1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:04,590 PROFESSOR: You have learned a lot in Calculus 2. 3 00:00:04,590 --> 00:00:12,180 Whether you took Calculus recently or long time ago, 4 00:00:12,180 --> 00:00:20,960 Chapter 9 is about vectors in r3 and eventually 5 00:00:20,960 --> 00:00:25,490 [? a plane ?] and operations with such vectors 6 00:00:25,490 --> 00:00:28,510 and the implications of the vectors 7 00:00:28,510 --> 00:00:32,880 in the equations of a line in space or plane 8 00:00:32,880 --> 00:00:35,180 in space-- stuff like that. 9 00:00:35,180 --> 00:00:42,820 Now, 9.1 to 9.5 was considered to be covered completely 10 00:00:42,820 --> 00:00:46,970 in Calc 2 here at Tech. 11 00:00:46,970 --> 00:00:52,400 However, lots of students come from South Plains College 12 00:00:52,400 --> 00:00:54,880 and [? Rio ?] College, lots of colleges 13 00:00:54,880 --> 00:00:59,220 where by the nature of the course Calculus 2, 14 00:00:59,220 --> 00:01:02,440 vectors in r3 are not covered. 15 00:01:02,440 --> 00:01:08,730 Therefore, I'd like to make an attempt to review 9.1 and 9.5 16 00:01:08,730 --> 00:01:12,820 quickly with the knowledge you have now as grown-ups 17 00:01:12,820 --> 00:01:15,260 in the area of vectors in r2. 18 00:01:15,260 --> 00:01:17,130 So again, what are vectors? 19 00:01:17,130 --> 00:01:20,700 They are oriented segments. 20 00:01:20,700 --> 00:01:22,940 Not only that they are oriented segments, 21 00:01:22,940 --> 00:01:29,840 but we make the distinction between a vector that is fixed 22 00:01:29,840 --> 00:01:33,960 in the sense that his origin is fixed-- we cannot move him-- 23 00:01:33,960 --> 00:01:36,870 and a free vector who is not married to the origin. 24 00:01:36,870 --> 00:01:40,830 He can shift by parallelism anywhere in space. 25 00:01:40,830 --> 00:01:43,020 And we call that a free vector. 26 00:01:43,020 --> 00:01:48,030 The distinction between those vectors would be vr of v bar. 27 00:01:48,030 --> 00:01:52,920 As you remember, v bar was the free guy, free vector, 28 00:01:52,920 --> 00:02:00,666 which is the-- actually, it's an equivalence class 29 00:02:00,666 --> 00:02:14,720 of all vectors that can be obtained from the generic v 30 00:02:14,720 --> 00:02:15,220 bounded. 31 00:02:15,220 --> 00:02:20,970 So I'm going to have to point by translation. 32 00:02:20,970 --> 00:02:25,090 So you have this kind of-- same magnitude 33 00:02:25,090 --> 00:02:29,411 for all vectors, same magnitude, same orientation, 34 00:02:29,411 --> 00:02:33,210 and parallel directions, parallel lines. 35 00:02:33,210 --> 00:02:35,380 What have we done to such a vector? 36 00:02:35,380 --> 00:02:38,690 As you remember very well, we decomposed him, 37 00:02:38,690 --> 00:02:44,940 being on the standard canonical basis, which for most of you 38 00:02:44,940 --> 00:02:50,140 engineers and engineering majors is denoted as ijk where 39 00:02:50,140 --> 00:02:55,850 ijk is an orthonormal frame with respect 40 00:02:55,850 --> 00:02:57,110 to the Cartesian coordinates. 41 00:02:57,110 --> 00:03:03,380 So i, j, and k will be their unit vectors 42 00:03:03,380 --> 00:03:08,775 on the x, y, z axes of coordinates, Cartesian axes 43 00:03:08,775 --> 00:03:09,730 of coordinates. 44 00:03:09,730 --> 00:03:14,530 So remember always that ijk are orthogonal to one another. 45 00:03:14,530 --> 00:03:17,460 Since this is review, I'd like to attract your attention 46 00:03:17,460 --> 00:03:25,320 to the fact that k is plus j. 47 00:03:25,320 --> 00:03:29,630 Think about it-- what happens you bring i over j. 48 00:03:29,630 --> 00:03:32,205 And you get k because you move up. 49 00:03:32,205 --> 00:03:35,219 Because it's like you are turning [INAUDIBLE] 50 00:03:35,219 --> 00:03:40,250 connection and the screw or whatever from the faucet 51 00:03:40,250 --> 00:03:41,690 is pointing upwards. 52 00:03:41,690 --> 00:03:43,530 It's like the right hand rule. 53 00:03:43,530 --> 00:03:48,560 If you would do the other way around, if you do j cross i, 54 00:03:48,560 --> 00:03:50,000 what are you going to have? 55 00:03:50,000 --> 00:03:55,330 Minus k-- so the properties of the cross product being 56 00:03:55,330 --> 00:03:57,740 antisymmetric are supposed to be, 57 00:03:57,740 --> 00:04:02,190 no, pay attention to the signs in all the exams that you have. 58 00:04:02,190 --> 00:04:05,150 What do we know about their respective products 59 00:04:05,150 --> 00:04:08,750 for vectors in space or in plain? 60 00:04:08,750 --> 00:04:13,530 If you have two vectors in their standard basis, 61 00:04:13,530 --> 00:04:17,680 you want i plus u2j plus u3k where 62 00:04:17,680 --> 00:04:24,480 ui is a real number and e1i plus v2j plus v3k where 63 00:04:24,480 --> 00:04:31,603 vi are [INAUDIBLE] real numbers the dot product or the scalar 64 00:04:31,603 --> 00:04:35,530 product-- now, I saw that in all your engineering and physics 65 00:04:35,530 --> 00:04:38,130 classes, you will use this notation. 66 00:04:38,130 --> 00:04:40,630 Mathematicians sometimes say, no, I'm 67 00:04:40,630 --> 00:04:42,870 going to use angular brackets because it's 68 00:04:42,870 --> 00:04:46,020 a scalar product in r3 or the scalar product 69 00:04:46,020 --> 00:04:47,910 and the dot product is the same thing, 70 00:04:47,910 --> 00:04:49,790 being that's the standard one here. 71 00:04:49,790 --> 00:04:54,670 You want v1 plus u2v2 plus u3v3. 72 00:04:54,670 --> 00:04:58,031 So what do you to remember what you do? 73 00:04:58,031 --> 00:04:59,955 First component plus first component 74 00:04:59,955 --> 00:05:02,365 times second component times second component 75 00:05:02,365 --> 00:05:05,950 plus third component times third component, OK? 76 00:05:05,950 --> 00:05:10,420 If you are in computer science, I 77 00:05:10,420 --> 00:05:13,150 saw that you use this notation. 78 00:05:13,150 --> 00:05:15,490 I was very happy to see that. 79 00:05:15,490 --> 00:05:16,570 the summation notation. 80 00:05:16,570 --> 00:05:20,240 But you don't have to use that in our class. 81 00:05:20,240 --> 00:05:24,410 Now, above the [? fresh ?] product of two vectors, 82 00:05:24,410 --> 00:05:29,100 you have the definition ijk the first row. 83 00:05:29,100 --> 00:05:31,800 So what you get is going to be a vector. 84 00:05:31,800 --> 00:05:36,230 Here, what you get is a scalar as a result. 85 00:05:36,230 --> 00:05:40,510 Here's what you get as a vector, as the result 86 00:05:40,510 --> 00:05:42,625 of the first product is a vector. 87 00:05:42,625 --> 00:05:48,029 So you have u1, u2, u3, v1, v2, v3. 88 00:05:48,029 --> 00:05:49,320 These are all friends of yours. 89 00:05:49,320 --> 00:05:56,840 I'm just reminding you the lucrative definitions. 90 00:05:56,840 --> 00:05:59,992 Now, some people said, yes, but I'd 91 00:05:59,992 --> 00:06:03,250 like to see the lucrative definitions that 92 00:06:03,250 --> 00:06:05,060 have to do with trig as well. 93 00:06:05,060 --> 00:06:06,130 OK, let's see. 94 00:06:06,130 --> 00:06:09,120 For those of you who asked me to remind you what they were, 95 00:06:09,120 --> 00:06:13,170 I will remind you what they were. 96 00:06:13,170 --> 00:06:20,480 For u.v, you get the same thing as writing magnitude u 97 00:06:20,480 --> 00:06:25,170 magnitude v and cosine of the angle between them 98 00:06:25,170 --> 00:06:27,380 no matter in which direction you take it 99 00:06:27,380 --> 00:06:29,370 because the cosine is the same. 100 00:06:29,370 --> 00:06:32,330 Cosine of pi is equal to cosine of negative phi 101 00:06:32,330 --> 00:06:34,500 or theta [INAUDIBLE]. 102 00:06:34,500 --> 00:06:35,930 How about the other one? 103 00:06:35,930 --> 00:06:38,482 Here's where one of you had a little bit 104 00:06:38,482 --> 00:06:41,590 of a misunderstanding. 105 00:06:41,590 --> 00:06:45,210 And I saw that happen in two finals, unfortunately. 106 00:06:45,210 --> 00:06:51,540 This is not the scalar vector that I'm right here. 107 00:06:51,540 --> 00:06:52,620 It's a vector. 108 00:06:52,620 --> 00:06:53,650 So what's missing? 109 00:06:53,650 --> 00:06:55,830 This is the scalar part. 110 00:06:55,830 --> 00:07:00,896 And then you have times e where e is the unit vector 111 00:07:00,896 --> 00:07:10,050 of the direction of the vector, the direction of u.v. 112 00:07:10,050 --> 00:07:12,480 Why I cannot use another notion? 113 00:07:12,480 --> 00:07:13,980 Because u is already taken. 114 00:07:13,980 --> 00:07:17,676 But e in itself should suggest to you 115 00:07:17,676 --> 00:07:22,090 that you have a unit vector, [? length of ?] one vector, OK? 116 00:07:22,090 --> 00:07:26,640 All right, what is the-- let's review a little 117 00:07:26,640 --> 00:07:29,280 bit the absolute value. 118 00:07:29,280 --> 00:07:31,610 Well, the absolute value is a scalar. 119 00:07:31,610 --> 00:07:34,430 So that scalar will be magnitude of your magnitude 120 00:07:34,430 --> 00:07:36,770 of [INAUDIBLE] sine of the angle. 121 00:07:36,770 --> 00:07:41,010 And do you guys remember the geometric interpretation 122 00:07:41,010 --> 00:07:42,556 of that? 123 00:07:42,556 --> 00:07:43,472 STUDENT: [INAUDIBLE] 124 00:07:43,472 --> 00:07:45,138 PROFESSOR: The area of the parallelogram 125 00:07:45,138 --> 00:07:48,380 based on the two vectors-- very good [INAUDIBLE]. 126 00:07:48,380 --> 00:07:50,870 U plus b is the area of the parallelogram 127 00:07:50,870 --> 00:07:56,980 that you would draw based on those two vectors. 128 00:07:56,980 --> 00:07:58,340 All right, good. 129 00:07:58,340 --> 00:08:00,730 Now, say goodbye, vectors. 130 00:08:00,730 --> 00:08:04,464 We've seen-- you've seen them through 9.4, 9.5. 131 00:08:04,464 --> 00:08:08,190 What was important to remember was 132 00:08:08,190 --> 00:08:10,850 that these vectors were the building 133 00:08:10,850 --> 00:08:14,630 blocks, the foundations, of the equations 134 00:08:14,630 --> 00:08:16,845 of the lines in space. 135 00:08:16,845 --> 00:08:18,860 That's your work [INAUDIBLE]. 136 00:08:18,860 --> 00:08:21,370 So what did we work with? 137 00:08:21,370 --> 00:08:26,110 Lines in space-- lines in space can be given in many ways. 138 00:08:26,110 --> 00:08:28,030 But now that you remember them, I'm 139 00:08:28,030 --> 00:08:35,600 going to give you the symmetric equation of a line in space. 140 00:08:35,600 --> 00:08:38,908 141 00:08:38,908 --> 00:08:42,640 OK, [INAUDIBLE] this can see [INAUDIBLE] 142 00:08:42,640 --> 00:08:44,350 included on the final. 143 00:08:44,350 --> 00:08:46,760 You are expected to know it. 144 00:08:46,760 --> 00:08:55,520 145 00:08:55,520 --> 00:08:59,190 So that is the symmetric equation, 146 00:08:59,190 --> 00:09:03,410 meaning the equation of a what? 147 00:09:03,410 --> 00:09:11,280 Of a line in space passing through or containing 148 00:09:11,280 --> 00:09:21,470 the point of p, not of coordinates x0, y0, [? z0, ?] 149 00:09:21,470 --> 00:09:29,820 and of direction [INAUDIBLE] in the sense of a vector. 150 00:09:29,820 --> 00:09:32,949 Now, if I were to draw such a line, 151 00:09:32,949 --> 00:09:35,344 I'm going to have the line over here. 152 00:09:35,344 --> 00:09:39,050 Going to have a vector for the point p0 on the line. 153 00:09:39,050 --> 00:09:41,970 I can put this free vector because he's free. 154 00:09:41,970 --> 00:09:43,720 He says, I'm a free guy. 155 00:09:43,720 --> 00:09:47,660 I can slide any way I want. 156 00:09:47,660 --> 00:09:52,190 So I'm going to have li plus mj plus mk. 157 00:09:52,190 --> 00:09:56,600 158 00:09:56,600 --> 00:09:58,070 This is the blue vector. 159 00:09:58,070 --> 00:10:01,570 Now, you don't have blue markers or blue pens, 160 00:10:01,570 --> 00:10:05,940 but you can still do a good job taking notes. 161 00:10:05,940 --> 00:10:09,570 Now somebody asked me just a week ago, 162 00:10:09,570 --> 00:10:11,970 saying that I've started doing review already 163 00:10:11,970 --> 00:10:16,340 and I don't understand what the difference is 164 00:10:16,340 --> 00:10:20,400 between the symmetric equation of a line in space 165 00:10:20,400 --> 00:10:24,840 and the parametric equations of a line in space. 166 00:10:24,840 --> 00:10:27,830 This is no essential difference. 167 00:10:27,830 --> 00:10:28,770 So what do we do? 168 00:10:28,770 --> 00:10:31,947 We denote this whole animal by t, a real number. 169 00:10:31,947 --> 00:10:35,220 170 00:10:35,220 --> 00:10:37,640 And then we erase the board. 171 00:10:37,640 --> 00:10:43,310 And then we write the three equations 172 00:10:43,310 --> 00:10:47,326 that govern-- I'm going to put if and only 173 00:10:47,326 --> 00:10:51,590 if xyz satisfy the following. 174 00:10:51,590 --> 00:10:53,360 So I'm going to have, what? 175 00:10:53,360 --> 00:11:07,881 X equals lt plus x0, y equals mt plus y0, n equals nt plus z0. 176 00:11:07,881 --> 00:11:11,600 Well, of course, we understand-- we know the meaning 177 00:11:11,600 --> 00:11:16,490 that lmn are like what [INAUDIBLE] physics direction 178 00:11:16,490 --> 00:11:18,690 cosines were telling me about it. 179 00:11:18,690 --> 00:11:21,960 And then x0, y0, z0 is a fixed point 180 00:11:21,960 --> 00:11:24,580 that belongs to that line. 181 00:11:24,580 --> 00:11:28,650 Now, since you know a little bit more than 182 00:11:28,650 --> 00:11:30,990 you knew in Calculus 2 when you saw 183 00:11:30,990 --> 00:11:35,580 that for the first time, what is the typical notation that we 184 00:11:35,580 --> 00:11:40,040 use all through Calc 3, all through the chapters? 185 00:11:40,040 --> 00:11:44,920 The position vector-- the position vector of the point 186 00:11:44,920 --> 00:11:53,930 on the line that is related to, what? 187 00:11:53,930 --> 00:11:57,280 So practically you have the origin here. 188 00:11:57,280 --> 00:12:04,200 [? Op0 ?] represent the vector x0i plus y0j plus e0k. 189 00:12:04,200 --> 00:12:06,880 So now you have a little bit of a different understanding 190 00:12:06,880 --> 00:12:08,910 of what's going on. 191 00:12:08,910 --> 00:12:15,300 And then after, let's say, t equals 1 hour, what do you do? 192 00:12:15,300 --> 00:12:18,100 You are adding the blue vector here. 193 00:12:18,100 --> 00:12:20,800 194 00:12:20,800 --> 00:12:23,950 Let's say at t equals 1, you are here. 195 00:12:23,950 --> 00:12:25,690 You are here at p1. 196 00:12:25,690 --> 00:12:29,920 So to get to p1, you have to add two vectors, right guys? 197 00:12:29,920 --> 00:12:34,030 This is the addition between the blue vector and the red vector. 198 00:12:34,030 --> 00:12:36,290 So what you get is your result. 199 00:12:36,290 --> 00:12:41,120 So if I am smart enough to understand my concepts are all 200 00:12:41,120 --> 00:12:44,110 connected, the position in this case 201 00:12:44,110 --> 00:12:49,860 will be r of t, which is-- I hate angular brackets, 202 00:12:49,860 --> 00:12:52,650 but just because you like them, I'm 203 00:12:52,650 --> 00:12:55,760 going to use them-- x of ty tz of tm 204 00:12:55,760 --> 00:12:58,140 to be consistent with the book. 205 00:12:58,140 --> 00:13:04,440 This is the same as xi plux yj plus zk. 206 00:13:04,440 --> 00:13:08,518 And what is this by the actual notations 207 00:13:08,518 --> 00:13:10,510 from the parametric equation? 208 00:13:10,510 --> 00:13:15,570 This is nothing but a certain lmn 209 00:13:15,570 --> 00:13:20,080 vector that is the vector li plus mj plus nk written 210 00:13:20,080 --> 00:13:21,750 with angular brackets because I know 211 00:13:21,750 --> 00:13:28,350 you like that times the time t plus the fixed vector x0, 212 00:13:28,350 --> 00:13:29,300 y0, z0. 213 00:13:29,300 --> 00:13:31,890 You can say, yeah, I thought it was a point. 214 00:13:31,890 --> 00:13:33,410 It is a point and a vector. 215 00:13:33,410 --> 00:13:41,960 You identified the point p0 with the position of the point p0 216 00:13:41,960 --> 00:13:44,660 starting with respect to the origin. 217 00:13:44,660 --> 00:13:47,240 So whether you're talking about mister p0 218 00:13:47,240 --> 00:13:51,900 being a point in space-- x0, y0, z0. 219 00:13:51,900 --> 00:13:54,810 Or you're talking about the [INAUDIBLE] position 220 00:13:54,810 --> 00:13:58,190 vector that [INAUDIBLE] is practically 221 00:13:58,190 --> 00:14:00,000 the same after identification. 222 00:14:00,000 --> 00:14:02,670 So you have something very nice. 223 00:14:02,670 --> 00:14:05,970 And if I asked you with the mind and the knowledge 224 00:14:05,970 --> 00:14:13,430 you have now what that does is mean-- r prime of t 225 00:14:13,430 --> 00:14:16,190 equals what? 226 00:14:16,190 --> 00:14:19,180 It's the velocity vector. 227 00:14:19,180 --> 00:14:22,295 And what is that as a vector? 228 00:14:22,295 --> 00:14:24,540 Do the differentiation. 229 00:14:24,540 --> 00:14:27,180 What do we get in terms of velocity vector? 230 00:14:27,180 --> 00:14:29,480 Prime with respect to t-- what do I get? 231 00:14:29,480 --> 00:14:30,146 STUDENT: Lmn. 232 00:14:30,146 --> 00:14:31,270 PROFESSOR: Lmn as a vector. 233 00:14:31,270 --> 00:14:33,860 But of course, as I hate angular notations, 234 00:14:33,860 --> 00:14:38,470 I will rewrite it-- li plus mj plus nk. 235 00:14:38,470 --> 00:14:40,190 So this is your velocity. 236 00:14:40,190 --> 00:14:44,155 What can you say about this type of motion? 237 00:14:44,155 --> 00:14:44,886 This is a-- 238 00:14:44,886 --> 00:14:46,510 STUDENT: [INAUDIBLE] constant velocity. 239 00:14:46,510 --> 00:14:48,835 PROFESSOR: Yeah, you have a constant velocity 240 00:14:48,835 --> 00:14:51,232 for this motion. 241 00:14:51,232 --> 00:14:56,356 If somebody would ask you you have-- 10 years from now, 242 00:14:56,356 --> 00:15:00,370 you have a boy who said, dad-- or a girl. 243 00:15:00,370 --> 00:15:01,513 let's not be biased. 244 00:15:01,513 --> 00:15:05,690 So he learns, math, good at math or physics, and says, 245 00:15:05,690 --> 00:15:08,370 what is the difference between velocity and speed? 246 00:15:08,370 --> 00:15:11,172 Well, most parents will say it's the same thing. 247 00:15:11,172 --> 00:15:14,406 Well, you're not most parents. 248 00:15:14,406 --> 00:15:16,290 You are educated parents. 249 00:15:16,290 --> 00:15:19,340 So this is-- don't tell your kid about vectors, 250 00:15:19,340 --> 00:15:22,670 but you can show them you have an oriented segment. 251 00:15:22,670 --> 00:15:26,390 So make your child run around around in circles and say, 252 00:15:26,390 --> 00:15:30,660 this is the velocity that's always tangent to the circle 253 00:15:30,660 --> 00:15:32,420 that you are running on. 254 00:15:32,420 --> 00:15:34,860 That's a velocity. 255 00:15:34,860 --> 00:15:38,890 And if they ask, well, they will catch 256 00:15:38,890 --> 00:15:41,990 the notions of acceleration and force faster than you 257 00:15:41,990 --> 00:15:44,020 because they see all these cartoons. 258 00:15:44,020 --> 00:15:48,360 And my son was telling me the other thing-- he's 10 259 00:15:48,360 --> 00:15:50,570 and I asked him, what the heck is that? 260 00:15:50,570 --> 00:15:53,910 It looked like an electromagnetic field 261 00:15:53,910 --> 00:15:55,780 surrounding some hero. 262 00:15:55,780 --> 00:15:59,700 And he said, mom, that's the force field of course. 263 00:15:59,700 --> 00:16:02,155 And I was thinking, force field? 264 00:16:02,155 --> 00:16:04,773 This is what I taught the other day when 265 00:16:04,773 --> 00:16:08,047 I was talking about [? crux. ?] Double integral of f.n 266 00:16:08,047 --> 00:16:10,011 [INAUDIBLE] f was the force field. 267 00:16:10,011 --> 00:16:13,950 So he was, like, talking about something very normal 268 00:16:13,950 --> 00:16:15,670 that you see every day. 269 00:16:15,670 --> 00:16:21,250 So do not underestimate your nephews, nieces, children. 270 00:16:21,250 --> 00:16:23,100 They will catch up on these things 271 00:16:23,100 --> 00:16:25,540 faster than you, which is good. 272 00:16:25,540 --> 00:16:30,820 Now, the speed in this case will be, what? 273 00:16:30,820 --> 00:16:33,170 What is the speed of this-- the speed 274 00:16:33,170 --> 00:16:36,345 of this motion, linear motion? 275 00:16:36,345 --> 00:16:38,132 STUDENT: Square root of l squared. 276 00:16:38,132 --> 00:16:42,880 PROFESSOR: Square root of l squared plus m squared plus n 277 00:16:42,880 --> 00:16:45,530 squared, which again is different from velocity. 278 00:16:45,530 --> 00:16:48,580 Velocity is a vector, speed is a scalar. 279 00:16:48,580 --> 00:16:50,640 Velocity is a vector, speed is a scalar. 280 00:16:50,640 --> 00:16:52,590 In general, doesn't have to be constant, 281 00:16:52,590 --> 00:16:54,510 but this is the blessing because lmn 282 00:16:54,510 --> 00:16:57,410 are given constants. [INAUDIBLE] in this case, 283 00:16:57,410 --> 00:16:59,660 you are on cruise control. 284 00:16:59,660 --> 00:17:02,240 You are moving on a line directly 285 00:17:02,240 --> 00:17:05,200 in your motion on cruise control driving 286 00:17:05,200 --> 00:17:07,835 to Amarillo at 60 miles an hour because you 287 00:17:07,835 --> 00:17:09,257 are afraid of the cops. 288 00:17:09,257 --> 00:17:11,060 And you are doing the right thing 289 00:17:11,060 --> 00:17:12,519 because don't mess with Texas. 290 00:17:12,519 --> 00:17:18,700 I have friends who came here to visit-- Texas, New Mexico, 291 00:17:18,700 --> 00:17:22,410 go to Santa Fe, go to Carlsbad Caverns. 292 00:17:22,410 --> 00:17:24,189 Many of them got caught. 293 00:17:24,189 --> 00:17:27,089 Many of them got tickets. 294 00:17:27,089 --> 00:17:29,270 So it's really serious. 295 00:17:29,270 --> 00:17:36,580 OK, that's go further and see what we 296 00:17:36,580 --> 00:17:40,750 remember about planes in space. 297 00:17:40,750 --> 00:17:44,120 Because planes in space are magic? 298 00:17:44,120 --> 00:17:44,980 No. 299 00:17:44,980 --> 00:17:47,640 Planes in space are very important. 300 00:17:47,640 --> 00:17:55,360 Planes in space are two dimensional objects 301 00:17:55,360 --> 00:18:00,750 embedded three dimensional [? area ?] spaces. 302 00:18:00,750 --> 00:18:02,550 This is what we're talking about. 303 00:18:02,550 --> 00:18:04,930 But even if you lived in a four dimensional 304 00:18:04,930 --> 00:18:08,220 space, five dimensional space, n dimensional space, 305 00:18:08,220 --> 00:18:10,340 in the space of your imagination, 306 00:18:10,340 --> 00:18:13,375 if you have this two dimensional object, 307 00:18:13,375 --> 00:18:16,420 it would still be called a plane. 308 00:18:16,420 --> 00:18:20,030 All right, so how about planes? 309 00:18:20,030 --> 00:18:22,880 What is their equation? 310 00:18:22,880 --> 00:18:29,980 In your case ax plus by plus cz plus d is the general equation. 311 00:18:29,980 --> 00:18:36,570 We now have a plane in r3. 312 00:18:36,570 --> 00:18:38,750 You should not forget about it. 313 00:18:38,750 --> 00:18:41,780 It's going to haunt you in the final 314 00:18:41,780 --> 00:18:45,250 and in other exams in your life through at least two 315 00:18:45,250 --> 00:18:47,540 or three different exercises. 316 00:18:47,540 --> 00:18:53,400 Now I'm going to ask you to do a simple exercise. 317 00:18:53,400 --> 00:19:07,831 What is the equation of the plane normal to the given line? 318 00:19:07,831 --> 00:19:09,350 And this is the given line. 319 00:19:09,350 --> 00:19:11,350 Look at it, how beautiful [INAUDIBLE]. 320 00:19:11,350 --> 00:19:25,850 And passing through-- that passes through the point 321 00:19:25,850 --> 00:19:32,410 another point-- x1, y1, z1-- that I give you. 322 00:19:32,410 --> 00:19:35,270 How do you solve solution? 323 00:19:35,270 --> 00:19:37,776 How do you solve this quickly? 324 00:19:37,776 --> 00:19:41,490 You should just remember what you learned 325 00:19:41,490 --> 00:19:43,660 and write that as soon as possible. 326 00:19:43,660 --> 00:19:46,772 Because, OK, this may be a little piece 327 00:19:46,772 --> 00:19:51,712 of a bigger problem in my exam. 328 00:19:51,712 --> 00:19:52,710 STUDENT: [INAUDIBLE] 329 00:19:52,710 --> 00:19:54,960 [? PROFESSOR: Who is ?] a? 330 00:19:54,960 --> 00:19:57,730 If this is normal to the line-- 331 00:19:57,730 --> 00:19:59,420 STUDENT: A is 1. 332 00:19:59,420 --> 00:20:04,480 PROFESSOR: You pick up abc exactly 333 00:20:04,480 --> 00:20:09,340 from the lmn of the line. 334 00:20:09,340 --> 00:20:12,000 Remember this was an essential piece of information. 335 00:20:12,000 --> 00:20:17,270 So the relationship between a line and its normal plane 336 00:20:17,270 --> 00:20:22,760 is that the direction of that line lmn 337 00:20:22,760 --> 00:20:27,900 gives the coefficients abc of the plane, all right? 338 00:20:27,900 --> 00:20:31,570 Don't forget that because you're going to stumble right into it 339 00:20:31,570 --> 00:20:35,990 in the exams [? lx ?] in the coming up-- in the one that's 340 00:20:35,990 --> 00:20:36,720 coming up. 341 00:20:36,720 --> 00:20:39,010 And c, is this good? 342 00:20:39,010 --> 00:20:41,940 No, I cannot say d and then look for d. 343 00:20:41,940 --> 00:20:44,710 I could-- I could [INAUDIBLE]. 344 00:20:44,710 --> 00:20:46,820 Whatever you want. 345 00:20:46,820 --> 00:20:48,570 But then it's more work for me. 346 00:20:48,570 --> 00:20:52,130 Look, I don't know-- suppose I don't know who d is. 347 00:20:52,130 --> 00:20:55,820 I have to make the plane satisfy-- 348 00:20:55,820 --> 00:20:59,092 make the point x1, y1, z1 satisfy 349 00:20:59,092 --> 00:21:02,070 the equation of the plane. 350 00:21:02,070 --> 00:21:03,970 And that is more work. 351 00:21:03,970 --> 00:21:06,550 I can do that if I forget. 352 00:21:06,550 --> 00:21:09,060 If I forget the theory, I can always do that. 353 00:21:09,060 --> 00:21:13,370 Subtract the two lines, subtract the second out of the first. 354 00:21:13,370 --> 00:21:15,330 I get something magic that I should 355 00:21:15,330 --> 00:21:19,585 have known from my previous knowledge, 356 00:21:19,585 --> 00:21:21,800 from a previous life-- no. 357 00:21:21,800 --> 00:21:29,320 L times x minus x1 plus m times y minus y1 plus z times 358 00:21:29,320 --> 00:21:30,560 z minus 1. 359 00:21:30,560 --> 00:21:34,830 And I notice that most of you-- you prove me on exams, 360 00:21:34,830 --> 00:21:38,490 you prove me on homework-- know that if you have 361 00:21:38,490 --> 00:21:44,290 the coefficients and you also have the point that 362 00:21:44,290 --> 00:21:48,120 is containing the plane, you can go ahead and write 363 00:21:48,120 --> 00:21:49,950 this equation from the start. 364 00:21:49,950 --> 00:21:54,050 So you know very well that x1, y1, z1 satisfies 365 00:21:54,050 --> 00:21:57,870 your [INAUDIBLE] the plane Then you can go ahead and write it. 366 00:21:57,870 --> 00:22:02,095 Save time on that exam Don't waste time. 367 00:22:02,095 --> 00:22:05,453 It's like a star test that's a four hour test. 368 00:22:05,453 --> 00:22:07,780 No, ours is only two hours and a half. 369 00:22:07,780 --> 00:22:11,370 But still, the pressure is about the same. 370 00:22:11,370 --> 00:22:15,720 So we have to remember these notions. 371 00:22:15,720 --> 00:22:18,810 We cannot survive without them. 372 00:22:18,810 --> 00:22:20,330 Let's move on. 373 00:22:20,330 --> 00:22:26,210 And one of you asked me. 374 00:22:26,210 --> 00:22:31,310 Do I need to know by heart the formula that 375 00:22:31,310 --> 00:22:35,340 give-- a formula that will give the distance between a point 376 00:22:35,340 --> 00:22:37,550 in space and a line in space? 377 00:22:37,550 --> 00:22:38,940 No, that is not assumed. 378 00:22:38,940 --> 00:22:41,290 You can build up to that one. 379 00:22:41,290 --> 00:22:42,350 It's not so immediate. 380 00:22:42,350 --> 00:22:44,435 It takes about 15 minutes. 381 00:22:44,435 --> 00:22:45,840 That's not a problem. 382 00:22:45,840 --> 00:22:47,890 What you are supposed to remember, 383 00:22:47,890 --> 00:22:53,550 though, is that the formula for distance between a given 384 00:22:53,550 --> 00:23:00,910 point in plane and a point in space and a given 385 00:23:00,910 --> 00:23:06,370 plane in space-- that was a long time ago that you knew that, 386 00:23:06,370 --> 00:23:09,465 but I said you should never for get it 387 00:23:09,465 --> 00:23:13,920 because it's similar to the formula 388 00:23:13,920 --> 00:23:21,210 for the distance between a point in plane and a line in plane. 389 00:23:21,210 --> 00:23:25,800 I'm not testing you, but I will-- I hope-- maybe I do. 390 00:23:25,800 --> 00:23:31,050 I hope that you remember how to write this as a fraction. 391 00:23:31,050 --> 00:23:34,090 I'm already giving you hits. 392 00:23:34,090 --> 00:23:34,590 What is-- 393 00:23:34,590 --> 00:23:35,970 STUDENT: [INAUDIBLE] 394 00:23:35,970 --> 00:23:38,150 PROFESSOR: Absolute value because it's a distance. 395 00:23:38,150 --> 00:23:39,090 STUDENT: [INAUDIBLE] 396 00:23:39,090 --> 00:23:40,296 PROFESSOR: Of what? 397 00:23:40,296 --> 00:23:40,796 STUDENT: Ax. 398 00:23:40,796 --> 00:23:41,555 399 00:23:41,555 --> 00:23:47,170 PROFESSOR: Ax0 plus by0 plus cz0-- 400 00:23:47,170 --> 00:23:48,130 STUDENT: Plus b. 401 00:23:48,130 --> 00:23:50,050 PROFESSOR: Plus b, O. Good. 402 00:23:50,050 --> 00:23:51,805 STUDENT: [INAUDIBLE] 403 00:23:51,805 --> 00:23:52,930 PROFESSOR: Square root of-- 404 00:23:52,930 --> 00:23:53,721 STUDENT: A squared. 405 00:23:53,721 --> 00:23:56,710 PROFESSOR: A squared plus b squared plus c squared. 406 00:23:56,710 --> 00:24:01,420 Right, so it's a generalization of the formula of the-- 407 00:24:01,420 --> 00:24:04,840 in plane if you have a point and a line that 408 00:24:04,840 --> 00:24:11,190 doesn't contain the point, you have a similar type of formula. 409 00:24:11,190 --> 00:24:16,560 Good, let's remember the basics of conics. 410 00:24:16,560 --> 00:24:20,230 Because I'm afraid that you forgot them from Calc 2 411 00:24:20,230 --> 00:24:24,860 and from analytic or trigonometry class. 412 00:24:24,860 --> 00:24:30,520 What were the standard conics that were used in this class 413 00:24:30,520 --> 00:24:34,418 and I would like you to never forget? 414 00:24:34,418 --> 00:24:38,780 Well, when you are in an exam, you 415 00:24:38,780 --> 00:24:42,860 may be asked the [INAUDIBLE] inside of an ellipse. 416 00:24:42,860 --> 00:24:44,655 But if you don't know the standard equation 417 00:24:44,655 --> 00:24:46,340 of an ellipse, that's bad. 418 00:24:46,340 --> 00:24:47,790 So you should. 419 00:24:47,790 --> 00:24:48,830 What is that? 420 00:24:48,830 --> 00:24:52,383 Ab are semi-axis. 421 00:24:52,383 --> 00:24:53,970 STUDENT: X squared over a squared. 422 00:24:53,970 --> 00:24:56,787 PROFESSOR: X squared over a squared plus y squared 423 00:24:56,787 --> 00:25:01,210 over b squared equals 1. 424 00:25:01,210 --> 00:25:05,920 Excellent, and what if I have-- I'm 425 00:25:05,920 --> 00:25:11,412 going to draw a rectangle with these kind 426 00:25:11,412 --> 00:25:14,160 of semi axes a and b. 427 00:25:14,160 --> 00:25:19,650 And I'm going to draw the diagonals-- the diagonals. 428 00:25:19,650 --> 00:25:23,360 And I'm going to draw a [INAUDIBLE] something 429 00:25:23,360 --> 00:25:27,580 that is touching, kissing at this point tangent to it. 430 00:25:27,580 --> 00:25:32,350 And it's asymptotic to the blue asymptotes. 431 00:25:32,350 --> 00:25:34,024 What is this animal? 432 00:25:34,024 --> 00:25:35,880 STUDENT: Hyperbola. 433 00:25:35,880 --> 00:25:38,940 PROFESSOR: The standard hyperbola? 434 00:25:38,940 --> 00:25:41,690 Tell me what-- it has these branches. 435 00:25:41,690 --> 00:25:42,717 The equation is what? 436 00:25:42,717 --> 00:25:43,550 STUDENT: [INAUDIBLE] 437 00:25:43,550 --> 00:25:46,560 PROFESSOR: X squared over a squared minus y squared 438 00:25:46,560 --> 00:25:49,140 over b squared equals 1. 439 00:25:49,140 --> 00:25:59,310 If I were to draw its brother-- oh-- 440 00:25:59,310 --> 00:26:05,890 that brother would be the conjugate, OK? 441 00:26:05,890 --> 00:26:11,490 And you would have to swap the sides of plus minus. 442 00:26:11,490 --> 00:26:14,490 And you'll get the conjugate. 443 00:26:14,490 --> 00:26:18,425 Quadrics-- OK, the parabola, I don't remind you the parabola 444 00:26:18,425 --> 00:26:20,850 because you see it everywhere. 445 00:26:20,850 --> 00:26:24,740 I'm going to review it when I work with some quadrics. 446 00:26:24,740 --> 00:26:30,735 So the [INAUDIBLE] quadrics-- and I really 447 00:26:30,735 --> 00:26:37,300 would like you to, if you feel the need to remind yourself 448 00:26:37,300 --> 00:26:43,220 when quadrics are, go to the so-called gallery of quadrics. 449 00:26:43,220 --> 00:26:48,730 Type these magic words as keywords in Google. 450 00:26:48,730 --> 00:26:52,580 And it's going to send you to a beautiful website 451 00:26:52,580 --> 00:26:57,160 from University of Minnesota that has a gallery of quadrics 452 00:26:57,160 --> 00:27:00,880 where not only do you see the most important quadrics 453 00:27:00,880 --> 00:27:06,840 in standard forms, but you also see the cross sections that you 454 00:27:06,840 --> 00:27:11,570 have when you curve those quardics with horizontal planes 455 00:27:11,570 --> 00:27:14,820 or other planes parallel to the planes of coordinates. 456 00:27:14,820 --> 00:27:18,040 So I don't know in which order to present them to you. 457 00:27:18,040 --> 00:27:22,940 But how about I present them to you in the order 458 00:27:22,940 --> 00:27:28,482 that they were mostly frequently used 459 00:27:28,482 --> 00:27:36,940 rather than starting with-- so ellipsoid and respectively 460 00:27:36,940 --> 00:27:39,688 a sphere. 461 00:27:39,688 --> 00:27:44,220 Depends if you like football-- American football or soccer. 462 00:27:44,220 --> 00:27:48,720 Well, let's see what the equations were. 463 00:27:48,720 --> 00:27:52,390 X squared over a squared plus y squared 464 00:27:52,390 --> 00:27:56,140 over b squared plus z squared over c squared 465 00:27:56,140 --> 00:27:58,760 equals 1 for the ellipsoids. 466 00:27:58,760 --> 00:28:06,230 If abc are equal and equal to r, what is that? 467 00:28:06,230 --> 00:28:12,455 That's a sphere of center origin-- standard sphere-- 468 00:28:12,455 --> 00:28:13,770 in radius . 469 00:28:13,770 --> 00:28:18,800 R These are your friends. 470 00:28:18,800 --> 00:28:20,610 Don't forget about them. 471 00:28:20,610 --> 00:28:28,220 When you draw the ellipsoid, remember 472 00:28:28,220 --> 00:28:32,220 that the first line, the dotted one, 473 00:28:32,220 --> 00:28:36,700 is an ellipse on the other behind the board. 474 00:28:36,700 --> 00:28:38,625 And that is obtained as x squared 475 00:28:38,625 --> 00:28:41,305 over a squared plus y squared over b squared equals 1. 476 00:28:41,305 --> 00:28:47,570 So it's going to be an intersection with z equals 0 477 00:28:47,570 --> 00:28:53,150 And similarly, you can take the plain that's x equals 0. 478 00:28:53,150 --> 00:28:56,270 And you get this ellipse, the plane that is y equals 0. 479 00:28:56,270 --> 00:28:57,940 And you get this ellipse. 480 00:28:57,940 --> 00:29:01,110 So those are all friends of yours. 481 00:29:01,110 --> 00:29:03,160 Remember that all the cross sections 482 00:29:03,160 --> 00:29:10,470 you have cutting with planes, the football, you have, what? 483 00:29:10,470 --> 00:29:12,140 Ellipses. 484 00:29:12,140 --> 00:29:15,240 That is easy and beautiful and it's not 485 00:29:15,240 --> 00:29:18,150 something you need a lot of thinking about. 486 00:29:18,150 --> 00:29:23,470 But let's move on some other guys that I'm afraid you forgot 487 00:29:23,470 --> 00:29:27,936 and you should not forget in any case. 488 00:29:27,936 --> 00:29:33,830 And the hyperboloids-- hyperboloids, 489 00:29:33,830 --> 00:29:42,830 the most standard ones, the classification 490 00:29:42,830 --> 00:29:48,540 that we had in the classroom was based on putting everybody 491 00:29:48,540 --> 00:29:49,880 to the left hand side. 492 00:29:49,880 --> 00:29:53,700 How many pluses, how many minuses you have had? 493 00:29:53,700 --> 00:29:57,300 If you have plus, plus, plus, minus or minus, minus, minus, 494 00:29:57,300 --> 00:30:01,350 plus, you have an uneven number of pluses and minus. 495 00:30:01,350 --> 00:30:03,516 That was the two-sheeted hyperbola. 496 00:30:03,516 --> 00:30:07,020 If you had an even number of pluses and minuses, 497 00:30:07,020 --> 00:30:09,970 that's the one sheet hyperbola. 498 00:30:09,970 --> 00:30:14,040 So let us remember how that went. 499 00:30:14,040 --> 00:30:21,410 Assuming that I love this one, this 500 00:30:21,410 --> 00:30:25,540 is the first one-- the first kind which 501 00:30:25,540 --> 00:30:31,150 is the one-sheeted hyperboloid. 502 00:30:31,150 --> 00:30:33,240 What is the symmetry axis? 503 00:30:33,240 --> 00:30:41,300 The surface of revolution-- What axis? 504 00:30:41,300 --> 00:30:44,600 Of axis 0x. 505 00:30:44,600 --> 00:30:47,962 So I'm going to go ahead and draw that. 506 00:30:47,962 --> 00:30:50,930 I'm going to draw as well as I can. 507 00:30:50,930 --> 00:30:53,316 I cannot draw very well today. 508 00:30:53,316 --> 00:30:56,102 Although I had three cups of coffee, doesn't matter. 509 00:30:56,102 --> 00:31:00,190 I'm still shaking when it comes to drawing. 510 00:31:00,190 --> 00:31:03,530 So in order to get the cross section, the first cross 511 00:31:03,530 --> 00:31:06,760 section, the red one, what do you guys do? 512 00:31:06,760 --> 00:31:08,110 STUDENT: [INAUDIBLE] 513 00:31:08,110 --> 00:31:09,669 PROFESSOR: It's a-- what? 514 00:31:09,669 --> 00:31:11,085 It's an ellipse because you said z 515 00:31:11,085 --> 00:31:13,650 equal to 0 just as you said now. 516 00:31:13,650 --> 00:31:18,110 So I get the ellipse of semi axis a and b. 517 00:31:18,110 --> 00:31:19,330 This is the x-axis. 518 00:31:19,330 --> 00:31:20,820 This is a. 519 00:31:20,820 --> 00:31:22,180 This is b. 520 00:31:22,180 --> 00:31:25,500 Well, it looks like horrible in b. 521 00:31:25,500 --> 00:31:28,870 And that's the [INAUDIBLE] we have. 522 00:31:28,870 --> 00:31:31,310 But now you say, but wait a minute. 523 00:31:31,310 --> 00:31:37,210 I would like to draw the cross section that corresponds to x 524 00:31:37,210 --> 00:31:38,040 equals 0. 525 00:31:38,040 --> 00:31:41,660 And that should be in the plane of the board. 526 00:31:41,660 --> 00:31:50,700 So if you set x to be 0, then you have the standard hyperbola 527 00:31:50,700 --> 00:31:53,450 based on semi axes b and c. 528 00:31:53,450 --> 00:31:55,605 Now, b, you believe me. 529 00:31:55,605 --> 00:31:59,762 But c, you don't believe me at all because you cannot see. 530 00:31:59,762 --> 00:32:05,130 So if I were to be proactive-- which right now I'm 531 00:32:05,130 --> 00:32:07,960 not very proactive, but I'll try-- 532 00:32:07,960 --> 00:32:12,614 I'm going to have to draw-- look, 533 00:32:12,614 --> 00:32:16,884 I'm not done even if I didn't have enough coffee. 534 00:32:16,884 --> 00:32:20,880 So the rectangle-- you see b and c here? 535 00:32:20,880 --> 00:32:22,370 OK, you see the asymptote? 536 00:32:22,370 --> 00:32:25,220 It was not a bad guess of the asymptote. 537 00:32:25,220 --> 00:32:28,700 This branch of the cross section looks like, really, 538 00:32:28,700 --> 00:32:30,510 a good branch for the asymptote. 539 00:32:30,510 --> 00:32:33,145 Good, and the other one in a similar way, 540 00:32:33,145 --> 00:32:35,300 you can find the other cross section, 541 00:32:35,300 --> 00:32:37,500 which is also a hyperbola. 542 00:32:37,500 --> 00:32:43,200 So your old friend which is one-sheeted hyperboloid, 543 00:32:43,200 --> 00:32:50,395 hyperboloid-- it sounds like a monster-- what 544 00:32:50,395 --> 00:32:53,730 was special about him? 545 00:32:53,730 --> 00:32:54,995 You have some extra credit. 546 00:32:54,995 --> 00:32:56,400 STUDENT: [INAUDIBLE] 547 00:32:56,400 --> 00:32:58,570 PROFESSOR: It's a [? ruled ?] surface generated 548 00:32:58,570 --> 00:33:01,292 by two families of lines. 549 00:33:01,292 --> 00:33:03,310 And thanks again for the model. 550 00:33:03,310 --> 00:33:05,630 I will keep it for the rest of my life. 551 00:33:05,630 --> 00:33:08,090 You got five bonus points because of that. 552 00:33:08,090 --> 00:33:10,690 I'm just-- well, this is something 553 00:33:10,690 --> 00:33:12,690 I will always remember. 554 00:33:12,690 --> 00:33:19,350 Number two, how do I write that two-sheeted hyperboloid 555 00:33:19,350 --> 00:33:23,260 if I wanted me to have the same axis of symmetry? 556 00:33:23,260 --> 00:33:25,325 It should be a surface of revolution 557 00:33:25,325 --> 00:33:29,530 consisting of two parts, two. 558 00:33:29,530 --> 00:33:30,970 They are disconnected, right? 559 00:33:30,970 --> 00:33:34,456 You have two sheets, two somethings, 560 00:33:34,456 --> 00:33:35,710 two connected components. 561 00:33:35,710 --> 00:33:38,690 562 00:33:38,690 --> 00:33:40,325 It's not hard at all. 563 00:33:40,325 --> 00:33:41,660 What do I need to do? 564 00:33:41,660 --> 00:33:44,550 565 00:33:44,550 --> 00:33:49,740 The same thing as here-- just change the minus to a plus. 566 00:33:49,740 --> 00:33:52,510 All righty, x squared over a squared plus y squared 567 00:33:52,510 --> 00:33:59,660 over b squared minus z squared over c squared plus 1 equals 0. 568 00:33:59,660 --> 00:34:04,276 Great, so I can go ahead and reminds you what that was. 569 00:34:04,276 --> 00:34:06,670 You didn't like it when you first, 570 00:34:06,670 --> 00:34:09,190 but maybe now you like it better. 571 00:34:09,190 --> 00:34:12,940 This is always yz. 572 00:34:12,940 --> 00:34:20,000 And I'm going to draw the two sheets. 573 00:34:20,000 --> 00:34:21,820 And I'm going to ask you eventually, 574 00:34:21,820 --> 00:34:26,310 because I am mean, how far apart they are. 575 00:34:26,310 --> 00:34:28,121 It's the surface of revolution. 576 00:34:28,121 --> 00:34:29,579 These two guys should be symmetric. 577 00:34:29,579 --> 00:34:32,199 578 00:34:32,199 --> 00:34:40,000 Well, so when I were-- if I were to take z equals 0, 579 00:34:40,000 --> 00:34:44,010 I would get no solution because this is impossible. 580 00:34:44,010 --> 00:34:48,614 I have a sum of squares equal 0, right? 581 00:34:48,614 --> 00:34:52,110 It's impossible to get 0 this way. 582 00:34:52,110 --> 00:34:57,945 When would I get 0 on the axis of rotation? 583 00:34:57,945 --> 00:35:01,570 Well, axis of rotation means forget about x and y. 584 00:35:01,570 --> 00:35:03,430 X is 0, y is 0. 585 00:35:03,430 --> 00:35:05,890 Z would be how much? 586 00:35:05,890 --> 00:35:06,700 STUDENT: C. 587 00:35:06,700 --> 00:35:07,702 PROFESSOR: Plus minus c. 588 00:35:07,702 --> 00:35:08,660 Plus minus-- very good. 589 00:35:08,660 --> 00:35:13,670 C, practically c, if c is positive, and minus c here. 590 00:35:13,670 --> 00:35:19,200 So I know how far apart they are, these two-- [INAUDIBLE] 591 00:35:19,200 --> 00:35:21,535 this is not [? x ?] [INAUDIBLE] minimum and the maximum 592 00:35:21,535 --> 00:35:23,090 over here. 593 00:35:23,090 --> 00:35:24,944 Now, one last question. 594 00:35:24,944 --> 00:35:28,160 Well-- OK, no. 595 00:35:28,160 --> 00:35:34,650 More questions-- when I were to intersect with, let's 596 00:35:34,650 --> 00:35:41,250 say, a z that is bigger than c, a z plane that 597 00:35:41,250 --> 00:35:46,810 is bigger than c over here, what am I going to get? 598 00:35:46,810 --> 00:35:47,547 No-- 599 00:35:47,547 --> 00:35:48,380 STUDENT: An ellipse. 600 00:35:48,380 --> 00:35:49,754 PROFESSOR: An elipse-- excellent. 601 00:35:49,754 --> 00:35:52,342 An ellipse here, an ellipse there everything 602 00:35:52,342 --> 00:35:53,310 is symmetrical. 603 00:35:53,310 --> 00:35:57,350 And finally, what if I take x to be 0? 604 00:35:57,350 --> 00:35:59,060 I'm in the plane of the board. 605 00:35:59,060 --> 00:36:00,790 I hide the x. 606 00:36:00,790 --> 00:36:02,220 I get this. 607 00:36:02,220 --> 00:36:04,640 What is this? 608 00:36:04,640 --> 00:36:11,480 A hyperbola in the plane of the board, which is yz. 609 00:36:11,480 --> 00:36:14,640 Y is going this way, z is going up. 610 00:36:14,640 --> 00:36:17,370 X doesn't exist anymore. 611 00:36:17,370 --> 00:36:19,700 So what kind of hyperbola is this? 612 00:36:19,700 --> 00:36:23,020 Do you like it? 613 00:36:23,020 --> 00:36:23,520 So-- 614 00:36:23,520 --> 00:36:26,430 STUDENT: [INAUDIBLE] 615 00:36:26,430 --> 00:36:28,790 PROFESSOR: Right, mean smart. 616 00:36:28,790 --> 00:36:31,670 go ahead and multiply by negative-- who said that? 617 00:36:31,670 --> 00:36:34,790 Zander, you got two extra points, extra [INAUDIBLE]. 618 00:36:34,790 --> 00:36:38,450 Minus y squared over b squared equals 1. 619 00:36:38,450 --> 00:36:39,620 What did he notice? 620 00:36:39,620 --> 00:36:41,540 What did he-- he gets my mind. 621 00:36:41,540 --> 00:36:45,910 I'm trying to say you have no hyperbola like that. 622 00:36:45,910 --> 00:36:48,630 So Zander said, I know what which ones. 623 00:36:48,630 --> 00:36:53,250 She wants these two branches to be the hyperbola. 624 00:36:53,250 --> 00:36:56,190 But that's a conjugate hyperbola. 625 00:36:56,190 --> 00:36:58,484 That is a conjugate hyperbola because you 626 00:36:58,484 --> 00:37:04,130 don't have y and z with minus between the squares and a y. 627 00:37:04,130 --> 00:37:10,160 So this is the conjugate hyperbola-- hyperbola-- 628 00:37:10,160 --> 00:37:13,610 that I'm going to draw. 629 00:37:13,610 --> 00:37:14,940 In what color? 630 00:37:14,940 --> 00:37:16,030 That's the question. 631 00:37:16,030 --> 00:37:18,950 It's really essential what color I'm going to use. 632 00:37:18,950 --> 00:37:23,320 So I'm going to use-- I'm going to use green. 633 00:37:23,320 --> 00:37:26,790 And this is the hyperbola we are talking about. 634 00:37:26,790 --> 00:37:32,250 It's a conjugate one drawn in the plane of the board. 635 00:37:32,250 --> 00:37:33,560 OK, all right. 636 00:37:33,560 --> 00:37:36,218 So if I wanted to drop those asymptotes, 637 00:37:36,218 --> 00:37:38,500 they will look very ugly. 638 00:37:38,500 --> 00:37:42,630 And I cannot do better, but that's [INAUDIBLE]. 639 00:37:42,630 --> 00:37:50,070 So we have reviewed the most awful quadrics. 640 00:37:50,070 --> 00:37:54,270 A friend of yours that by now all of you love 641 00:37:54,270 --> 00:37:57,020 is mister paraboloid. 642 00:37:57,020 --> 00:38:01,530 You have used that in all sorts of examples. 643 00:38:01,530 --> 00:38:06,040 I'm going to remind you what the standard one was 644 00:38:06,040 --> 00:38:08,980 that we used before. 645 00:38:08,980 --> 00:38:19,210 So [INAUDIBLE] paraboloids, elliptic paraboloid. 646 00:38:19,210 --> 00:38:22,650 647 00:38:22,650 --> 00:38:25,870 Circular paraboloid is just the particular case. 648 00:38:25,870 --> 00:38:29,020 649 00:38:29,020 --> 00:38:32,981 The elliptic paraboloid that you're used to 650 00:38:32,981 --> 00:38:37,460 is the following-- z equals x squared over a squared 651 00:38:37,460 --> 00:38:41,810 plus y squared over b squared. 652 00:38:41,810 --> 00:38:44,640 They may be positive if you want. 653 00:38:44,640 --> 00:38:49,480 They don't-- in general, they are not equal. 654 00:38:49,480 --> 00:38:53,900 The circular paraboloid-- well, you simply 655 00:38:53,900 --> 00:38:58,577 assume that a and b are equal. 656 00:38:58,577 --> 00:39:04,230 And then you put-- you want a c squared or an r squared. 657 00:39:04,230 --> 00:39:06,660 Let's put an r squared on top. 658 00:39:06,660 --> 00:39:10,425 It really doesn't matter what you're putting there. 659 00:39:10,425 --> 00:39:11,630 Can I draw? 660 00:39:11,630 --> 00:39:15,930 Hopefully, hopefully, hopefully I can draw. 661 00:39:15,930 --> 00:39:18,710 It looks like a valley whose minimum 662 00:39:18,710 --> 00:39:22,160 is at the origin I'm going to draw 663 00:39:22,160 --> 00:39:29,680 so that the intersection with the horizontal plane 664 00:39:29,680 --> 00:39:32,170 will be visible to you. 665 00:39:32,170 --> 00:39:36,820 And I take this z greater than 0. 666 00:39:36,820 --> 00:39:39,407 And then I'm going to have some sort of ellipse. 667 00:39:39,407 --> 00:39:42,180 668 00:39:42,180 --> 00:39:44,535 Under that, there is nothing. 669 00:39:44,535 --> 00:39:46,110 Under the origin, there is nothing 670 00:39:46,110 --> 00:39:50,210 because z is going to be positive at x equals 0, 671 00:39:50,210 --> 00:39:53,170 y equals 0, and passing through the origin-- very 672 00:39:53,170 --> 00:39:56,280 nice and [? sassy ?] Quadric. 673 00:39:56,280 --> 00:40:01,790 There is one that occurred in many examples like a nightmare. 674 00:40:01,790 --> 00:40:04,070 And it was based on that one. 675 00:40:04,070 --> 00:40:05,910 And I'm going to draw-- no, no, no. 676 00:40:05,910 --> 00:40:07,730 I'm going to write it and you draw it 677 00:40:07,730 --> 00:40:09,560 with the eyes of your imagination 678 00:40:09,560 --> 00:40:11,760 and see what that is. 679 00:40:11,760 --> 00:40:16,670 Because you are, again, going to bump into it into the exam. 680 00:40:16,670 --> 00:40:20,760 We had all sorts of patches of that. 681 00:40:20,760 --> 00:40:22,450 Look at the areas of the patch. 682 00:40:22,450 --> 00:40:25,860 And you cannot get rid of that. 683 00:40:25,860 --> 00:40:29,025 It's haunting your dreams. 684 00:40:29,025 --> 00:40:29,730 What is this? 685 00:40:29,730 --> 00:40:31,350 STUDENT: [INAUDIBLE] 686 00:40:31,350 --> 00:40:34,390 PROFESSOR: Upside down paraboloid-- 687 00:40:34,390 --> 00:40:35,870 what is the vertex? 688 00:40:35,870 --> 00:40:37,500 Where is the vertex at? 689 00:40:37,500 --> 00:40:38,400 STUDENT: 0, 0, 1. 690 00:40:38,400 --> 00:40:41,870 PROFESSOR: 0, 0, 1-- very good. 691 00:40:41,870 --> 00:40:45,970 What's special about it? 692 00:40:45,970 --> 00:40:49,490 So assume that I would draw the-- I 693 00:40:49,490 --> 00:40:56,258 would draw it to compute the normal to the surface. 694 00:40:56,258 --> 00:40:58,234 How would I do that? 695 00:40:58,234 --> 00:40:59,716 STUDENT: [INAUDIBLE] 696 00:40:59,716 --> 00:41:00,704 PROFESSOR: Uh, yeah. 697 00:41:00,704 --> 00:41:02,412 Well, it's a little bit more complicated. 698 00:41:02,412 --> 00:41:04,900 I would have to shift everybody to once side, 699 00:41:04,900 --> 00:41:07,962 the side that I have a certain increase in form [? than to ?] 700 00:41:07,962 --> 00:41:09,890 the gradient [? to stuff ?] like that. 701 00:41:09,890 --> 00:41:15,776 So don't forget about this type of project is an essential one. 702 00:41:15,776 --> 00:41:18,752 703 00:41:18,752 --> 00:41:21,300 Am I missing anybody important? 704 00:41:21,300 --> 00:41:22,600 Yes. 705 00:41:22,600 --> 00:41:23,910 We live in tests. 706 00:41:23,910 --> 00:41:27,763 We cannot say goodbye to the last section of the chapter 707 00:41:27,763 --> 00:41:32,655 nine, which is 9.7, without meeting again our friend 708 00:41:32,655 --> 00:41:35,220 the saddle, right? 709 00:41:35,220 --> 00:41:39,050 The saddle is-- this is elliptic paraboloid. 710 00:41:39,050 --> 00:41:42,200 And the last very important quadric 711 00:41:42,200 --> 00:41:46,890 that I wanted to talk about today is the-- 712 00:41:46,890 --> 00:41:50,850 713 00:41:50,850 --> 00:41:52,205 STUDENT: What about a cone? 714 00:41:52,205 --> 00:41:52,830 PROFESSOR: Huh? 715 00:41:52,830 --> 00:41:53,820 STUDENT: How about a cone? 716 00:41:53,820 --> 00:41:55,320 PROFESSOR: Oh, a cone is too easy. 717 00:41:55,320 --> 00:41:58,720 But yeah, let's talk about the cone as well. 718 00:41:58,720 --> 00:42:01,600 Give me an example of the standard cone. 719 00:42:01,600 --> 00:42:04,240 Thank you, [INAUDIBLE]. 720 00:42:04,240 --> 00:42:06,510 X squared-- well-- 721 00:42:06,510 --> 00:42:08,010 STUDENT: T squared equals x squared. 722 00:42:08,010 --> 00:42:11,450 PROFESSOR: I'm going to draw it first 723 00:42:11,450 --> 00:42:14,390 so that you know what I want. 724 00:42:14,390 --> 00:42:16,250 Unless I draw it, how would you know what 725 00:42:16,250 --> 00:42:19,070 to invent or to come up with? 726 00:42:19,070 --> 00:42:23,200 It's not an ice cream cone-- it's a double cone. 727 00:42:23,200 --> 00:42:27,010 So I can have a positive z and a negative z-- two 728 00:42:27,010 --> 00:42:29,770 different sheets that are symmetric with one another. 729 00:42:29,770 --> 00:42:33,570 730 00:42:33,570 --> 00:42:35,740 So how do I write that? 731 00:42:35,740 --> 00:42:36,722 STUDENT: T squared. 732 00:42:36,722 --> 00:42:37,680 PROFESSOR: Yes, equals? 733 00:42:37,680 --> 00:42:38,513 STUDENT: [INAUDIBLE] 734 00:42:38,513 --> 00:42:41,300 735 00:42:41,300 --> 00:42:45,930 PROFESSOR: Well, would you like it to be like most of those 736 00:42:45,930 --> 00:42:50,772 that we see in the examples in the book, right? 737 00:42:50,772 --> 00:42:54,200 But it doesn't have to be like that. 738 00:42:54,200 --> 00:42:56,640 Of course, if it's like that, of course 739 00:42:56,640 --> 00:42:59,240 you realize z-- set the z. 740 00:42:59,240 --> 00:43:00,890 Set the plane and altitude. 741 00:43:00,890 --> 00:43:03,320 Then you're going to have circle, circle, circle, 742 00:43:03,320 --> 00:43:07,340 circle-- circle after circle of different radii 743 00:43:07,340 --> 00:43:08,810 as cross sections. 744 00:43:08,810 --> 00:43:10,450 Z could also be negative. 745 00:43:10,450 --> 00:43:13,960 Except for the case of the origin, where you have 0, 0, 0. 746 00:43:13,960 --> 00:43:17,890 747 00:43:17,890 --> 00:43:21,720 Now, if you were to set x equals 0, of course 748 00:43:21,720 --> 00:43:24,260 you would get y equals plus minus 749 00:43:24,260 --> 00:43:30,050 z, which are exactly these lines, the red lines that I'm 750 00:43:30,050 --> 00:43:31,050 drawing in this picture. 751 00:43:31,050 --> 00:43:34,332 752 00:43:34,332 --> 00:43:35,910 So practically, this is, what? 753 00:43:35,910 --> 00:43:37,240 Called a what? 754 00:43:37,240 --> 00:43:39,450 A circular cone. 755 00:43:39,450 --> 00:43:42,090 If I wanted to make it more interesting, 756 00:43:42,090 --> 00:43:45,560 I would put a squared and b squared. 757 00:43:45,560 --> 00:43:48,360 And it would be an elliptic cone. 758 00:43:48,360 --> 00:43:51,740 And we stayed away from that as much as we could. 759 00:43:51,740 --> 00:43:55,160 We brought it up now because Zander asked about it. 760 00:43:55,160 --> 00:43:57,950 So how about the number four, number five, whatever it 761 00:43:57,950 --> 00:43:59,070 is-- number four? 762 00:43:59,070 --> 00:44:03,090 The [INAUDIBLE] what was the typical equation 763 00:44:03,090 --> 00:44:15,715 of the hyperbolic paraboloid that I had in mind? 764 00:44:15,715 --> 00:44:16,548 STUDENT: [INAUDIBLE] 765 00:44:16,548 --> 00:44:19,992 766 00:44:19,992 --> 00:44:31,970 PROFESSOR: Z equals x squared minus y squared, very good. 767 00:44:31,970 --> 00:44:34,530 So I will try again and draw it. 768 00:44:34,530 --> 00:44:36,770 It's not so easy to draw. 769 00:44:36,770 --> 00:44:40,450 770 00:44:40,450 --> 00:44:47,200 If I were to choose x to be 0 and draw exactly 771 00:44:47,200 --> 00:44:51,140 in the plane of the board, z equals minus y squared 772 00:44:51,140 --> 00:44:59,550 would be some coordinate line, right? 773 00:44:59,550 --> 00:45:02,000 This is what we call such a thing. 774 00:45:02,000 --> 00:45:06,220 If we fix the x to be x0, we get a coordinate line. 775 00:45:06,220 --> 00:45:09,800 If we fix the y to be y0, we get another coordinate line. 776 00:45:09,800 --> 00:45:12,360 There are two families of lines. 777 00:45:12,360 --> 00:45:17,160 Why is z equals minus y squared drawn in this board, 778 00:45:17,160 --> 00:45:19,315 on the board, in this plane? 779 00:45:19,315 --> 00:45:23,530 It's a parabola that opens upside down. 780 00:45:23,530 --> 00:45:34,030 OK, so you have something like this which you are drawing, 781 00:45:34,030 --> 00:45:36,050 right? 782 00:45:36,050 --> 00:45:39,880 And then what if y would be 0? 783 00:45:39,880 --> 00:45:43,470 Then you get z equals x0. 784 00:45:43,470 --> 00:45:48,402 So it's going to a parabola that opens up. 785 00:45:48,402 --> 00:45:53,720 Then I have to locate myself and draw it on that wall. 786 00:45:53,720 --> 00:45:54,720 But I can't. 787 00:45:54,720 --> 00:45:57,695 Because if I do that, I'm going to get in trouble. 788 00:45:57,695 --> 00:46:01,740 So I better draw it like this in perspective. 789 00:46:01,740 --> 00:46:04,720 And you guys should imagine what we have. 790 00:46:04,720 --> 00:46:12,079 So if we were to cut down with a knife, 791 00:46:12,079 --> 00:46:15,572 we would get-- we will still get these parabolas 792 00:46:15,572 --> 00:46:18,070 that all point down. 793 00:46:18,070 --> 00:46:22,165 And in those directions, these are just the highest parts 794 00:46:22,165 --> 00:46:23,590 of the saddle. 795 00:46:23,590 --> 00:46:27,030 And let's say this would be the lowest part of the saddle. 796 00:46:27,030 --> 00:46:32,280 Where-- where is the part of the rider? 797 00:46:32,280 --> 00:46:34,990 A guy's butt is here. 798 00:46:34,990 --> 00:46:38,760 And his leg is following the shape of the saddle going down. 799 00:46:38,760 --> 00:46:43,322 That's the cowboy boot, OK? 800 00:46:43,322 --> 00:46:46,530 And he is-- hold on. 801 00:46:46,530 --> 00:46:52,659 I don't know how-- what's the attitude of the [INAUDIBLE]? 802 00:46:52,659 --> 00:46:54,615 Well, it doesn't look like a cowboy hat. 803 00:46:54,615 --> 00:46:57,549 But anyway, I'm sorry. 804 00:46:57,549 --> 00:46:59,505 He looks a little bit Vietnamese. 805 00:46:59,505 --> 00:47:03,417 That was not the intention. 806 00:47:03,417 --> 00:47:05,862 STUDENT: [INAUDIBLE] 807 00:47:05,862 --> 00:47:07,840 PROFESSOR: Then let him be Mexican. 808 00:47:07,840 --> 00:47:11,160 Half of the population in this town are Mexican. 809 00:47:11,160 --> 00:47:14,500 So this is his leg that goes down. 810 00:47:14,500 --> 00:47:18,910 811 00:47:18,910 --> 00:47:20,380 OK, very good. 812 00:47:20,380 --> 00:47:23,810 Look-- he even has a-- what do you call that? 813 00:47:23,810 --> 00:47:24,790 That's so beautiful. 814 00:47:24,790 --> 00:47:30,625 In the Mexican culture, they make those embroidered by hand 815 00:47:30,625 --> 00:47:33,050 with many colors belts. 816 00:47:33,050 --> 00:47:35,640 But there are some special belts. 817 00:47:35,640 --> 00:47:39,450 OK-- depends on the area of Mexico You visit. 818 00:47:39,450 --> 00:47:41,260 I liked several of them. 819 00:47:41,260 --> 00:47:42,800 They're so beautiful. 820 00:47:42,800 --> 00:47:45,585 But my favorite one is, of course, 821 00:47:45,585 --> 00:47:50,535 the Rivera Maya, which is where you go to the Chichen Itza, 822 00:47:50,535 --> 00:47:54,645 to the mystic areas, to the sea, and eat the good food 823 00:47:54,645 --> 00:48:00,205 and go to Cozumel and forget about school for a week. 824 00:48:00,205 --> 00:48:02,680 That is paradise for me. 825 00:48:02,680 --> 00:48:05,155 But [INAUDIBLE] is not bad either. 826 00:48:05,155 --> 00:48:08,125 If I were to choose where to live and I had money, 827 00:48:08,125 --> 00:48:11,110 I would live in Cozumel for the rest of my life. 828 00:48:11,110 --> 00:48:15,610 OK, so this is the saddle that is oriented 829 00:48:15,610 --> 00:48:18,450 so that you have a parabola going in this direction, 830 00:48:18,450 --> 00:48:23,160 going up, a parabola going down in this direction. 831 00:48:23,160 --> 00:48:28,090 What is magic about a saddle point? 832 00:48:28,090 --> 00:48:29,479 Do you remember? 833 00:48:29,479 --> 00:48:33,160 STUDENT: It was [INAUDIBLE] 834 00:48:33,160 --> 00:48:36,315 PROFESSOR: It's-- one direction is like a max and one direction 835 00:48:36,315 --> 00:48:37,450 is like a min. 836 00:48:37,450 --> 00:48:41,340 So when you compute that discriminant, you get negative. 837 00:48:41,340 --> 00:48:44,672 You get like the product between the curvatures. 838 00:48:44,672 --> 00:48:46,713 One curvature in one direction would be positive. 839 00:48:46,713 --> 00:48:48,390 The other one would be negative. 840 00:48:48,390 --> 00:48:52,880 So it's like getting the product plus minus equals y, [? yes. ?] 841 00:48:52,880 --> 00:48:55,620 But again, we will talk about this sometimes later. 842 00:48:55,620 --> 00:48:58,840 You don't even have to know that. 843 00:48:58,840 --> 00:49:01,840 Shall we say goodbye to This? 844 00:49:01,840 --> 00:49:03,780 I guess it's time. 845 00:49:03,780 --> 00:49:08,270 And chapter nine is now fresh in your memory. 846 00:49:08,270 --> 00:49:11,866 You would be really to start chapter 10. 847 00:49:11,866 --> 00:49:14,800 848 00:49:14,800 --> 00:49:19,510 What I want to do, I want to-- without being recorded. 849 00:49:19,510 --> 00:49:20,543