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 --> 00:00:19,650 at ocw.mit.edu. 8 00:00:31,852 --> 00:00:33,157 HERBERT GROSS: Hi. 9 00:00:33,157 --> 00:00:35,240 There's a cute book called The Point and the Line. 10 00:00:37,960 --> 00:00:40,670 It was also made it into an animated cartoon 11 00:00:40,670 --> 00:00:45,570 that ends with the line, "To the vector belongs the spoils." 12 00:00:45,570 --> 00:00:47,730 And in this particular course, we're 13 00:00:47,730 --> 00:00:52,500 going to find that both the traditional meaning of vector, 14 00:00:52,500 --> 00:00:55,920 plus a modern meaning of vector, plays 15 00:00:55,920 --> 00:00:59,450 a very fundamental role in the study of calculus 16 00:00:59,450 --> 00:01:01,070 of several variables. 17 00:01:01,070 --> 00:01:03,930 In fact, the modern definition which we will ultimately 18 00:01:03,930 --> 00:01:06,700 get to later in the course, is the one 19 00:01:06,700 --> 00:01:09,820 that will be of utmost importance to us. 20 00:01:09,820 --> 00:01:12,420 But in the meantime, to better appreciate 21 00:01:12,420 --> 00:01:15,840 what this will be like, we will start with a more conventional 22 00:01:15,840 --> 00:01:17,580 treatment of vectors. 23 00:01:17,580 --> 00:01:19,430 Hopefully one fringe benefit of this 24 00:01:19,430 --> 00:01:23,860 will be to revisit what vectors really are. 25 00:01:23,860 --> 00:01:28,560 And to get on with this, let me again point out 26 00:01:28,560 --> 00:01:33,670 that we will be giving an overview of what the unit is 27 00:01:33,670 --> 00:01:35,910 all about in this lecture. 28 00:01:35,910 --> 00:01:38,100 And the details will again be left 29 00:01:38,100 --> 00:01:42,670 to the supplementary notes, the text, and the exercises. 30 00:01:42,670 --> 00:01:46,270 I call this lecture "'Arrow' Arithmetic." 31 00:01:46,270 --> 00:01:48,830 And the reason for calling this "'Arrow' Arithmetic" 32 00:01:48,830 --> 00:01:52,040 hopefully will become clear in a moment. 33 00:01:52,040 --> 00:01:54,000 For the time being, let's very quickly 34 00:01:54,000 --> 00:01:56,650 review what a vector is. 35 00:01:56,650 --> 00:01:58,290 I think most of us know this, but just 36 00:01:58,290 --> 00:02:00,620 to make sure there are no misinterpretations, 37 00:02:00,620 --> 00:02:05,840 a vector is any quantity which depends on a direction as well 38 00:02:05,840 --> 00:02:07,870 as a magnitude. 39 00:02:07,870 --> 00:02:10,020 In other words, a vector quantity 40 00:02:10,020 --> 00:02:13,960 is one which in order to specify uniquely, 41 00:02:13,960 --> 00:02:17,980 we must tell what its magnitude is-- its size; 42 00:02:17,980 --> 00:02:23,070 we must tell what direction it's in, 43 00:02:23,070 --> 00:02:27,520 and finally something which I call the sense wherein 44 00:02:27,520 --> 00:02:33,050 I distinguish between the same direction being traversed 45 00:02:33,050 --> 00:02:34,650 in two different ways. 46 00:02:34,650 --> 00:02:37,340 And perhaps the best way to emphasize this 47 00:02:37,340 --> 00:02:40,340 is in terms of a one-dimensional analogy 48 00:02:40,340 --> 00:02:44,200 that we have already studied very, very early 49 00:02:44,200 --> 00:02:46,040 in our mathematical careers. 50 00:02:46,040 --> 00:02:49,720 I'm thinking of the idea of visualizing signed numbers 51 00:02:49,720 --> 00:02:51,920 in terms of the number line. 52 00:02:51,920 --> 00:02:54,730 In other words, recall that the number line was essentially 53 00:02:54,730 --> 00:02:56,240 the x-axis. 54 00:02:56,240 --> 00:03:01,560 And starting at some origin, which we called say, 0, 55 00:03:01,560 --> 00:03:04,650 we numbered 1, 2, et cetera 56 00:03:04,650 --> 00:03:10,110 if we moved in the left-to-right direction; and negative 1, 57 00:03:10,110 --> 00:03:14,460 and negative 2, et cetera, if we went from right to left. 58 00:03:14,460 --> 00:03:18,820 In other words, even though the direction of the x-axis 59 00:03:18,820 --> 00:03:21,730 is still the same, whether we go from left 60 00:03:21,730 --> 00:03:23,850 to right or from right to left, we 61 00:03:23,850 --> 00:03:27,400 distinguish between the two motions in terms of-- 62 00:03:27,400 --> 00:03:30,380 and this is what we mean by the sense of a vector. 63 00:03:30,380 --> 00:03:32,950 Some people say, well the direction 64 00:03:32,950 --> 00:03:36,450 includes the sense; I prefer to have these two words dangling 65 00:03:36,450 --> 00:03:39,970 around, so I can put them at my disposal in any way 66 00:03:39,970 --> 00:03:41,300 that I see fit. 67 00:03:41,300 --> 00:03:45,710 But roughly speaking then, this is all a vector is. 68 00:03:45,710 --> 00:03:48,860 Now the idea is why arrows? 69 00:03:48,860 --> 00:03:52,490 And the answer is-- and here's a little ratio 70 00:03:52,490 --> 00:03:56,300 I put down here-- an arrow is to a vector 71 00:03:56,300 --> 00:03:59,384 as a length is to a scalar, whereby scalar, I 72 00:03:59,384 --> 00:04:00,800 mean a number, something which has 73 00:04:00,800 --> 00:04:02,910 a magnitude, but no direction. 74 00:04:02,910 --> 00:04:05,340 Remember when we first started talking 75 00:04:05,340 --> 00:04:08,520 about numerical arithmetic, and talked 76 00:04:08,520 --> 00:04:12,410 about graphing, and visualizing numbers as lengths. 77 00:04:12,410 --> 00:04:15,040 We picked the length to represent the number, 78 00:04:15,040 --> 00:04:18,040 and at that particular time, since there 79 00:04:18,040 --> 00:04:20,880 was only one degree of freedom-- namely the x-axis-- 80 00:04:20,880 --> 00:04:22,760 we didn't emphasize direction. 81 00:04:22,760 --> 00:04:25,490 But the idea is to carry this analogy further. 82 00:04:25,490 --> 00:04:29,260 What one does is when one has a quantity which 83 00:04:29,260 --> 00:04:32,920 is a vector, an arrow-- a traditional arrow-- 84 00:04:32,920 --> 00:04:36,020 a directed line segment with a given sense. 85 00:04:36,020 --> 00:04:38,200 The arrowhead indicating the sense 86 00:04:38,200 --> 00:04:41,130 is perhaps the best way of drawing 87 00:04:41,130 --> 00:04:43,210 what we mean by a vector. 88 00:04:43,210 --> 00:04:45,900 Namely, I have just a few random examples here. 89 00:04:45,900 --> 00:04:50,960 Here's a collection of things which are arrows. 90 00:04:50,960 --> 00:04:53,530 They're line segments in a given direction, 91 00:04:53,530 --> 00:04:56,990 and the arrowhead indicates a particular sense. 92 00:04:56,990 --> 00:04:59,960 Notice by the way, that the concept of a vector 93 00:04:59,960 --> 00:05:02,390 does not depend on an arrow, anymore 94 00:05:02,390 --> 00:05:05,360 than the concept of a number depends on a length. 95 00:05:05,360 --> 00:05:09,110 The arrow is simply a convenient geometric representation 96 00:05:09,110 --> 00:05:12,330 for visualizing the vector. 97 00:05:12,330 --> 00:05:15,990 Now hopefully, in terms of the last unit, 98 00:05:15,990 --> 00:05:18,890 you now have the feeling that in mathematics, 99 00:05:18,890 --> 00:05:21,640 to have an arithmetic, one needs more 100 00:05:21,640 --> 00:05:23,380 than a collection of objects. 101 00:05:23,380 --> 00:05:26,190 That to have a structure, one needs a collection 102 00:05:26,190 --> 00:05:30,190 of objects together with certain rules, and definitions, 103 00:05:30,190 --> 00:05:31,300 and what have you. 104 00:05:31,300 --> 00:05:34,560 And consequently, it becomes rather mandatory 105 00:05:34,560 --> 00:05:40,160 that somehow or other we be able to categorize a pair of vectors 106 00:05:40,160 --> 00:05:43,460 as to whether they shall be called equal or not. 107 00:05:43,460 --> 00:05:45,410 See, what shall we mean by even saying 108 00:05:45,410 --> 00:05:47,770 the two vectors shall be equal? 109 00:05:47,770 --> 00:05:49,480 The equality of vectors. 110 00:05:49,480 --> 00:05:51,260 And I claim the following-- and let 111 00:05:51,260 --> 00:05:53,510 me just state this cold-bloodedly 112 00:05:53,510 --> 00:05:56,940 until we come to the end, and show 113 00:05:56,940 --> 00:05:58,420 what we're driving at again. 114 00:05:58,420 --> 00:06:01,270 To say that the vector A equals the vector B means 115 00:06:01,270 --> 00:06:05,480 that first of all, the magnitude of A equals the magnitude of B. 116 00:06:05,480 --> 00:06:08,570 By the way, again as in the one-dimensional case-- when 117 00:06:08,570 --> 00:06:11,580 we called the absolute value the length-- 118 00:06:11,580 --> 00:06:13,570 we do the same thing in terms of arrows. 119 00:06:13,570 --> 00:06:15,710 We abbreviate the magnitude of an arrow 120 00:06:15,710 --> 00:06:18,660 by writing the absolute value of the arrow. 121 00:06:18,660 --> 00:06:20,640 In other words, the absolute value of A 122 00:06:20,640 --> 00:06:23,430 must equal the absolute value of B. 123 00:06:23,430 --> 00:06:28,540 Secondly, the arrow A must be parallel to the arrow B. 124 00:06:28,540 --> 00:06:33,330 And thirdly, A and B must have the same sense. 125 00:06:33,330 --> 00:06:34,950 And again to show you what this is, 126 00:06:34,950 --> 00:06:37,810 remember I said to you a moment ago that the arrows are just 127 00:06:37,810 --> 00:06:42,980 a convenient pictorial way of representing a vector. 128 00:06:42,980 --> 00:06:46,870 Let's observe that these three properties here 129 00:06:46,870 --> 00:06:53,390 juxtaposition very nicely with these three properties here. 130 00:06:53,390 --> 00:06:56,080 In other words, notice that our definition of equality 131 00:06:56,080 --> 00:07:01,010 of arrows captures exactly what three ingredients go 132 00:07:01,010 --> 00:07:03,180 into defining a vector. 133 00:07:03,180 --> 00:07:05,780 Since all we're talking about is magnitude, direction, 134 00:07:05,780 --> 00:07:10,310 and sense, how can we distinguish validly 135 00:07:10,310 --> 00:07:13,910 between two vectors other than in terms of magnitude, 136 00:07:13,910 --> 00:07:15,491 direction, and sense. 137 00:07:15,491 --> 00:07:15,990 OK? 138 00:07:20,140 --> 00:07:22,800 By the way, let's be very careful about this. 139 00:07:22,800 --> 00:07:25,470 As long as the only way that we can determine 140 00:07:25,470 --> 00:07:28,620 the difference between two vectors-- or arrows-- 141 00:07:28,620 --> 00:07:30,730 is in terms of their magnitude, direction, 142 00:07:30,730 --> 00:07:33,590 and sense, what this means is that if you 143 00:07:33,590 --> 00:07:36,290 have two vectors which are parallel, 144 00:07:36,290 --> 00:07:38,180 have the same length-- two arrows, 145 00:07:38,180 --> 00:07:42,580 same length, same direction, and the same sense, 146 00:07:42,580 --> 00:07:44,970 they must be called equal. 147 00:07:44,970 --> 00:07:47,270 You see, in other words, vectors do not 148 00:07:47,270 --> 00:07:49,680 have to coincide to be equal. 149 00:07:49,680 --> 00:07:51,690 Unlike the case of straight lines. 150 00:07:51,690 --> 00:07:54,000 In other words, if I were to draw 151 00:07:54,000 --> 00:07:57,570 a pair of parallel lines here of equal length, 152 00:07:57,570 --> 00:08:00,130 these would be called different lines. 153 00:08:00,130 --> 00:08:05,360 As soon as I do this these are called equal arrows. 154 00:08:05,360 --> 00:08:06,130 Why? 155 00:08:06,130 --> 00:08:08,300 Because of our definition of equality. 156 00:08:08,300 --> 00:08:10,690 And if somebody says, "But I don't like that definition," 157 00:08:10,690 --> 00:08:12,064 well this is the same as the rule 158 00:08:12,064 --> 00:08:13,890 that three strikes is an out in baseball. 159 00:08:13,890 --> 00:08:16,056 If somebody doesn't like that rule, you say to them, 160 00:08:16,056 --> 00:08:18,250 lookit, go and play your own game, 161 00:08:18,250 --> 00:08:21,067 and make as many strikes is an out as you want, but don't call 162 00:08:21,067 --> 00:08:23,650 that game baseball, because that will confuse it with the game 163 00:08:23,650 --> 00:08:25,800 that we're playing here, which is different. 164 00:08:25,800 --> 00:08:29,440 In other words, we make up our rules to govern, 165 00:08:29,440 --> 00:08:32,679 as I said in the last lecture, what we believe to be reality. 166 00:08:32,679 --> 00:08:35,380 And we believe that for how we're going to use vectors 167 00:08:35,380 --> 00:08:37,990 this is a very, very natural definition. 168 00:08:37,990 --> 00:08:40,440 And this is not the first time in mathematics 169 00:08:40,440 --> 00:08:42,049 that we've done something like this. 170 00:08:42,049 --> 00:08:44,710 Way back around the third or fourth grade, here's 171 00:08:44,710 --> 00:08:49,750 a little analogy, you get used to saying 1/2 equals 3/6. 172 00:08:49,750 --> 00:08:54,000 But obviously, the ordered number pair 1 comma 2, 173 00:08:54,000 --> 00:08:56,330 and 3 comma 6 look different. 174 00:08:56,330 --> 00:08:58,580 It certainly makes a difference physically 175 00:08:58,580 --> 00:09:01,120 whether you cut a pie in half and take one piece 176 00:09:01,120 --> 00:09:04,680 or cut the pie into six pieces and take three of them. 177 00:09:04,680 --> 00:09:06,400 The physical process is different. 178 00:09:06,400 --> 00:09:08,180 The two fractions don't look alike. 179 00:09:08,180 --> 00:09:09,970 Why do we call them equal? 180 00:09:09,970 --> 00:09:13,460 And it means that in terms of what we want ratios to mean, 181 00:09:13,460 --> 00:09:16,640 the number which you must multiply by 2 to get 1, 182 00:09:16,640 --> 00:09:20,100 is the same number that you must multiply by 6 to get 3, 183 00:09:20,100 --> 00:09:22,870 and consequently, let's just keep 184 00:09:22,870 --> 00:09:24,580 the equality in that sense. 185 00:09:24,580 --> 00:09:26,340 Who cares whether they look alike or not? 186 00:09:26,340 --> 00:09:28,920 Let's call them equal if they capture 187 00:09:28,920 --> 00:09:30,289 the important characteristic. 188 00:09:30,289 --> 00:09:31,830 And that's the way this game is going 189 00:09:31,830 --> 00:09:33,350 to be played all the time. 190 00:09:33,350 --> 00:09:37,160 We will make up our definitions to fit our mood, 191 00:09:37,160 --> 00:09:40,710 and hopefully the mood that suits me best will suit you 192 00:09:40,710 --> 00:09:44,210 best, because I will try to pick the reasons that you yourselves 193 00:09:44,210 --> 00:09:47,250 would have liked to have seen motivate 194 00:09:47,250 --> 00:09:49,270 why we did the things that we did 195 00:09:49,270 --> 00:09:50,920 or that we're going to do here. 196 00:09:50,920 --> 00:09:53,680 So let me talk about the "addition" of vectors, 197 00:09:53,680 --> 00:09:56,020 and let me put the word "Addition" in quotation marks 198 00:09:56,020 --> 00:09:59,560 here to indicate that all I mean by the addition of vectors-- 199 00:09:59,560 --> 00:10:02,340 and perhaps I should still say addition of arrows-- 200 00:10:02,340 --> 00:10:06,010 is I would like a rule that tells me how to combine two 201 00:10:06,010 --> 00:10:09,660 arrows to form another arrow. 202 00:10:09,660 --> 00:10:13,400 Now again I can make up hundreds of rules that tell me how, 203 00:10:13,400 --> 00:10:16,390 for two given arrows, I'm going to form a third arrow 204 00:10:16,390 --> 00:10:17,990 from the given two. 205 00:10:17,990 --> 00:10:21,040 But since our aim is to solve problems of the real world, 206 00:10:21,040 --> 00:10:23,810 why not pick a definition? 207 00:10:23,810 --> 00:10:26,890 Why not pick a definition that has already 208 00:10:26,890 --> 00:10:29,590 been used in physics and in other applications? 209 00:10:29,590 --> 00:10:31,110 The thing called the resultant. 210 00:10:31,110 --> 00:10:35,370 For example, given a point O, and two forces, 211 00:10:35,370 --> 00:10:40,670 represented as arrows, A and B, acting at O, we have agreed-- 212 00:10:40,670 --> 00:10:44,660 or you may recall-- that the resultant force is 213 00:10:44,660 --> 00:10:47,870 the vector that forms the diagonal 214 00:10:47,870 --> 00:10:52,650 of the parallelogram which has A and B as consecutive edges. 215 00:10:52,650 --> 00:10:56,600 In other words, this vector here was what in the old days 216 00:10:56,600 --> 00:10:58,970 was called the resultant. 217 00:10:58,970 --> 00:11:00,860 Because that has such a nice usage, 218 00:11:00,860 --> 00:11:03,625 I am going to call that the sum of two vectors. 219 00:11:06,580 --> 00:11:08,770 By the sum of two vectors I mean what physically 220 00:11:08,770 --> 00:11:10,510 used to be the resultant. 221 00:11:10,510 --> 00:11:12,790 And a very nice way of remembering the rule, 222 00:11:12,790 --> 00:11:16,390 is remember that this vector B is equal to this vector, 223 00:11:16,390 --> 00:11:18,150 because this is a parallelogram. 224 00:11:18,150 --> 00:11:22,300 These are equal in magnitude, parallel in direction, 225 00:11:22,300 --> 00:11:23,820 and have the same sense. 226 00:11:23,820 --> 00:11:26,320 Notice that if I want a quick recipe that 227 00:11:26,320 --> 00:11:30,560 was convenient for adding two vectors A and B, you say what? 228 00:11:30,560 --> 00:11:34,430 Place the tail of the second next to the head of the first, 229 00:11:34,430 --> 00:11:37,220 and then draw the vector-- the arrow-- 230 00:11:37,220 --> 00:11:40,030 that goes from the tail of the first 231 00:11:40,030 --> 00:11:42,270 to the head of the second. 232 00:11:42,270 --> 00:11:45,640 And this will be the definition of the addition of arrows. 233 00:11:45,640 --> 00:11:49,630 And again, this is done in much more detail in our written 234 00:11:49,630 --> 00:11:51,010 material. 235 00:11:51,010 --> 00:11:54,440 By the way, notice again, a parallelism 236 00:11:54,440 --> 00:11:57,660 between the structure of vectors and the structure 237 00:11:57,660 --> 00:11:59,470 of ordinary arithmetic. 238 00:11:59,470 --> 00:12:04,230 In the same way that, given two numbers, they have one sum, 239 00:12:04,230 --> 00:12:07,090 but that a given sum can be formed 240 00:12:07,090 --> 00:12:10,350 by different combinations of numbers, 241 00:12:10,350 --> 00:12:15,800 that two given vectors, which I call here x_1 and y_1, 242 00:12:15,800 --> 00:12:20,620 give A as a sum, but notice also that the vector A that is 243 00:12:20,620 --> 00:12:25,590 sum of the vectors x_2 and y_2. 244 00:12:25,590 --> 00:12:28,100 To generalize this, if I picked any point 245 00:12:28,100 --> 00:12:33,240 I wanted on the blackboard here, and drew any two lines, one 246 00:12:33,240 --> 00:12:37,060 going from the tail of A to this point, 247 00:12:37,060 --> 00:12:40,180 and one going from this point to the head of A. Notice 248 00:12:40,180 --> 00:12:42,900 that the resulting two vectors here-- 249 00:12:42,900 --> 00:12:46,970 this vector plus this vector, by definition of vector addition, 250 00:12:46,970 --> 00:12:49,080 head to tail, from the tail of the first 251 00:12:49,080 --> 00:12:52,200 to the head of the second, would still be the vector A. 252 00:12:52,200 --> 00:12:55,270 In other words, in terms of the diagram that I drew originally, 253 00:12:55,270 --> 00:12:58,550 notice that A can be written as x_1 plus y_1, 254 00:12:58,550 --> 00:13:02,430 it can also be written as x_2 plus y_2. 255 00:13:02,430 --> 00:13:05,460 And notice again the similarity of the structure 256 00:13:05,460 --> 00:13:10,140 between arithmetic and vectors, and notice the arbitrariness 257 00:13:10,140 --> 00:13:13,780 with which we make up our definitions. 258 00:13:13,780 --> 00:13:16,740 We make up the definitions to be our servant, 259 00:13:16,740 --> 00:13:19,620 rather than for us to become the servant of the subject. 260 00:13:19,620 --> 00:13:22,330 We model the definitions-- the truth 261 00:13:22,330 --> 00:13:27,240 that we call-- after what we believe happens in real life. 262 00:13:27,240 --> 00:13:31,110 By the way, a very convenient coordinate system 263 00:13:31,110 --> 00:13:36,380 is our old friend the x and y Cartesian coordinate system. 264 00:13:36,380 --> 00:13:39,310 And the reason for this is as follows. 265 00:13:39,310 --> 00:13:43,620 Given any vector A, we can always 266 00:13:43,620 --> 00:13:48,110 assume that the vector originates at the origin. 267 00:13:48,110 --> 00:13:49,790 And the reason for that is we've already 268 00:13:49,790 --> 00:13:52,100 seen that two vectors are equal if they 269 00:13:52,100 --> 00:13:54,650 have the same magnitude, direction, and sense 270 00:13:54,650 --> 00:13:57,100 so wherever A was in the plane of the blackboard, 271 00:13:57,100 --> 00:13:59,610 we can shift it parallel to itself, 272 00:13:59,610 --> 00:14:01,740 so its tail starts at the origin. 273 00:14:01,740 --> 00:14:03,980 Let's assume that the vector A, the arrow 274 00:14:03,980 --> 00:14:07,010 A, which starts at the origin, terminates at the point 275 00:14:07,010 --> 00:14:09,140 a comma b. 276 00:14:09,140 --> 00:14:13,760 Let i be a unit vector-- meaning a vector that 277 00:14:13,760 --> 00:14:16,610 has one unit of length-- in the direction 278 00:14:16,610 --> 00:14:18,340 of the positive x-axis. 279 00:14:18,340 --> 00:14:21,860 Let j be a unit vector which is in the direction 280 00:14:21,860 --> 00:14:24,340 of the positive y-axis, and our claim 281 00:14:24,340 --> 00:14:27,360 is that our definition of addition-- and you've 282 00:14:27,360 --> 00:14:29,290 all seen this, I'm pretty sure. 283 00:14:29,290 --> 00:14:32,750 Is that the vector A can now be written as what? 284 00:14:32,750 --> 00:14:39,340 A is equal to a*i plus b*j. 285 00:14:39,340 --> 00:14:42,330 Where you see, if you've seen this notation before 286 00:14:42,330 --> 00:14:44,080 notice that in terms of our game, 287 00:14:44,080 --> 00:14:47,200 we are now forced to introduce a new operation. 288 00:14:47,200 --> 00:14:49,710 Because we claim in our game that it's validity 289 00:14:49,710 --> 00:14:50,890 that's important. 290 00:14:50,890 --> 00:14:52,760 Validity, not truth. 291 00:14:52,760 --> 00:14:56,130 If all I can use are the rules of my game, for the first time, 292 00:14:56,130 --> 00:14:57,690 what do I see here? 293 00:14:57,690 --> 00:15:00,650 I see a number multiplying the vector. 294 00:15:00,650 --> 00:15:02,800 And if I don't have rules of my game 295 00:15:02,800 --> 00:15:05,870 that tell me what a number multiplied by a vector means, 296 00:15:05,870 --> 00:15:08,670 in terms of the logic machine, the logic machine 297 00:15:08,670 --> 00:15:11,730 can't do a thing to give me inescapable conclusions. 298 00:15:11,730 --> 00:15:16,380 Without assumptions, without rules, there can be no proof. 299 00:15:16,380 --> 00:15:18,760 So I must make up a new definition, 300 00:15:18,760 --> 00:15:21,670 and this definition is called scalar multiplication. 301 00:15:21,670 --> 00:15:25,010 And it's going to tell me how to multiply a scalar-- a number-- 302 00:15:25,010 --> 00:15:26,120 by a vector. 303 00:15:26,120 --> 00:15:30,750 If c is a number, and V is a vector, by c times V, 304 00:15:30,750 --> 00:15:34,210 I mean a vector, which has-- I have to be careful, 305 00:15:34,210 --> 00:15:36,320 see c could be a negative number. 306 00:15:36,320 --> 00:15:39,500 And since you think of length as being positive, 307 00:15:39,500 --> 00:15:41,390 I take the absolute value of c just 308 00:15:41,390 --> 00:15:43,431 to get the magnitude of the number in here. 309 00:15:43,431 --> 00:15:43,930 You see? 310 00:15:43,930 --> 00:15:45,960 In other words, this is a vector which 311 00:15:45,960 --> 00:15:53,260 has the magnitude of c times the magnitude of V. Meaning it's 312 00:15:53,260 --> 00:15:56,660 c times as large as V if c is positive, 313 00:15:56,660 --> 00:16:00,590 and minus c times as large as V if c is negative. 314 00:16:00,590 --> 00:16:03,110 The same meaning of absolute value as always. 315 00:16:03,110 --> 00:16:07,400 For example, 2 times V would be a vector whose magnitude 316 00:16:07,400 --> 00:16:12,050 was twice that of the vector V. Negative 2 times V would also 317 00:16:12,050 --> 00:16:14,590 be a vector whose magnitude is twice 318 00:16:14,590 --> 00:16:17,621 that of V. The other property is what? 319 00:16:17,621 --> 00:16:19,120 I've just told the magnitude, I also 320 00:16:19,120 --> 00:16:21,410 have to give you the direction and the sense. 321 00:16:21,410 --> 00:16:26,680 c times v is in the same direction as V. Minus 2 times 322 00:16:26,680 --> 00:16:31,380 V, and plus 2 times V both have the direction the V, 323 00:16:31,380 --> 00:16:32,300 and both are what? 324 00:16:32,300 --> 00:16:33,990 Twice the magnitude of V. 325 00:16:33,990 --> 00:16:36,680 The only difference between the minus is this. 326 00:16:36,680 --> 00:16:40,810 That c times V has the same sense. 327 00:16:40,810 --> 00:16:42,290 Same sense as what? 328 00:16:42,290 --> 00:16:46,960 Same sense as V, if c is positive. 329 00:16:46,960 --> 00:16:50,330 That means if c is negative, it has the opposite sense. 330 00:16:50,330 --> 00:16:53,980 In other words, for example, 2 times V 331 00:16:53,980 --> 00:16:59,890 has the same sense as V; minus 2V has the opposite sense of V. 332 00:16:59,890 --> 00:17:02,210 But the important point is this-- 333 00:17:02,210 --> 00:17:06,670 that vector arithmetic shares many structural properties 334 00:17:06,670 --> 00:17:08,300 of regular arithmetic. 335 00:17:08,300 --> 00:17:10,920 That is very crucial to understand. 336 00:17:10,920 --> 00:17:13,250 I have a friend of mine who doesn't 337 00:17:13,250 --> 00:17:16,140 like teaching students who are having 338 00:17:16,140 --> 00:17:18,619 a few problems with ordinary arithmetic, 339 00:17:18,619 --> 00:17:20,970 and likes to teach advanced courses. 340 00:17:20,970 --> 00:17:23,375 And he often has said that the trouble 341 00:17:23,375 --> 00:17:25,000 with the average high school curriculum 342 00:17:25,000 --> 00:17:27,150 for the average student, is that by the time 343 00:17:27,150 --> 00:17:29,590 the student is through, he knows three things: 344 00:17:29,590 --> 00:17:32,290 If you see a sign, change it; if you see a decimal point, 345 00:17:32,290 --> 00:17:34,902 move it; and if you see a fraction, invert it. 346 00:17:34,902 --> 00:17:36,610 But he should have gone one step further: 347 00:17:36,610 --> 00:17:38,620 when the average old-time engineering 348 00:17:38,620 --> 00:17:42,566 students saw a vector, he only knew one thing about it. 349 00:17:42,566 --> 00:17:44,190 And the one thing that he knew about it 350 00:17:44,190 --> 00:17:47,640 was, if you see a vector, break it down into components. 351 00:17:47,640 --> 00:17:50,390 And after that, that was the end of what they ever did 352 00:17:50,390 --> 00:17:51,870 with vectors in the old days. 353 00:17:51,870 --> 00:17:54,430 But the thing that is crucial to understand here 354 00:17:54,430 --> 00:17:56,590 is that we will use vector arithmetic 355 00:17:56,590 --> 00:17:58,240 in a far more powerful way. 356 00:17:58,240 --> 00:18:02,090 We will use the structure of vector arithmetic. 357 00:18:02,090 --> 00:18:05,120 In the same way that we formed algebra 358 00:18:05,120 --> 00:18:07,370 from numerical arithmetic, we will 359 00:18:07,370 --> 00:18:11,360 form vector algebra, and vector calculus 360 00:18:11,360 --> 00:18:14,486 from vector arithmetic. 361 00:18:14,486 --> 00:18:15,610 Let me give you an example. 362 00:18:15,610 --> 00:18:17,900 We already know that for two numbers a and b, 363 00:18:17,900 --> 00:18:20,130 a plus b equals b plus a. 364 00:18:20,130 --> 00:18:22,940 My claim is that from our definition of arrows 365 00:18:22,940 --> 00:18:25,200 and the like, it should also be a rule 366 00:18:25,200 --> 00:18:29,310 that for arrows, A plus B should equal B plus A. 367 00:18:29,310 --> 00:18:34,140 Or, for numbers, we already know that A plus B plus C equals 368 00:18:34,140 --> 00:18:36,800 A plus B plus C. 369 00:18:36,800 --> 00:18:39,200 That doesn't look right. 370 00:18:39,200 --> 00:18:41,420 By the way, what I wrote there wasn't false, 371 00:18:41,420 --> 00:18:42,620 it was certainly true. 372 00:18:42,620 --> 00:18:45,766 But it was too trivially true. 373 00:18:45,766 --> 00:18:47,390 I guess what I wanted to emphasize here 374 00:18:47,390 --> 00:18:50,610 was that the voice inflection made no difference. 375 00:18:50,610 --> 00:18:54,470 Now again, notice how we verify the rules of the game. 376 00:18:54,470 --> 00:18:58,040 I have to make up rules that I will call basic truths. 377 00:18:58,040 --> 00:19:00,240 Those rules-- or axioms-- will be 378 00:19:00,240 --> 00:19:03,640 modeled after what I believe to be true in reality. 379 00:19:03,640 --> 00:19:06,980 Now, notice in reality I've formed this whole idea 380 00:19:06,980 --> 00:19:08,640 in terms of what? 381 00:19:08,640 --> 00:19:09,390 Arrows. 382 00:19:09,390 --> 00:19:11,240 I defined what addition meant. 383 00:19:11,240 --> 00:19:14,060 Let's take a look and see what A plus B means. 384 00:19:14,060 --> 00:19:22,520 If this is the vector A, and this is the vector B, then 385 00:19:22,520 --> 00:19:25,200 what is the vector A plus B? 386 00:19:25,200 --> 00:19:28,690 By definition of how you add two vectors, 387 00:19:28,690 --> 00:19:35,240 this vector here would be A plus B. On the other hand, 388 00:19:35,240 --> 00:19:38,230 I could take the vector A, which is down here, 389 00:19:38,230 --> 00:19:41,520 move it parallel to itself so it originates here, 390 00:19:41,520 --> 00:19:44,040 mark it off the same length. 391 00:19:44,040 --> 00:19:45,950 This is still the vector A. 392 00:19:45,950 --> 00:19:51,320 If I now draw the arrow that goes from the tail of B 393 00:19:51,320 --> 00:19:55,250 to the head of A, what vector is this? 394 00:19:55,250 --> 00:19:58,840 Well by definition, it's B plus A. 395 00:19:58,840 --> 00:20:02,180 How are B plus A and A plus B related? 396 00:20:02,180 --> 00:20:04,490 They are opposite sides of a parallelogram, 397 00:20:04,490 --> 00:20:07,400 therefore they have the same magnitude, and same direction. 398 00:20:07,400 --> 00:20:10,510 And the picture shows us that they have the same sense. 399 00:20:10,510 --> 00:20:12,180 Since this is true in the picture, 400 00:20:12,180 --> 00:20:15,130 why not claim that one of the rules 401 00:20:15,130 --> 00:20:17,050 of our game of vector arithmetic-- 402 00:20:17,050 --> 00:20:19,620 even though legally you shouldn't use pictures, 403 00:20:19,620 --> 00:20:21,380 but pictures do give us the motivation. 404 00:20:21,380 --> 00:20:24,140 Why not say, OK, we'll accept the rule 405 00:20:24,140 --> 00:20:27,300 that A plus B equals B plus A. How 406 00:20:27,300 --> 00:20:29,250 about our second rule over here? 407 00:20:29,250 --> 00:20:31,100 Let's take three vectors. 408 00:20:31,100 --> 00:20:37,990 Let me take A over here, let me put B over here, 409 00:20:37,990 --> 00:20:39,729 and let me put C over here. 410 00:20:39,729 --> 00:20:41,770 And I hope that this will come out legible enough 411 00:20:41,770 --> 00:20:43,830 for all of us to see. 412 00:20:43,830 --> 00:20:47,170 I claim on the one hand that I can think of A plus B 413 00:20:47,170 --> 00:20:49,010 plus C in two different ways. 414 00:20:49,010 --> 00:20:52,900 In other words, to use the way that says A plus (B plus C) 415 00:20:52,900 --> 00:20:56,860 from this diagram, how do I get the vector B plus C? 416 00:20:56,860 --> 00:20:59,570 Since B and C are lined up in the right position head 417 00:20:59,570 --> 00:21:02,850 to tail, the vector B plus C is the one 418 00:21:02,850 --> 00:21:05,970 that goes from the tail of B to the head of C. 419 00:21:05,970 --> 00:21:09,700 So I call this vector B plus C. 420 00:21:09,700 --> 00:21:14,010 Now, what is A plus (B plus C)? 421 00:21:14,010 --> 00:21:18,110 It's this vector plus this one. 422 00:21:18,110 --> 00:21:21,150 In other words, this vector here could 423 00:21:21,150 --> 00:21:28,080 be labeled A plus (B plus C). 424 00:21:28,080 --> 00:21:32,660 On the other hand, what is the vector A plus B? 425 00:21:32,660 --> 00:21:37,760 A plus B is this vector. 426 00:21:37,760 --> 00:21:42,090 Because it goes from the tail of A to the head of B 427 00:21:42,090 --> 00:21:45,100 and consequently, (A plus B) plus 428 00:21:45,100 --> 00:21:48,850 C is the vector that goes from the tail of A plus B 429 00:21:48,850 --> 00:21:52,800 to the head of C. In other words, what vector is that? 430 00:21:52,800 --> 00:21:55,910 It's this vector, but what name does that have? 431 00:21:55,910 --> 00:21:57,050 We just saw. 432 00:21:57,050 --> 00:22:04,080 That's (A plus B) plus C. And now we see what? 433 00:22:04,080 --> 00:22:07,100 We have two different names for the same arrow. 434 00:22:07,100 --> 00:22:10,630 In particular then, these two different names represent what? 435 00:22:10,630 --> 00:22:13,630 Two arrows which have the same magnitude, sense, 436 00:22:13,630 --> 00:22:16,030 and direction, and therefore they're equal. 437 00:22:16,030 --> 00:22:18,680 And that's why we make up rules like this. 438 00:22:18,680 --> 00:22:21,160 And by the way, the analogy continues on, 439 00:22:21,160 --> 00:22:24,345 and this is done in much more detail in our notes, 440 00:22:24,345 --> 00:22:27,560 but I just thought I would like to say a few more 441 00:22:27,560 --> 00:22:28,960 words in general. 442 00:22:28,960 --> 00:22:32,530 For example, one talks about the zero vector. 443 00:22:32,530 --> 00:22:34,540 And I write that with a 0 with an arrow over it, 444 00:22:34,540 --> 00:22:36,410 to indicate the zero arrow. 445 00:22:36,410 --> 00:22:39,690 Maybe your intuition tells you that the zero vector should 446 00:22:39,690 --> 00:22:41,930 somehow be connected with a zero number, 447 00:22:41,930 --> 00:22:43,870 but the point that I want to drive at 448 00:22:43,870 --> 00:22:47,200 is an analogy to what we did in the previous lecture, when 449 00:22:47,200 --> 00:22:50,840 we showed why b to the 0 has to be defined to be 1. 450 00:22:50,840 --> 00:22:53,280 The idea is that an ordinary arithmetic, what 451 00:22:53,280 --> 00:22:55,640 is the beauty of the number 0? 452 00:22:55,640 --> 00:22:58,930 The beauty of the numbers 0 is that with respect to addition, 453 00:22:58,930 --> 00:23:01,590 it does not change the identity of any number. 454 00:23:01,590 --> 00:23:04,510 Any number plus 0 is that number back again. 455 00:23:04,510 --> 00:23:08,390 In particular then, why not keep the same structure for vectors? 456 00:23:08,390 --> 00:23:09,580 And it's our choice to make. 457 00:23:09,580 --> 00:23:12,990 We would like to say OK, let's define the zero vector to be 458 00:23:12,990 --> 00:23:15,680 such that if A is any other vector, 459 00:23:15,680 --> 00:23:19,880 A plus 0 will still be A. Well let's 460 00:23:19,880 --> 00:23:21,510 get an idea of what that means. 461 00:23:21,510 --> 00:23:25,060 Let B denote the magnitude of the zero vector. 462 00:23:25,060 --> 00:23:26,570 What this means is what? 463 00:23:26,570 --> 00:23:28,880 How do I add 0 to A? 464 00:23:28,880 --> 00:23:34,890 I start with the vector A, and it originates here. 465 00:23:34,890 --> 00:23:37,650 And terminates at the point p. 466 00:23:37,650 --> 00:23:40,500 Now how do I add on the zero vector? 467 00:23:40,500 --> 00:23:43,920 The zero vector, all I know is it's going to start at P, 468 00:23:43,920 --> 00:23:46,010 and be length b. 469 00:23:46,010 --> 00:23:47,570 So the surest thing I can say is, 470 00:23:47,570 --> 00:23:52,000 lookit, let me draw a circle centered at P, with the radius 471 00:23:52,000 --> 00:23:54,500 equal to b. 472 00:23:54,500 --> 00:23:56,360 In other words the radius of this circle 473 00:23:56,360 --> 00:24:00,100 here, the radius of this circle is b. 474 00:24:00,100 --> 00:24:02,750 All I know is that whatever the zero vector is, 475 00:24:02,750 --> 00:24:06,790 if its magnitude is b, the zero vector must originated at P, 476 00:24:06,790 --> 00:24:09,290 and terminate someplace on this circle. 477 00:24:09,290 --> 00:24:12,740 Because that's the locus of all vectors of length b 478 00:24:12,740 --> 00:24:14,250 which originate at P. 479 00:24:14,250 --> 00:24:17,390 For the sake of argument, let's say that the zero vector 480 00:24:17,390 --> 00:24:20,540 happens to be PQ. 481 00:24:20,540 --> 00:24:27,200 Then A plus 0 is the vector that goes from this point to Q. 482 00:24:27,200 --> 00:24:31,340 But A plus 0 must be A. And since A and A plus 0 483 00:24:31,340 --> 00:24:33,597 originate at the same point, the only way 484 00:24:33,597 --> 00:24:35,430 they can have the same magnitude, direction, 485 00:24:35,430 --> 00:24:38,060 and sense, is that they must coincide. 486 00:24:38,060 --> 00:24:41,670 In other words, in particular, P must equal Q. 487 00:24:41,670 --> 00:24:45,850 And as soon as P equals Q, the magnitude of the zero vector 488 00:24:45,850 --> 00:24:48,210 must be the number 0. 489 00:24:48,210 --> 00:24:51,280 In other words, in terms of our game structure again, 490 00:24:51,280 --> 00:24:54,260 if we want this structural to be obeyed, 491 00:24:54,260 --> 00:24:57,350 and again I must emphasize this: The choice is ours to make. 492 00:24:57,350 --> 00:24:59,920 But if we want the structural rule to be obeyed, 493 00:24:59,920 --> 00:25:04,100 we must define the zero vector to be the vector which 494 00:25:04,100 --> 00:25:06,400 has zero magnitude. 495 00:25:06,400 --> 00:25:07,952 As natural as it seems, that's not 496 00:25:07,952 --> 00:25:09,160 the real reason for doing it. 497 00:25:09,160 --> 00:25:12,910 We do it because it gives us the structure that we want. 498 00:25:12,910 --> 00:25:16,710 The same is true, for example, about the additive inverse. 499 00:25:16,710 --> 00:25:20,040 Remember in ordinary arithmetic, the additive inverse 500 00:25:20,040 --> 00:25:20,810 meant what? 501 00:25:20,810 --> 00:25:23,160 The number that must be added to the given number 502 00:25:23,160 --> 00:25:24,860 to give the zero number. 503 00:25:24,860 --> 00:25:26,540 Or the number 0. 504 00:25:26,540 --> 00:25:30,240 In vectors, we should define negative A 505 00:25:30,240 --> 00:25:32,620 to be the vector which when added to A 506 00:25:32,620 --> 00:25:35,270 gives the zero vector. 507 00:25:35,270 --> 00:25:37,330 Well, lookit, any vector that I add to A-- 508 00:25:37,330 --> 00:25:39,490 let me call that vector B. Any vector 509 00:25:39,490 --> 00:25:49,780 that I add to A-- which I call A plus B-- looks like this. 510 00:25:49,780 --> 00:25:52,680 Well what is the only way that the vector A 511 00:25:52,680 --> 00:25:55,680 plus B can be the zero vector? 512 00:25:55,680 --> 00:25:57,640 Well the zero vector has no length. 513 00:25:57,640 --> 00:25:59,780 That means the only way that's possible 514 00:25:59,780 --> 00:26:03,280 is if the tail of A plus B coincides 515 00:26:03,280 --> 00:26:07,690 with the head of A plus B. And the only way that can happen, 516 00:26:07,690 --> 00:26:10,190 is if B terminates right here. 517 00:26:10,190 --> 00:26:14,250 In other words, the vector that we call negative A 518 00:26:14,250 --> 00:26:20,380 is going to be the vector which has the same magnitude 519 00:26:20,380 --> 00:26:23,090 and direction as A, but the opposite sense. 520 00:26:23,090 --> 00:26:24,990 And we're not saying it's self evident. 521 00:26:24,990 --> 00:26:26,730 All we're saying is look-- if you 522 00:26:26,730 --> 00:26:29,060 want this structural property to be true, 523 00:26:29,060 --> 00:26:32,460 you have no choice but to define negative A to be 524 00:26:32,460 --> 00:26:36,100 the vector which has the same magnitude and direction as A, 525 00:26:36,100 --> 00:26:37,550 but the opposite sense. 526 00:26:37,550 --> 00:26:39,680 And by the way, as soon as you do that, 527 00:26:39,680 --> 00:26:42,350 there are no more guessing games involved in how 528 00:26:42,350 --> 00:26:43,940 you subtract two vectors. 529 00:26:43,940 --> 00:26:46,060 For, just in the same way that we did it 530 00:26:46,060 --> 00:26:49,820 in ordinary arithmetic, we define the difference of two 531 00:26:49,820 --> 00:26:52,670 vectors A minus B to mean the vector 532 00:26:52,670 --> 00:26:57,800 A added to the vector negative B. And what is negative B? 533 00:26:57,800 --> 00:27:01,230 The vector which just has the opposite sense of the vector 534 00:27:01,230 --> 00:27:04,450 B. And again, these computational details 535 00:27:04,450 --> 00:27:09,090 are left to the notes, and to the exercises. 536 00:27:09,090 --> 00:27:10,850 But the main overview that I want 537 00:27:10,850 --> 00:27:13,140 you to get from this lecture is to see 538 00:27:13,140 --> 00:27:15,960 that there are more to vectors than dividing them up 539 00:27:15,960 --> 00:27:17,150 into components. 540 00:27:17,150 --> 00:27:19,910 That there is an arithmetic imposed on them, 541 00:27:19,910 --> 00:27:22,590 and that arithmetic that's impose on vectors 542 00:27:22,590 --> 00:27:24,590 is just as real as the arithmetic 543 00:27:24,590 --> 00:27:26,610 of the ordinary real numbers. 544 00:27:26,610 --> 00:27:29,160 And to reinforce this, in our next lesson, 545 00:27:29,160 --> 00:27:33,310 we will talk about vectors in three-dimensional space, where 546 00:27:33,310 --> 00:27:35,860 the geometry becomes tougher, but the structure 547 00:27:35,860 --> 00:27:37,720 remains exactly the same. 548 00:27:37,720 --> 00:27:41,020 But until next time, good bye. 549 00:27:41,020 --> 00:27:43,390 Funding for the publication of this video 550 00:27:43,390 --> 00:27:48,270 was provided by the Gabriella and Paul Rosenbaum foundation. 551 00:27:48,270 --> 00:27:52,440 Help OCW continue to provide free and open access to MIT 552 00:27:52,440 --> 00:28:00,150 courses, by making a donation at ocw.mit.edu/donate.