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 00:00:33,590 --> 00:00:34,590 PROFESSOR: Hi. 9 00:00:34,590 --> 00:00:36,470 Our lecture today is on the one hand, 10 00:00:36,470 --> 00:00:40,790 deceptively simple, and on the other hand, deceptively hard. 11 00:00:40,790 --> 00:00:43,110 That from a certain point of view, 12 00:00:43,110 --> 00:00:46,620 to talk about the equations of lines and planes 13 00:00:46,620 --> 00:00:50,200 is, first of all, a topic that many people 14 00:00:50,200 --> 00:00:52,130 say, "I'm not interested in studying geometry. 15 00:00:52,130 --> 00:00:54,230 When are we going to get back to calculus?" 16 00:00:54,230 --> 00:00:58,120 And this is very analogous to our problem of part one 17 00:00:58,120 --> 00:01:00,270 in calculus, if you recall, where 18 00:01:00,270 --> 00:01:02,490 we started with analytic geometry, 19 00:01:02,490 --> 00:01:04,470 and the question came up, why do we 20 00:01:04,470 --> 00:01:07,610 need geometry when what we really wanted to study 21 00:01:07,610 --> 00:01:08,470 was calculus? 22 00:01:08,470 --> 00:01:11,400 And the idea of graphs played a very important role 23 00:01:11,400 --> 00:01:12,060 in calculus. 24 00:01:12,060 --> 00:01:16,070 We found out, for example, also, that the relatively harmless 25 00:01:16,070 --> 00:01:19,090 straight line was a rather crucial curve. 26 00:01:19,090 --> 00:01:23,400 Namely, given any curve, no matter how random in the plane, 27 00:01:23,400 --> 00:01:25,640 provided only that it was smooth, 28 00:01:25,640 --> 00:01:28,140 we were able to approximate that curve 29 00:01:28,140 --> 00:01:32,170 by a sequence of tangent lines to various points. 30 00:01:32,170 --> 00:01:35,530 And in a similar way, one finds that planes 31 00:01:35,530 --> 00:01:39,830 will do for functions of two real variables what 32 00:01:39,830 --> 00:01:42,620 the equations of lines did for functions 33 00:01:42,620 --> 00:01:44,990 of a single real variable. 34 00:01:44,990 --> 00:01:47,940 As I say, we'll talk about that in more detail as we go along. 35 00:01:47,940 --> 00:01:50,630 The other nice part about this lesson 36 00:01:50,630 --> 00:01:54,350 is the fact that we can now take some of our vector properties 37 00:01:54,350 --> 00:01:56,350 that we've learned from the past lessons 38 00:01:56,350 --> 00:01:59,190 and hit what I call the home run ball, once 39 00:01:59,190 --> 00:02:02,400 and for all now, finish off what we've started. 40 00:02:02,400 --> 00:02:06,090 And that is, in the study of three-dimensional geometry, 41 00:02:06,090 --> 00:02:09,360 the crucial characteristics are those building blocks 42 00:02:09,360 --> 00:02:11,220 called lines and planes. 43 00:02:11,220 --> 00:02:13,610 And by finding convenient recipes 44 00:02:13,610 --> 00:02:15,810 for expressing lines and planes, we'll 45 00:02:15,810 --> 00:02:18,190 be most of the way home as far as using 46 00:02:18,190 --> 00:02:19,640 this material is concerned. 47 00:02:19,640 --> 00:02:22,690 Again, we'll hit mainly the highlights today. 48 00:02:22,690 --> 00:02:25,940 The remainder of the material will be covered, 49 00:02:25,940 --> 00:02:29,280 I hope, adequately between the text and the exercises. 50 00:02:29,280 --> 00:02:32,740 At any rate then, the lesson today 51 00:02:32,740 --> 00:02:36,050 is "Equations of Lines and Planes." 52 00:02:36,050 --> 00:02:39,880 And to refresh what I just said before, the little ratio-- 53 00:02:39,880 --> 00:02:45,510 planes are to surfaces what lines are to curves-- that we 54 00:02:45,510 --> 00:02:47,990 can approximate curves by tangent lines, 55 00:02:47,990 --> 00:02:52,240 we can approximate smooth surfaces by tangent planes. 56 00:02:52,240 --> 00:02:54,630 Now what we would like to do is go back 57 00:02:54,630 --> 00:02:58,322 to Cartesian coordinates and find the equation of a plane. 58 00:02:58,322 --> 00:03:00,280 The first question that we asked before we even 59 00:03:00,280 --> 00:03:03,920 use any coordinate system is just how much information do 60 00:03:03,920 --> 00:03:06,950 you need before you can determine a plane? 61 00:03:06,950 --> 00:03:10,880 Well one way we know of is to know three points in the plane. 62 00:03:10,880 --> 00:03:13,560 Another way is analogous to the line case, 63 00:03:13,560 --> 00:03:16,980 where to determine a line, you needed a point and a slope. 64 00:03:16,980 --> 00:03:20,410 One way of determining a plane is to know what? 65 00:03:20,410 --> 00:03:24,510 A point in the plane and the direction of the plane. 66 00:03:24,510 --> 00:03:26,950 And one way of getting the direction of the plane 67 00:03:26,950 --> 00:03:30,644 is to fix a normal, a perpendicular to the plane. 68 00:03:30,644 --> 00:03:32,810 In other words, the approach that we're going to use 69 00:03:32,810 --> 00:03:35,220 is, let's suppose that we know a point that we 70 00:03:35,220 --> 00:03:36,710 want to be in our plane. 71 00:03:36,710 --> 00:03:38,890 And we'll call that point P_0. 72 00:03:38,890 --> 00:03:43,120 We'll denote it generically by (x_0, y_0, z_0). 73 00:03:43,120 --> 00:03:48,100 And let's let capital N denote the vector A*i plus B*j plus 74 00:03:48,100 --> 00:03:51,280 C*k, which is perpendicular to our plane. 75 00:03:51,280 --> 00:03:51,780 OK? 76 00:03:51,780 --> 00:03:54,850 So we're given a vector perpendicular to the plane, 77 00:03:54,850 --> 00:03:56,930 we're given a point in the plane, 78 00:03:56,930 --> 00:03:58,710 and now in Cartesian coordinates, 79 00:03:58,710 --> 00:04:01,090 we would like to know the equation of the plane. 80 00:04:01,090 --> 00:04:04,460 And as we do so often, we simply come back to our little diagram 81 00:04:04,460 --> 00:04:07,200 here that will utilize vector geometry. 82 00:04:07,200 --> 00:04:10,510 What we say is, OK, here's a little diagram here. 83 00:04:10,510 --> 00:04:12,660 Here's P_0 in the plane. 84 00:04:12,660 --> 00:04:15,030 Here's N, normal to the plane. 85 00:04:15,030 --> 00:04:17,120 Now we pick any point P whatsoever, 86 00:04:17,120 --> 00:04:21,279 any point in the whole world whatsoever, as long 87 00:04:21,279 --> 00:04:23,770 as it's in space, three-dimensional space. 88 00:04:23,770 --> 00:04:26,740 And what we say is-- what does it mean for the point P 89 00:04:26,740 --> 00:04:27,980 to be in the plane? 90 00:04:27,980 --> 00:04:33,880 Well, in vector language, for the point P to be in the plane, 91 00:04:33,880 --> 00:04:38,590 I think it's rather obvious that P_0 P had better 92 00:04:38,590 --> 00:04:42,800 be perpendicular to N. On the other hand, 93 00:04:42,800 --> 00:04:45,010 in dot product language, what does it 94 00:04:45,010 --> 00:04:48,380 mean to say that N is perpendicular to P_0 P? 95 00:04:48,380 --> 00:04:54,410 That says that the dot product of N and P_0 P must be 0. 96 00:04:54,410 --> 00:04:57,900 Now just to economize space, let me utilize the notation 97 00:04:57,900 --> 00:04:59,640 that we've talked about in the notes 98 00:04:59,640 --> 00:05:02,970 and in the exercises where in Cartesian coordinates 99 00:05:02,970 --> 00:05:07,180 I'll abbreviate a vector in i, j, and k components 100 00:05:07,180 --> 00:05:09,095 just by writing down its components. 101 00:05:09,095 --> 00:05:12,450 In other words, let me abbreviate N by A comma B comma 102 00:05:12,450 --> 00:05:16,840 C, which stands for A*i plus B*j plus C*k, et cetera. 103 00:05:16,840 --> 00:05:19,940 Notice the beauty, now, of our Cartesian coordinate system. 104 00:05:19,940 --> 00:05:23,510 N is the vector whose components are A, B, and C. What about P_0 105 00:05:23,510 --> 00:05:24,470 P? 106 00:05:24,470 --> 00:05:27,000 Well, that's the vector whose components 107 00:05:27,000 --> 00:05:32,000 are x minus x_0, y minus y_0, and z minus z_0, 108 00:05:32,000 --> 00:05:34,590 that beauty of Cartesian coordinates again. 109 00:05:34,590 --> 00:05:37,350 Consequently, I want the dot product of these two vectors 110 00:05:37,350 --> 00:05:38,550 to be 0. 111 00:05:38,550 --> 00:05:40,770 But in Cartesian coordinates, we know 112 00:05:40,770 --> 00:05:43,300 that to dot two vectors, we simply 113 00:05:43,300 --> 00:05:46,490 multiply corresponding components and add, 114 00:05:46,490 --> 00:05:48,190 and that results in what? 115 00:05:48,190 --> 00:05:54,360 A times (x minus x_0) plus B times (y minus y_0), 116 00:05:54,360 --> 00:05:59,100 plus C times (z minus z_0) equals 0. 117 00:05:59,100 --> 00:06:02,700 By the way, each of these steps is reversible, 118 00:06:02,700 --> 00:06:07,770 meaning that under these given conditions, the point (x, y, z) 119 00:06:07,770 --> 00:06:10,700 is in the given plane if and only 120 00:06:10,700 --> 00:06:14,180 if this equation here is satisfied. 121 00:06:14,180 --> 00:06:18,000 Another way of saying that is what? 122 00:06:18,000 --> 00:06:26,140 If A, B, C, x_0, y_0, and z_0 are given constants, 123 00:06:26,140 --> 00:06:32,320 then A times x minus x_0 plus B times y minus y_0 plus C times 124 00:06:32,320 --> 00:06:37,790 z minus z_0 equals 0 is the equation of the plane which 125 00:06:37,790 --> 00:06:42,550 passes through the point x_0, y_0, z_0, 126 00:06:42,550 --> 00:06:48,880 and which has the vector A*i plus B*j plus C*k as its normal 127 00:06:48,880 --> 00:06:49,910 vector. 128 00:06:49,910 --> 00:06:52,130 And perhaps the easiest way to illustrate this 129 00:06:52,130 --> 00:06:54,660 is again, by means of an example. 130 00:06:54,660 --> 00:06:57,730 Let me simply write down a linear expression. 131 00:06:57,730 --> 00:07:02,860 I'll write down 2 times x minus 1 plus 3 times y plus 2 132 00:07:02,860 --> 00:07:05,680 plus 4 times z minus 5. 133 00:07:05,680 --> 00:07:08,730 My claim is that this is the special case where 134 00:07:08,730 --> 00:07:14,070 A, B, and C are played by 2, 3, and 4, respectively; 135 00:07:14,070 --> 00:07:21,020 x_0, y_0, and z_0 are played by 1, negative 2, and negative 5-- 136 00:07:21,020 --> 00:07:24,230 the same thing as in our previous study 137 00:07:24,230 --> 00:07:27,610 of ordinary two-dimensional geometry. 138 00:07:27,610 --> 00:07:30,040 Remember that the standard form of our equation 139 00:07:30,040 --> 00:07:31,640 uses a minus sign. 140 00:07:31,640 --> 00:07:33,830 Consequently, to use the equation, 141 00:07:33,830 --> 00:07:37,320 where we see y plus 2, we should rewrite that 142 00:07:37,320 --> 00:07:40,220 as y minus minus 2. 143 00:07:40,220 --> 00:07:43,520 And what we're saying is that this equation is then what? 144 00:07:43,520 --> 00:07:50,130 It passes through the point (1, -2, 5). 145 00:07:50,130 --> 00:07:57,250 And has as its normal the vector 2*i plus 3*j plus 4*k. 146 00:07:57,250 --> 00:08:00,170 Actually, this is not a very difficult concept 147 00:08:00,170 --> 00:08:05,070 once you try a few examples and see what's happening over here. 148 00:08:05,070 --> 00:08:06,840 By the way, before I go on, I just 149 00:08:06,840 --> 00:08:09,820 want to make a note that I'm going to return to later on. 150 00:08:09,820 --> 00:08:11,710 Notice, by the way, there is nothing 151 00:08:11,710 --> 00:08:15,160 sacred about the right-hand side of this equation being 0. 152 00:08:15,160 --> 00:08:18,890 Notice that somehow or other, the really important factor 153 00:08:18,890 --> 00:08:23,060 was 2x plus 3y plus 4z. 154 00:08:23,060 --> 00:08:27,380 In other words, notice that the other terms led to a constant 155 00:08:27,380 --> 00:08:29,960 which could have been transposed onto the right-hand side 156 00:08:29,960 --> 00:08:33,000 of the equation, and that somehow or other, 157 00:08:33,000 --> 00:08:34,830 I can change the constant. 158 00:08:34,830 --> 00:08:38,190 But if these multipliers in front-- the 2, 3, and 4-- 159 00:08:38,190 --> 00:08:39,567 stay the same, see? 160 00:08:39,567 --> 00:08:40,900 Notice what I'm driving at here. 161 00:08:40,900 --> 00:08:43,940 No matter how I-- let me go back up here for a second. 162 00:08:43,940 --> 00:08:49,860 No matter how I change x_0, y_0, z_0, whatever plane this is, 163 00:08:49,860 --> 00:08:56,010 it still has A*i plus B*j plus C*k as a normal vector. 164 00:08:56,010 --> 00:09:00,320 What I can change is what point the plane passes through. 165 00:09:00,320 --> 00:09:04,290 In other words, somehow or other if I leave A, B, and C fixed, 166 00:09:04,290 --> 00:09:07,900 but I vary x_0, y_0, z_0, I generate 167 00:09:07,900 --> 00:09:10,430 a family of parallel planes. 168 00:09:10,430 --> 00:09:14,012 And that can be restated somewhat differently. 169 00:09:14,012 --> 00:09:15,970 Well I guess if it wasn't somewhat differently, 170 00:09:15,970 --> 00:09:18,430 that wouldn't be called "restated," would it? 171 00:09:18,430 --> 00:09:19,750 Let's just note this way. 172 00:09:19,750 --> 00:09:25,840 For fixed A, B, and C, the equation A*x plus B*y plus C*z 173 00:09:25,840 --> 00:09:29,180 equals D. See, forget about the 0 on the right-hand side. 174 00:09:29,180 --> 00:09:31,280 Let D be an arbitrary constant. 175 00:09:31,280 --> 00:09:33,470 What I'm saying is that this equation 176 00:09:33,470 --> 00:09:37,750 is a family of parallel planes. 177 00:09:37,750 --> 00:09:40,560 And why is it a family of parallel planes? 178 00:09:40,560 --> 00:09:46,200 Because every plane in this family has, as a normal vector, 179 00:09:46,200 --> 00:09:50,120 A*i plus B*j plus C*k. 180 00:09:50,120 --> 00:09:54,826 A*i plus B*j plus C*k. 181 00:09:54,826 --> 00:09:55,690 OK? 182 00:09:55,690 --> 00:09:57,690 The second point I would like to emphasize 183 00:09:57,690 --> 00:09:59,780 about the equation of our plane is 184 00:09:59,780 --> 00:10:03,616 that it's called a linear equation, meaning-- 185 00:10:03,616 --> 00:10:06,240 and I don't know why I suddenly switched to small letters here, 186 00:10:06,240 --> 00:10:08,198 but that certainly doesn't make any difference. 187 00:10:08,198 --> 00:10:10,900 With a, b, c, and d as constants, 188 00:10:10,900 --> 00:10:15,640 observe that a*x plus b*y plus c*z equals d is what we call 189 00:10:15,640 --> 00:10:19,720 a linear algebraic equation in the variables x, y, and z. 190 00:10:19,720 --> 00:10:24,690 Namely, each variable appears multiplied only by a constant. 191 00:10:24,690 --> 00:10:26,990 And we add these things up. 192 00:10:26,990 --> 00:10:31,830 And notice that in a way, a plane should be linear, 193 00:10:31,830 --> 00:10:34,310 meaning there's no curvature to a plane 194 00:10:34,310 --> 00:10:36,000 once it's fixed in space. 195 00:10:36,000 --> 00:10:40,220 And this sort of generalizes the idea of the line. 196 00:10:40,220 --> 00:10:44,250 Remember, the general definition of a line was of the form what? 197 00:10:44,250 --> 00:10:47,350 a*x plus b*y equals a constant. 198 00:10:47,350 --> 00:10:50,280 That was a two-dimensional linear equation. 199 00:10:50,280 --> 00:10:53,354 The plane is a three-dimensional linear equation. 200 00:10:53,354 --> 00:10:54,770 And one of the subjects that we'll 201 00:10:54,770 --> 00:10:57,700 return to later in the course, but I just mention it 202 00:10:57,700 --> 00:10:59,990 in passing now, is that even though you 203 00:10:59,990 --> 00:11:03,040 can't draw in more than three-dimensional space, 204 00:11:03,040 --> 00:11:05,620 if you have 15 variables, you can certainly 205 00:11:05,620 --> 00:11:09,220 have a linear equation in 15 unknowns. 206 00:11:09,220 --> 00:11:12,010 And the interesting point in calculus of several variables 207 00:11:12,010 --> 00:11:15,980 is that even when you run out of pictures-- when you can't draw 208 00:11:15,980 --> 00:11:18,690 the situation-- the linear equation 209 00:11:18,690 --> 00:11:22,960 plays a very, very special role in the development of calculus 210 00:11:22,960 --> 00:11:26,050 of several variables, analogous to what a line does 211 00:11:26,050 --> 00:11:28,870 for a curve in the case of one variable 212 00:11:28,870 --> 00:11:30,920 and what a plane does for a surface 213 00:11:30,920 --> 00:11:33,235 in the case of two variables. 214 00:11:33,235 --> 00:11:35,580 What I wanted to emphasize though, also-- 215 00:11:35,580 --> 00:11:38,350 and we'll come back to this in very short order-- 216 00:11:38,350 --> 00:11:42,980 is that a plane has two degrees of freedom. 217 00:11:42,980 --> 00:11:44,660 Meaning what? 218 00:11:44,660 --> 00:11:48,977 That in a plane, observe that given the linear equation, 219 00:11:48,977 --> 00:11:49,560 you have what? 220 00:11:49,560 --> 00:11:51,750 One linear equation and three unknowns. 221 00:11:51,750 --> 00:11:56,310 It requires that you pick two of the variables 222 00:11:56,310 --> 00:11:58,570 before the rest of the equation is determined. 223 00:11:58,570 --> 00:12:03,770 In other words, if given x + 2y + 3z = 6, if I say, 224 00:12:03,770 --> 00:12:06,850 let x be 15, notice that I have what? 225 00:12:06,850 --> 00:12:13,210 15 + 2y + 3z = 6, which gives me one equation and the two 226 00:12:13,210 --> 00:12:14,910 unknowns y and z. 227 00:12:14,910 --> 00:12:17,680 That to actually uniquely fix anything, 228 00:12:17,680 --> 00:12:21,350 I must specify what two of the three variables are. 229 00:12:21,350 --> 00:12:23,040 In other words, in this equation, 230 00:12:23,040 --> 00:12:26,490 I can choose any two of the three variables at random, 231 00:12:26,490 --> 00:12:28,350 and solve for the third. 232 00:12:28,350 --> 00:12:31,900 By the way, if anybody is having difficulty 233 00:12:31,900 --> 00:12:34,090 understanding the difference between the 6 234 00:12:34,090 --> 00:12:36,850 being on the right-hand side as we have it now 235 00:12:36,850 --> 00:12:40,310 and the 0 as we used it originally, 236 00:12:40,310 --> 00:12:45,110 notice that we can very quickly find a point which 237 00:12:45,110 --> 00:12:47,200 this plane passes through. 238 00:12:47,200 --> 00:12:49,870 For example, among other things, just set y 239 00:12:49,870 --> 00:12:53,350 and z equal to 0, in which case x is 6. 240 00:12:53,350 --> 00:12:58,690 So certainly one point in this plane is (6, 0, 0). 241 00:12:58,690 --> 00:13:01,920 What is a vector perpendicular to this plane? 242 00:13:01,920 --> 00:13:06,700 It's the vector 1i plus 2j plus 3k. 243 00:13:06,700 --> 00:13:09,790 So using the standard form, we could write what? 244 00:13:09,790 --> 00:13:20,490 x minus 6 plus 2 times y minus 0 plus 3 times z minus 0 245 00:13:20,490 --> 00:13:21,620 equals 0. 246 00:13:21,620 --> 00:13:24,450 Notice, by the way, that algebraically, 247 00:13:24,450 --> 00:13:27,650 these two equations are equivalent. 248 00:13:27,650 --> 00:13:30,230 But that in this form, this tells me what? 249 00:13:30,230 --> 00:13:33,770 This specifies a point that the plane passes through, 250 00:13:33,770 --> 00:13:34,300 namely what? 251 00:13:34,300 --> 00:13:37,680 This is the plane that passes through (6, 0, 0), 252 00:13:37,680 --> 00:13:43,250 and has the vector i plus 2j plus 3k as a normal. 253 00:13:43,250 --> 00:13:45,740 By the way, you could do this in different ways. 254 00:13:45,740 --> 00:13:48,050 Some person might say, why couldn't you 255 00:13:48,050 --> 00:13:54,210 have transposed the 6 over here, then taken 3z minus 6? 256 00:13:54,210 --> 00:13:54,710 You see? 257 00:13:54,710 --> 00:13:58,510 Why don't you let x and y be 0 and solve this equation, 258 00:13:58,510 --> 00:14:00,380 and get that z equals 2? 259 00:14:00,380 --> 00:14:05,580 In other words, isn't (0, 0, 2) also a point in the plane? 260 00:14:05,580 --> 00:14:07,650 And the answer is, yes it is. 261 00:14:07,650 --> 00:14:10,080 And you could have written the equation now as what? 262 00:14:10,080 --> 00:14:17,110 x minus 0 plus twice y minus 0 plus 3 times z minus 2 263 00:14:17,110 --> 00:14:18,310 equals 0. 264 00:14:18,310 --> 00:14:19,360 That would be what? 265 00:14:19,360 --> 00:14:23,760 The equation of the plane that passed through (0, 0, 2), 266 00:14:23,760 --> 00:14:28,250 and had as its normal i plus 2j plus 3k. 267 00:14:28,250 --> 00:14:33,520 Of course what happens is that (6, 0, 0) and (0, 0, 2) 268 00:14:33,520 --> 00:14:35,270 belong to the same plane. 269 00:14:35,270 --> 00:14:37,410 I mean, that's another thing to keep in mind here, 270 00:14:37,410 --> 00:14:39,580 that the plane that we're talking about 271 00:14:39,580 --> 00:14:41,620 passes through more than one point. 272 00:14:41,620 --> 00:14:47,820 So x_0, y_0, z_0 can be played by an infinity of choices. 273 00:14:47,820 --> 00:14:50,030 At any rate, let's let that go now 274 00:14:50,030 --> 00:14:53,870 as the equation of our plane, and let's talk now 275 00:14:53,870 --> 00:14:55,480 about the equation of a line. 276 00:14:55,480 --> 00:14:56,600 That's a plane. 277 00:14:56,600 --> 00:14:59,260 Let's talk about the equation of a line. 278 00:14:59,260 --> 00:15:01,070 How do we determine a line? 279 00:15:01,070 --> 00:15:02,830 In two-dimensional space, we said 280 00:15:02,830 --> 00:15:06,150 we needed to know a point on the line and the slope. 281 00:15:06,150 --> 00:15:08,010 And another way of saying that is, 282 00:15:08,010 --> 00:15:10,430 we need to know a point on the line 283 00:15:10,430 --> 00:15:13,990 and we would like to know a line parallel to the given line. 284 00:15:13,990 --> 00:15:16,790 In vector language, what we say is, OK, 285 00:15:16,790 --> 00:15:19,200 let's suppose we're given the line l. 286 00:15:19,200 --> 00:15:21,685 And we know that the vector V, whose components 287 00:15:21,685 --> 00:15:25,800 are A, B, and C, that that vector is parallel to l 288 00:15:25,800 --> 00:15:28,550 and that the point P_0 whose coordinates are 289 00:15:28,550 --> 00:15:32,940 (x_0, y_0, z_0), that that point is on the line l. 290 00:15:32,940 --> 00:15:36,320 Then the question is, how do we find the equation of the line 291 00:15:36,320 --> 00:15:37,030 l? 292 00:15:37,030 --> 00:15:41,150 And again, vector methods come to our aid very nicely. 293 00:15:41,150 --> 00:15:46,330 What we say is, let's pick any other point P in space. 294 00:15:46,330 --> 00:15:47,500 All right? 295 00:15:47,500 --> 00:15:52,390 What does it mean if the point P is on the line l? 296 00:15:52,390 --> 00:15:55,060 If the point P is on the line l, since we 297 00:15:55,060 --> 00:15:57,120 want to use vector methods, let's 298 00:15:57,120 --> 00:16:01,750 simply observe that since l is parallel to V, 299 00:16:01,750 --> 00:16:06,300 the vector P_0 P, being parallel to V, 300 00:16:06,300 --> 00:16:09,120 must be a scalar multiple of V. That's 301 00:16:09,120 --> 00:16:12,860 what parallel means for vectors, scalar multiple. 302 00:16:12,860 --> 00:16:17,110 So P_0 p is equal to some constant times V. 303 00:16:17,110 --> 00:16:19,470 And let me pause here for a moment 304 00:16:19,470 --> 00:16:24,120 to point out that this constant is really a variable. 305 00:16:24,120 --> 00:16:24,870 That sounds awful. 306 00:16:24,870 --> 00:16:26,470 How can a constant be a variable? 307 00:16:26,470 --> 00:16:30,510 What I mean of course, is that P was any point in this line. 308 00:16:30,510 --> 00:16:34,700 Notice that t determines the length of P_0 P, 309 00:16:34,700 --> 00:16:38,370 and how long P_0 P is, is going to depend on where I choose 310 00:16:38,370 --> 00:16:41,340 P. In other words, for different choices of P, 311 00:16:41,340 --> 00:16:43,260 I get a different scalar multiple. 312 00:16:43,260 --> 00:16:47,900 And by the way, if I choose P on the wrong side of P_0, 313 00:16:47,900 --> 00:16:49,950 as I've deliberately done over here, 314 00:16:49,950 --> 00:16:53,805 notice that P_0 P has the opposite sense of V. 315 00:16:53,805 --> 00:16:56,070 So that t can even be negative. 316 00:16:56,070 --> 00:16:59,430 In other words, not only is t a variable, but if it's negative, 317 00:16:59,430 --> 00:17:02,570 it means that P_0 P has the opposite sense of V. 318 00:17:02,570 --> 00:17:04,819 If it's positive, they have the same sense. 319 00:17:04,819 --> 00:17:06,589 But I'm not going to belabor that point. 320 00:17:06,589 --> 00:17:09,660 What I'm now going to do is, in Cartesian coordinates, 321 00:17:09,660 --> 00:17:11,700 see what this equation tells me. 322 00:17:11,700 --> 00:17:13,829 And right away, it tells me what? 323 00:17:13,829 --> 00:17:19,250 That P_0 P is that vector whose components are x minus x_0, y 324 00:17:19,250 --> 00:17:22,760 minus y_0, and z minus z_0. 325 00:17:22,760 --> 00:17:25,050 What vector is t times V? 326 00:17:25,050 --> 00:17:29,440 Well, V, we saw, had as components, A, B, and C. 327 00:17:29,440 --> 00:17:32,350 And in Cartesian coordinates, multiplying a vector 328 00:17:32,350 --> 00:17:37,540 by a scalar simply multiplies each component by that scalar. 329 00:17:37,540 --> 00:17:41,050 So in other words, t times V is the vector whose components are 330 00:17:41,050 --> 00:17:44,630 t*A, t*B, and t*C. 331 00:17:44,630 --> 00:17:47,270 We also know, in Cartesian coordinates, 332 00:17:47,270 --> 00:17:49,800 that the only way that two vectors can be equal 333 00:17:49,800 --> 00:17:51,840 is component by component. 334 00:17:51,840 --> 00:17:53,380 And that tells us what? 335 00:17:53,380 --> 00:17:56,510 That x minus x_0 must equal t times 336 00:17:56,510 --> 00:18:01,750 A. y minus y_0 must equal t times B. 337 00:18:01,750 --> 00:18:06,180 And z minus z_0 must equal t times C. That's 338 00:18:06,180 --> 00:18:08,790 these three equations here. 339 00:18:08,790 --> 00:18:10,550 What do all of these three equations 340 00:18:10,550 --> 00:18:13,020 have in common numerically? 341 00:18:13,020 --> 00:18:15,370 They all have the factor t. 342 00:18:15,370 --> 00:18:18,640 And consequently, I can solve each of these three equations 343 00:18:18,640 --> 00:18:20,620 for t. 344 00:18:20,620 --> 00:18:21,460 Namely, what? 345 00:18:21,460 --> 00:18:23,950 Divide both sides of this equation by A, 346 00:18:23,950 --> 00:18:27,550 both sides of this equation by B, both sides of this equation 347 00:18:27,550 --> 00:18:32,140 by C, being very careful that neither A, B, nor C are 0. 348 00:18:32,140 --> 00:18:35,930 By the way, if they are 0, straightforward ramifications 349 00:18:35,930 --> 00:18:38,920 take place that we'll leave for the textbook to explain. 350 00:18:38,920 --> 00:18:40,570 Don't worry about that part right now. 351 00:18:40,570 --> 00:18:42,550 We don't want to get bogged down in that. 352 00:18:42,550 --> 00:18:46,050 But at any rate, if we now go from here 353 00:18:46,050 --> 00:18:48,800 to see what that says, we now wind up 354 00:18:48,800 --> 00:18:51,720 with the standard equation of the straight line. 355 00:18:51,720 --> 00:18:59,250 Namely, if you have x minus x_0 over A equals y minus y_0 356 00:18:59,250 --> 00:19:05,080 over B equals z minus z_0 over C equals some constant t, 357 00:19:05,080 --> 00:19:09,110 that particular form is called the standard equation 358 00:19:09,110 --> 00:19:10,140 for a straight line. 359 00:19:10,140 --> 00:19:11,700 What straight line is it? 360 00:19:11,700 --> 00:19:14,880 It's the line which passes through the point (x_0, y_0, 361 00:19:14,880 --> 00:19:21,910 z_0) and is parallel to the vector A*i plus B*j plus C*k. 362 00:19:21,910 --> 00:19:26,200 By means of an example, x minus 1 over 4 363 00:19:26,200 --> 00:19:31,590 equals y minus 5 over 3 equals z minus 6 over 7 364 00:19:31,590 --> 00:19:35,620 is the equation-- it's one equation, really. 365 00:19:35,620 --> 00:19:38,557 It's the equation of a line which has what property? 366 00:19:38,557 --> 00:19:40,140 It passes through the point (1, 5, 6). 367 00:19:42,680 --> 00:19:48,670 And it's parallel to the vector 4i plus 3j plus 7k. 368 00:19:48,670 --> 00:19:53,887 And by the way, I have to be very, very on my guard here. 369 00:19:53,887 --> 00:19:55,470 There's something very deceptive here. 370 00:19:55,470 --> 00:19:57,580 The equation of a plane and a line 371 00:19:57,580 --> 00:19:59,400 are very, very much different. 372 00:19:59,400 --> 00:20:02,420 But they look enough alike so it may confuse you. 373 00:20:02,420 --> 00:20:04,360 You know, it reminds me of my daughter, 374 00:20:04,360 --> 00:20:07,069 who I get a lot of stories from, was eating a sandwich one day. 375 00:20:07,069 --> 00:20:09,360 And I asked her what kind of a sandwich she was eating. 376 00:20:09,360 --> 00:20:11,443 And she said it was like a peanut butter and jelly 377 00:20:11,443 --> 00:20:12,139 sandwich. 378 00:20:12,139 --> 00:20:14,680 And I never heard of a sandwich that was like a peanut butter 379 00:20:14,680 --> 00:20:15,471 and jelly sandwich. 380 00:20:15,471 --> 00:20:18,614 So I looked at it to see what it was, and it was ham and cheese. 381 00:20:18,614 --> 00:20:20,780 And I say, why did you say it was like peanut butter 382 00:20:20,780 --> 00:20:21,279 and jelly? 383 00:20:21,279 --> 00:20:23,140 And she says well, it was two things in it. 384 00:20:23,140 --> 00:20:23,470 All right? 385 00:20:23,470 --> 00:20:23,969 Lookit. 386 00:20:23,969 --> 00:20:25,850 The equation of a line and the plane 387 00:20:25,850 --> 00:20:29,450 have three things in it-- x, y, and z. 388 00:20:29,450 --> 00:20:32,470 But to juxtaposition these, let me write down the two 389 00:20:32,470 --> 00:20:35,810 things that may look confusing. 390 00:20:35,810 --> 00:20:37,806 Let's suppose I write this down. 391 00:20:37,806 --> 00:20:38,930 You see what I'm doing now? 392 00:20:38,930 --> 00:20:42,660 What I'm doing now is I'm changing the equal signs here 393 00:20:42,660 --> 00:20:45,940 to plus signs and bringing up the denominators here. 394 00:20:45,940 --> 00:20:47,350 See, this is a line. 395 00:20:47,350 --> 00:20:48,910 This is a plane. 396 00:20:48,910 --> 00:20:50,910 What plane is this? 397 00:20:50,910 --> 00:20:55,640 This is the plane which passes through the point 1 comma 398 00:20:55,640 --> 00:21:02,700 5, comma 6, and has the line of the vector 4i plus 3j plus 7k 399 00:21:02,700 --> 00:21:04,890 as its normal. 400 00:21:04,890 --> 00:21:06,460 How can I best explain this to you 401 00:21:06,460 --> 00:21:08,210 to keep this straight in your mind? 402 00:21:08,210 --> 00:21:10,390 Well, I think the easiest way-- and again, 403 00:21:10,390 --> 00:21:12,040 notice what I'm saying, see the x, 404 00:21:12,040 --> 00:21:14,540 y's, and z's here, the x, y's and z's here. 405 00:21:14,540 --> 00:21:15,550 Which is which? 406 00:21:15,550 --> 00:21:18,620 The easiest way is to keep track of degrees of freedom. 407 00:21:18,620 --> 00:21:20,870 Remember in the plane, we said lookit. 408 00:21:20,870 --> 00:21:23,470 You can pick two of the variables at random 409 00:21:23,470 --> 00:21:25,310 and solve for the third. 410 00:21:25,310 --> 00:21:28,970 I claim in this system-- in this system here, 411 00:21:28,970 --> 00:21:31,330 there is only one degree of freedom. 412 00:21:31,330 --> 00:21:35,160 The line has one degree of freedom. 413 00:21:35,160 --> 00:21:36,610 Namely, let's repeat this example, 414 00:21:36,610 --> 00:21:39,130 so we don't have to keep looking back to the board here. 415 00:21:39,130 --> 00:21:43,270 Let's take x minus 1 over 4 equals y minus 5 over 3 416 00:21:43,270 --> 00:21:45,662 equals z minus 6 over 7. 417 00:21:45,662 --> 00:21:47,620 And since we don't like to work with fractions, 418 00:21:47,620 --> 00:21:51,310 I'll pick a number that works out nicely. 419 00:21:51,310 --> 00:21:54,570 I say, OK, let's see what happens when x is 9. 420 00:21:54,570 --> 00:21:56,070 Now here's the whole point. 421 00:21:56,070 --> 00:22:01,210 As soon as I say that x equals 9, as soon as I let x equal 9, 422 00:22:01,210 --> 00:22:04,140 this is fixed. 423 00:22:04,140 --> 00:22:04,640 Right? 424 00:22:04,640 --> 00:22:07,480 In fact, what does it become fixed as soon as I do this? 425 00:22:07,480 --> 00:22:11,480 As soon as x equals 9, x minus 1 over 4 is 2. 426 00:22:11,480 --> 00:22:16,627 Now notice that y minus 5 over 3 has to equal 2. 427 00:22:16,627 --> 00:22:17,960 Well I have no more choice then. 428 00:22:17,960 --> 00:22:23,510 If y minus 5 over 3 has to equal 2, and also z minus 6 over 7 429 00:22:23,510 --> 00:22:26,950 has to equal 2-- you see what I'm saying here? 430 00:22:26,950 --> 00:22:31,740 This fixes the fact that y must be 11, and that z must be 20. 431 00:22:31,740 --> 00:22:33,980 In other words, the choice of x equals 432 00:22:33,980 --> 00:22:39,080 9 forces me to make y equal 11 and z equal 20. 433 00:22:39,080 --> 00:22:41,010 One degree of freedom. 434 00:22:41,010 --> 00:22:43,030 And by the way, if you want to see this thing 435 00:22:43,030 --> 00:22:45,440 from a geometrical point of view, what we're saying 436 00:22:45,440 --> 00:22:50,800 is, visualize this line cutting through space, all right? 437 00:22:50,800 --> 00:22:53,120 Notice that directly on that line, 438 00:22:53,120 --> 00:22:56,430 only one point will have its x-coordinate equal to 9. 439 00:22:56,430 --> 00:22:58,110 And what we're saying is, the point 440 00:22:58,110 --> 00:23:00,870 on that line whose x-coordinate is 9 441 00:23:00,870 --> 00:23:05,080 is the point 9 comma 11 comma 20. 442 00:23:05,080 --> 00:23:05,710 OK? 443 00:23:05,710 --> 00:23:09,650 One degree of freedom again. 444 00:23:09,650 --> 00:23:11,450 That's very, very crucial for you to see. 445 00:23:11,450 --> 00:23:14,500 By the way, I guess one thing that bothers a lot of students 446 00:23:14,500 --> 00:23:17,220 is the fact that they read this as two separate equations. 447 00:23:17,220 --> 00:23:20,820 They say, you know, why isn't this x minus 1 over 4 448 00:23:20,820 --> 00:23:22,780 equals y minus 5 over 3? 449 00:23:22,780 --> 00:23:25,100 Why can't I treat that as one equation? 450 00:23:25,100 --> 00:23:30,240 Or why couldn't I take y minus 5 over 3 equals z minus 6 over 7? 451 00:23:30,240 --> 00:23:33,180 Or why couldn't I take x minus 1 over 4 452 00:23:33,180 --> 00:23:35,870 and say that equals z minus 6 over 7? 453 00:23:35,870 --> 00:23:38,890 And the answer is, that by itself isn't enough. 454 00:23:38,890 --> 00:23:41,210 But rather than give you a negative answer, 455 00:23:41,210 --> 00:23:44,030 let me give you a positive one. 456 00:23:44,030 --> 00:23:47,560 Let me close today's lesson with this particular illustration. 457 00:23:47,560 --> 00:23:53,510 Suppose we had solved x minus 1 over 4 equals y minus 5 over 3. 458 00:23:53,510 --> 00:23:56,880 What we would have obtained is the equation 459 00:23:56,880 --> 00:23:59,380 4y minus 3x equals 17. 460 00:23:59,380 --> 00:24:01,070 Now this is very dangerous. 461 00:24:01,070 --> 00:24:04,870 When you look at the equation 4y minus 3x equals 17, 462 00:24:04,870 --> 00:24:06,450 I'll bet you dollars to doughnuts 463 00:24:06,450 --> 00:24:12,370 you tend to think of this as a line rather than as a plane. 464 00:24:12,370 --> 00:24:14,840 But the interesting thing is, notice 465 00:24:14,840 --> 00:24:17,240 that the way we got this equation 466 00:24:17,240 --> 00:24:20,840 was ignoring the z-coordinate of our points. 467 00:24:20,840 --> 00:24:22,800 And what we're really saying is, let's forget 468 00:24:22,800 --> 00:24:24,410 about the z-coordinate. 469 00:24:24,410 --> 00:24:27,680 In other words, 4y minus 3x equals 470 00:24:27,680 --> 00:24:32,900 17 may be viewed as a line, but in this particular case, 471 00:24:32,900 --> 00:24:33,740 it's a plane. 472 00:24:33,740 --> 00:24:35,270 In fact, what plane is it? 473 00:24:35,270 --> 00:24:39,470 It's the plane that goes through the line 4y minus 3x 474 00:24:39,470 --> 00:24:42,360 equals 17, which lies on the xy-plane. 475 00:24:42,360 --> 00:24:45,060 It's the plane that goes through that line perpendicular 476 00:24:45,060 --> 00:24:46,310 to the xy-plane. 477 00:24:46,310 --> 00:24:50,350 By the way, again, if you go back to part one of this course 478 00:24:50,350 --> 00:24:53,540 where we stress sets, the language of sets 479 00:24:53,540 --> 00:24:55,900 comes to our rescue very nicely. 480 00:24:55,900 --> 00:24:59,870 The difference between whether 4y minus 3x equals 481 00:24:59,870 --> 00:25:03,010 17 is a plane or whether it's a line 482 00:25:03,010 --> 00:25:08,100 hinges on whether we're talking about the set of pairs 483 00:25:08,100 --> 00:25:12,640 x comma y such that 4y minus 3x equals 17, 484 00:25:12,640 --> 00:25:16,560 or whether we're talking about the set of triplets (x, y, z) 485 00:25:16,560 --> 00:25:20,330 such that 4y minus 3x equals 17. 486 00:25:20,330 --> 00:25:22,520 In this particular example, we're 487 00:25:22,520 --> 00:25:24,230 talking about points in space. 488 00:25:24,230 --> 00:25:26,290 In other words, our universe of discourse 489 00:25:26,290 --> 00:25:31,680 are the points x comma y comma z, not the points x comma y. 490 00:25:31,680 --> 00:25:34,650 Well anyway, rather than to belabor this point, what 491 00:25:34,650 --> 00:25:39,050 I'm saying is, in the same way that this equation represents 492 00:25:39,050 --> 00:25:47,780 a plane, in a similar way, had we equated y minus 5 over 3 493 00:25:47,780 --> 00:25:50,380 equals z minus 6 over 7, we would've 494 00:25:50,380 --> 00:25:57,120 obtained the plane 7y minus 3z equals 17. 495 00:25:57,120 --> 00:26:00,430 So if you don't like to look at our set of three equations, 496 00:26:00,430 --> 00:26:03,280 if you'd like to look at these three equations 497 00:26:03,280 --> 00:26:06,780 in pairs-- you see, if you want to look at these three 498 00:26:06,780 --> 00:26:09,220 equations in pairs, another way of saying 499 00:26:09,220 --> 00:26:15,220 it is this, that the triple equality-- x minus 1 over 4 500 00:26:15,220 --> 00:26:19,620 equals y minus 5 over 3 equals z minus 6 over 7-- 501 00:26:19,620 --> 00:26:23,430 that that may be viewed as the intersection of the two 502 00:26:23,430 --> 00:26:27,880 planes-- namely the plane determined by this equation 503 00:26:27,880 --> 00:26:30,540 and the plane determined by this equation. 504 00:26:30,540 --> 00:26:33,690 Of course, someone can also say, isn't there a plane 505 00:26:33,690 --> 00:26:37,610 determined by this one and this one? 506 00:26:37,610 --> 00:26:39,430 And the answer is yes, there is. 507 00:26:39,430 --> 00:26:42,410 Notice that whereas you have three equalities here, 508 00:26:42,410 --> 00:26:44,470 only two of them are independent. 509 00:26:44,470 --> 00:26:46,590 Namely, as soon as the first equals 510 00:26:46,590 --> 00:26:49,260 the second and the second equals the third, 511 00:26:49,260 --> 00:26:52,080 the first must equal the third. 512 00:26:52,080 --> 00:26:52,880 OK? 513 00:26:52,880 --> 00:26:56,160 But the whole idea is, can you now see the difference? 514 00:26:56,160 --> 00:26:58,780 The easiest way I know of to distinguish 515 00:26:58,780 --> 00:27:02,860 the difference between the equation of a line and a plane. 516 00:27:02,860 --> 00:27:05,790 The plane has two degrees of freedom. 517 00:27:05,790 --> 00:27:08,320 The line has but one degree of freedom. 518 00:27:08,320 --> 00:27:11,240 And that triple equality says as soon as you've 519 00:27:11,240 --> 00:27:13,220 picked one of the unknowns, you've 520 00:27:13,220 --> 00:27:15,050 determined all of the others. 521 00:27:15,050 --> 00:27:16,770 Whereas that string of plus signs 522 00:27:16,770 --> 00:27:19,220 says that once you've determined one, 523 00:27:19,220 --> 00:27:21,760 you still have some freedom left. 524 00:27:21,760 --> 00:27:25,140 Now what we're going to do is next time 525 00:27:25,140 --> 00:27:27,510 start a new phase of vectors. 526 00:27:27,510 --> 00:27:29,980 For the time being, what we have now done 527 00:27:29,980 --> 00:27:32,040 is finished, at least for the moment, 528 00:27:32,040 --> 00:27:35,950 our preliminary investigation of three-dimensional space 529 00:27:35,950 --> 00:27:39,590 as seen through the eyes of Cartesian coordinates. 530 00:27:39,590 --> 00:27:41,860 At any rate, until next time, goodbye. 531 00:27:44,420 --> 00:27:46,790 Funding for the publication of this video 532 00:27:46,790 --> 00:27:51,670 was provided by the Gabriella and Paul Rosenbaum Foundation. 533 00:27:51,670 --> 00:27:55,850 Help OCW continue to provide free and open access to MIT 534 00:27:55,850 --> 00:28:00,260 courses by making a donation at ocw.mit.edu/donate.