:32,940 Perfectly well-defined system, isn't it? 315 00:15:32,940 --> 00:15:36,010 In other words, if I tell you the range and bearing, 316 00:15:36,010 --> 00:15:38,750 I've certainly given you as much information 317 00:15:38,750 --> 00:15:41,610 as if I told you the i and j components. 318 00:15:41,610 --> 00:15:44,940 Now here's the important point, the real kicker to this thing. 319 00:15:44,940 --> 00:15:49,180 And that is that the vector A minus B-- and A minus B 320 00:15:49,180 --> 00:15:50,030 is what? 321 00:15:50,030 --> 00:15:51,070 It's this vector here. 322 00:15:55,400 --> 00:15:57,840 And by the way, notice how vector arithmetic goes. 323 00:15:57,840 --> 00:15:59,950 How could you tell quickly whether this 324 00:15:59,950 --> 00:16:02,400 is A minus B or B minus A? 325 00:16:02,400 --> 00:16:05,040 We talked about this in the previous unit, 326 00:16:05,040 --> 00:16:07,130 and had exercises and discussion on it. 327 00:16:07,130 --> 00:16:10,600 But notice again, the structure of vector arithmetic. 328 00:16:10,600 --> 00:16:14,380 If you look at addition, what is this vector here? 329 00:16:14,380 --> 00:16:17,830 This is the vector which must be added on to B 330 00:16:17,830 --> 00:16:22,610 to give the vector A. And that, by definition, is called what? 331 00:16:22,610 --> 00:16:25,430 A minus B. A minus B is what vector? 332 00:16:25,430 --> 00:16:30,200 The vector you must add on to B to yield A. Subtraction 333 00:16:30,200 --> 00:16:32,020 is still the inverse of addition. 334 00:16:32,020 --> 00:16:36,260 And hopefully by now, you should be becoming much less fearful 335 00:16:36,260 --> 00:16:39,300 of this vector notation, that structurally, it's 336 00:16:39,300 --> 00:16:43,810 the same as our first unit of arithmetic. 337 00:16:43,810 --> 00:16:47,780 Well, numerical arithmetic, OK, the more regular type, 338 00:16:47,780 --> 00:16:49,470 ordinary type of arithmetic. 339 00:16:49,470 --> 00:16:51,130 But here's the important point. 340 00:16:51,130 --> 00:16:55,070 This vector is still A minus B. But I 341 00:16:55,070 --> 00:17:00,400 claim that it's trivial to see that this vector is not 342 00:17:00,400 --> 00:17:03,800 the vector whose magnitude is r_2 minus r_1 343 00:17:03,800 --> 00:17:07,720 and whose angle of bearing is theta_2 minus theta_1. 344 00:17:07,720 --> 00:17:10,940 Now the word "trivial" is a very misleading word in mathematics. 345 00:17:10,940 --> 00:17:13,420 One of our famous mathematical anecdotes 346 00:17:13,420 --> 00:17:16,276 is the professor who said that a proof was trivial. 347 00:17:16,276 --> 00:17:18,359 The student says, "It doesn't seem trivial to me." 348 00:17:18,359 --> 00:17:20,810 Their professor says, "Well, it is trivial." 349 00:17:20,810 --> 00:17:22,490 He didn't quite see it in a hurry. 350 00:17:22,490 --> 00:17:23,400 He says, "Wait here." 351 00:17:23,400 --> 00:17:25,030 He ran down to his office. 352 00:17:25,030 --> 00:17:28,260 Came up three hours later with a ream of papers 353 00:17:28,260 --> 00:17:30,880 that he had written on, bathed in perspiration, 354 00:17:30,880 --> 00:17:31,637 and a big smile. 355 00:17:31,637 --> 00:17:32,720 And he said, "I was right. 356 00:17:32,720 --> 00:17:34,020 It was trivial." 357 00:17:34,020 --> 00:17:36,170 So just because what I think is trivial 358 00:17:36,170 --> 00:17:38,130 may not be what you think is trivial, 359 00:17:38,130 --> 00:17:43,060 let me go through this statement in more computational detail. 360 00:17:43,060 --> 00:17:46,610 What I'm saying is, let v denote the vector that we 361 00:17:46,610 --> 00:17:49,120 called A minus B before. 362 00:17:49,120 --> 00:17:52,670 Notice that in terms of what the vectors A and B are, 363 00:17:52,670 --> 00:17:54,840 this length is r_1. 364 00:17:54,840 --> 00:17:56,650 This length is r_2. 365 00:17:56,650 --> 00:17:59,670 And notice that since this whole angle was theta_2, 366 00:17:59,670 --> 00:18:03,630 and the angle from here to here was theta_1, this angle in here 367 00:18:03,630 --> 00:18:06,140 must be the difference between those two angles, which 368 00:18:06,140 --> 00:18:08,380 is theta_2 minus theta_1. 369 00:18:08,380 --> 00:18:11,320 Now notice that this is still a triangle. 370 00:18:11,320 --> 00:18:13,120 The length of this triangle is still 371 00:18:13,120 --> 00:18:16,850 expressible in terms of these sides and the included angle. 372 00:18:16,850 --> 00:18:20,110 But you may recall from plane trigonometry and our refresher 373 00:18:20,110 --> 00:18:22,630 of that in the first part of our course-- part 374 00:18:22,630 --> 00:18:25,660 one-- that to find the third side of a triangle, 375 00:18:25,660 --> 00:18:27,850 given two sides and the included angle, 376 00:18:27,850 --> 00:18:30,010 one must use the Law of Cosines. 377 00:18:30,010 --> 00:18:33,490 In other words, the magnitude of v has what property? 378 00:18:33,490 --> 00:18:36,860 That its square is equal to the sum of the squares of these two 379 00:18:36,860 --> 00:18:40,870 sides minus twice the product of these two 380 00:18:40,870 --> 00:18:44,220 lengths times the cosine of the included angle. 381 00:18:44,220 --> 00:18:49,170 In other words, this is just-- I hope this shows up all right-- 382 00:18:49,170 --> 00:18:51,650 this is the Law of Cosines. 383 00:18:51,650 --> 00:18:54,350 And if it's difficult to read, don't bother reading it. 384 00:18:54,350 --> 00:18:56,720 It's still the Law of Cosines, and hopefully you 385 00:18:56,720 --> 00:18:58,070 recognize it as such. 386 00:18:58,070 --> 00:19:00,590 In other words, to find the magnitude of v, 387 00:19:00,590 --> 00:19:03,880 it's not just r_1 or r_2 minus r_1. 388 00:19:03,880 --> 00:19:08,380 It's the square root of r_1 squared plus r_2 squared minus 389 00:19:08,380 --> 00:19:11,790 2r_1*r_2 cosine theta_2 minus theta_1. 390 00:19:11,790 --> 00:19:14,080 Can you use polar coordinates if you want? 391 00:19:14,080 --> 00:19:15,700 The answer is you bet you can. 392 00:19:15,700 --> 00:19:18,540 But when you do use it, make sure that whenever you're 393 00:19:18,540 --> 00:19:21,150 going to find the magnitude, that you don't just 394 00:19:21,150 --> 00:19:22,470 subtract the two magnitudes. 395 00:19:22,470 --> 00:19:23,870 You have to use this. 396 00:19:23,870 --> 00:19:26,770 And by the way, notice I haven't even gone into this part, 397 00:19:26,770 --> 00:19:28,780 because it's irrelevant from the point of view 398 00:19:28,780 --> 00:19:30,610 that I'm trying to emphasize now. 399 00:19:30,610 --> 00:19:33,700 This mess just gives you the magnitude of a vector. 400 00:19:33,700 --> 00:19:36,040 It doesn't even tell you what direction it's in. 401 00:19:36,040 --> 00:19:38,830 I have to still do a heck of a lot of geometry 402 00:19:38,830 --> 00:19:43,060 if I want to find out what this angle here 403 00:19:43,060 --> 00:19:46,130 is-- a lot of work involved. 404 00:19:46,130 --> 00:19:47,990 I can use polar coordinates, but I 405 00:19:47,990 --> 00:19:50,770 lose some of the luxury of Cartesian coordinates. 406 00:19:50,770 --> 00:19:55,010 In fact, notice that the vector whose magnitude 407 00:19:55,010 --> 00:19:58,040 is r_2 minus r_1 and whose angle of bearing 408 00:19:58,040 --> 00:20:01,750 is theta_2 minus theta_1, can also be computed. 409 00:20:01,750 --> 00:20:05,850 Namely, how do we find theta_2 minus theta_1? 410 00:20:05,850 --> 00:20:08,980 Well up here, we have the angle theta_2. 411 00:20:08,980 --> 00:20:10,580 We have the angle theta_1. 412 00:20:10,580 --> 00:20:12,960 We computed theta_2 minus theta_1. 413 00:20:12,960 --> 00:20:17,210 I can now mark off that angle here, theta_2 minus theta_1. 414 00:20:17,210 --> 00:20:20,820 I can now take a circle of radius r_1-- 415 00:20:20,820 --> 00:20:23,410 well, why even say a circle-- mark off the length 416 00:20:23,410 --> 00:20:27,590 r_1 onto r_2, assuming r_2 is the greater of the two lengths. 417 00:20:27,590 --> 00:20:30,690 That difference will be r_2 minus r_1. 418 00:20:30,690 --> 00:20:33,290 In other words, if I swing an arc over here, 419 00:20:33,290 --> 00:20:35,360 this would r_1 also. 420 00:20:35,360 --> 00:20:39,100 This distance here would be r_2 minus r_1. 421 00:20:39,100 --> 00:20:42,990 I take that distance, and mark it off down here. 422 00:20:42,990 --> 00:20:44,570 And what vector is this? 423 00:20:44,570 --> 00:20:47,100 This is the vector, which in polar coordinates 424 00:20:47,100 --> 00:20:51,690 would have its magnitude equal to r_2 minus r_1, 425 00:20:51,690 --> 00:20:55,980 and have its angle equal to theta_2 minus theta_1. 426 00:20:55,980 --> 00:20:58,340 And all I want you to see is no matter 427 00:20:58,340 --> 00:21:00,780 how you slice it, just from this picture alone, 428 00:21:00,780 --> 00:21:06,530 this vector is not the same as this vector. 429 00:21:06,530 --> 00:21:08,590 And by the way, don't make the mistake of saying, 430 00:21:08,590 --> 00:21:10,230 gee, the way you've draw them, they 431 00:21:10,230 --> 00:21:12,250 look like they could be the same length. 432 00:21:12,250 --> 00:21:15,560 Remember that even if by coincidence these two vectors 433 00:21:15,560 --> 00:21:19,000 had the same length, you must remember 434 00:21:19,000 --> 00:21:22,850 that vector equality requires not just the same magnitude, 435 00:21:22,850 --> 00:21:24,480 but the same direction. 436 00:21:24,480 --> 00:21:27,090 And even with the same direction, the same sense. 437 00:21:27,090 --> 00:21:29,940 What should be obvious is, at least 438 00:21:29,940 --> 00:21:33,420 based on this one picture, that the direction of this vector 439 00:21:33,420 --> 00:21:37,450 certainly is not the same as the direction of this vector. 440 00:21:37,450 --> 00:21:39,920 In other words, I guess I've said this many times in part 441 00:21:39,920 --> 00:21:43,790 one, and even though it comes out like phony facetiousness, 442 00:21:43,790 --> 00:21:45,430 I mean it quite sincerely. 443 00:21:45,430 --> 00:21:48,010 Given two vectors in polar coordinates, 444 00:21:48,010 --> 00:21:52,690 one certainly has the right to invent 445 00:21:52,690 --> 00:21:57,910 the vector r_2 minus r_1 comma theta_2 minus theta_1. 446 00:21:57,910 --> 00:22:00,230 You certainly have the right to do that, 447 00:22:00,230 --> 00:22:03,100 but you don't have the right to call that A minus B. What 448 00:22:03,100 --> 00:22:05,680 I meant by being facetious is, you have the right 449 00:22:05,680 --> 00:22:08,060 to call it that, but you're going to be wrong. 450 00:22:08,060 --> 00:22:10,920 Because A minus B has already been defined. 451 00:22:10,920 --> 00:22:13,180 And all we have now is a choice of what 452 00:22:13,180 --> 00:22:17,650 the answer looks like in a particular coordinate system. 453 00:22:17,650 --> 00:22:19,880 And by the way, just as a quick review, 454 00:22:19,880 --> 00:22:22,430 assuming that you've studied different number bases at one 455 00:22:22,430 --> 00:22:25,220 time in your careers, this has come up many times 456 00:22:25,220 --> 00:22:27,420 in the mathematics curriculum under the heading 457 00:22:27,420 --> 00:22:30,680 of "Number-versus-Numeral," or more generally, 458 00:22:30,680 --> 00:22:34,330 I call it the heading of "Name-versus-Concept." 459 00:22:34,330 --> 00:22:36,610 For example, consider the number 6. 460 00:22:36,610 --> 00:22:40,450 6 is this many, and no matter how you slice it again, 461 00:22:40,450 --> 00:22:41,890 this many is an even number. 462 00:22:41,890 --> 00:22:46,950 It breaks up into bundles of 2 with none left over. 463 00:22:46,950 --> 00:22:47,960 OK? 464 00:22:47,960 --> 00:22:52,130 If you were to write 6 as a base 5 numeral, it's what? 465 00:22:52,130 --> 00:22:55,260 One bundle of 5 with 1 left over. 466 00:22:55,260 --> 00:22:58,380 In other words, notice that the number 6 467 00:22:58,380 --> 00:23:01,200 ends in an odd digit in base 5. 468 00:23:01,200 --> 00:23:03,800 In other words, the test for evenness 469 00:23:03,800 --> 00:23:06,890 by looking at the last digit depends on the number base. 470 00:23:06,890 --> 00:23:09,110 In other words, in an even base, a number 471 00:23:09,110 --> 00:23:11,710 is even if its units digit is even. 472 00:23:11,710 --> 00:23:14,800 In an odd base, a number may or may not 473 00:23:14,800 --> 00:23:18,589 be even, depending on whether its units digit is even. 474 00:23:18,589 --> 00:23:20,380 In other words, the actual test that you're 475 00:23:20,380 --> 00:23:24,430 using-- the actual convenient computational property-- 476 00:23:24,430 --> 00:23:25,900 depends on the base. 477 00:23:25,900 --> 00:23:29,260 But the important point is that 6 is always even, 478 00:23:29,260 --> 00:23:30,890 independently of a number base. 479 00:23:30,890 --> 00:23:34,050 In fact, the number 6 as six tally marks 480 00:23:34,050 --> 00:23:38,644 was invented long before place-value enumeration. 481 00:23:38,644 --> 00:23:39,810 You see what I'm driving at? 482 00:23:39,810 --> 00:23:42,990 In other words, the number concept stays the same. 483 00:23:42,990 --> 00:23:44,800 The concept stays the same. 484 00:23:44,800 --> 00:23:46,570 What the thing looks like in terms 485 00:23:46,570 --> 00:23:48,670 of a particular set of names, that's 486 00:23:48,670 --> 00:23:51,510 what depends on the names that you choose. 487 00:23:51,510 --> 00:23:52,397 All right? 488 00:23:52,397 --> 00:23:54,230 I think that is very, very important to see. 489 00:23:54,230 --> 00:23:56,760 The vector properties do not change. 490 00:23:56,760 --> 00:23:59,450 The convenient computational recipes 491 00:23:59,450 --> 00:24:02,320 do depend on what coordinate system you're choosing. 492 00:24:02,320 --> 00:24:04,280 And that's one of the reasons-- the same as 493 00:24:04,280 --> 00:24:07,640 in part one of this course-- why we find it very convenient, 494 00:24:07,640 --> 00:24:10,510 whenever possible, to use Cartesian coordinates. 495 00:24:10,510 --> 00:24:11,305 OK? 496 00:24:11,305 --> 00:24:14,225 And let me just summarize this some more. 497 00:24:17,240 --> 00:24:22,460 Structurally, this is very important here. 498 00:24:22,460 --> 00:24:26,700 Structurally, arrow arithmetic is the same for both two 499 00:24:26,700 --> 00:24:28,410 and three dimensions. 500 00:24:28,410 --> 00:24:30,890 For example, without belaboring the point, 501 00:24:30,890 --> 00:24:33,040 whether we're dealing with two-dimensional arrows 502 00:24:33,040 --> 00:24:36,260 or three-dimensional arrows, A plus B is B plus A. 503 00:24:36,260 --> 00:24:41,230 A plus (B plus C) is equal to (A plus B) plus C. 504 00:24:41,230 --> 00:24:44,520 and let me just say, et cetera, and not belabor this particular 505 00:24:44,520 --> 00:24:48,180 point, that structurally, we cannot tell the difference 506 00:24:48,180 --> 00:24:51,910 between two-dimensional vectors and three-dimensional vectors. 507 00:24:51,910 --> 00:24:56,170 What we can tell the difference between, I guess, 508 00:24:56,170 --> 00:24:57,860 is the geometry. 509 00:24:57,860 --> 00:25:01,070 The geometry may be more difficult to visualize 510 00:25:01,070 --> 00:25:03,260 in three-space than in two-space. 511 00:25:03,260 --> 00:25:07,710 But structurally, we cannot tell the difference. 512 00:25:07,710 --> 00:25:09,840 And I guess this is what this thing is all about, 513 00:25:09,840 --> 00:25:13,380 that as you read the textbook and do your assignment, 514 00:25:13,380 --> 00:25:15,990 there is going to be a tendency on your part 515 00:25:15,990 --> 00:25:18,370 to try to rely on diagrams. 516 00:25:18,370 --> 00:25:22,130 And boy, you have to be an expert in descriptive geometry 517 00:25:22,130 --> 00:25:26,720 to be able to take a view of something in three dimensions, 518 00:25:26,720 --> 00:25:31,430 and try to get its true lengths, as you look along various axes 519 00:25:31,430 --> 00:25:33,080 and lines of sight. 520 00:25:33,080 --> 00:25:35,890 The beauty is going to be that as we continue on 521 00:25:35,890 --> 00:25:40,160 with this course, we will use geometry for motivation. 522 00:25:40,160 --> 00:25:42,780 But once we have motivated things by geometry, 523 00:25:42,780 --> 00:25:47,010 we will extract those rules that we can use even in cases where 524 00:25:47,010 --> 00:25:48,490 we can't see the picture. 525 00:25:48,490 --> 00:25:51,290 In much the same way, going back to our first lecture 526 00:25:51,290 --> 00:25:54,380 when we talked about why b to the 0 equals 1, 527 00:25:54,380 --> 00:25:57,910 we made up our rules to make sure 528 00:25:57,910 --> 00:26:03,840 that they would conform to the nice computational recipes that 529 00:26:03,840 --> 00:26:06,970 were prevalent in the more simple cases. 530 00:26:06,970 --> 00:26:09,480 And we are going to play this thing to the hilt. 531 00:26:09,480 --> 00:26:12,460 We're going to really exploit this and explore it 532 00:26:12,460 --> 00:26:14,980 in future lectures, and in fact, throughout the rest 533 00:26:14,980 --> 00:26:15,880 of this course. 534 00:26:15,880 --> 00:26:18,100 But more about that next time. 535 00:26:18,100 --> 00:26:22,140 And until next time, good bye. 536 00:26:22,140 --> 00:26:24,510 Funding for the publication of this video 537 00:26:24,510 --> 00:26:29,390 was provided by the Gabriella and Paul Rosenbaum Foundation. 538 00:26:29,390 --> 00:26:33,560 Help OCW continue to provide free and open access to MIT 539 00:26:33,560 --> 00:26:37,978 courses by making a donation at ocw.mit.edu/donate.