WEBVTT 00:00:00.000 --> 00:00:00.500 00:00:00.500 --> 00:00:04.590 PROFESSOR: You have learned a lot in Calculus 2. 00:00:04.590 --> 00:00:12.180 Whether you took Calculus recently or long time ago, 00:00:12.180 --> 00:00:20.960 Chapter 9 is about vectors in r3 and eventually 00:00:20.960 --> 00:00:25.490 [? a plane ?] and operations with such vectors 00:00:25.490 --> 00:00:28.510 and the implications of the vectors 00:00:28.510 --> 00:00:32.880 in the equations of a line in space or plane 00:00:32.880 --> 00:00:35.180 in space-- stuff like that. 00:00:35.180 --> 00:00:42.820 Now, 9.1 to 9.5 was considered to be covered completely 00:00:42.820 --> 00:00:46.970 in Calc 2 here at Tech. 00:00:46.970 --> 00:00:52.400 However, lots of students come from South Plains College 00:00:52.400 --> 00:00:54.880 and [? Rio ?] College, lots of colleges 00:00:54.880 --> 00:00:59.220 where by the nature of the course Calculus 2, 00:00:59.220 --> 00:01:02.440 vectors in r3 are not covered. 00:01:02.440 --> 00:01:08.730 Therefore, I'd like to make an attempt to review 9.1 and 9.5 00:01:08.730 --> 00:01:12.820 quickly with the knowledge you have now as grown-ups 00:01:12.820 --> 00:01:15.260 in the area of vectors in r2. 00:01:15.260 --> 00:01:17.130 So again, what are vectors? 00:01:17.130 --> 00:01:20.700 They are oriented segments. 00:01:20.700 --> 00:01:22.940 Not only that they are oriented segments, 00:01:22.940 --> 00:01:29.840 but we make the distinction between a vector that is fixed 00:01:29.840 --> 00:01:33.960 in the sense that his origin is fixed-- we cannot move him-- 00:01:33.960 --> 00:01:36.870 and a free vector who is not married to the origin. 00:01:36.870 --> 00:01:40.830 He can shift by parallelism anywhere in space. 00:01:40.830 --> 00:01:43.020 And we call that a free vector. 00:01:43.020 --> 00:01:48.030 The distinction between those vectors would be vr of v bar. 00:01:48.030 --> 00:01:52.920 As you remember, v bar was the free guy, free vector, 00:01:52.920 --> 00:02:00.666 which is the-- actually, it's an equivalence class 00:02:00.666 --> 00:02:14.720 of all vectors that can be obtained from the generic v 00:02:14.720 --> 00:02:15.220 bounded. 00:02:15.220 --> 00:02:20.970 So I'm going to have to point by translation. 00:02:20.970 --> 00:02:25.090 So you have this kind of-- same magnitude 00:02:25.090 --> 00:02:29.411 for all vectors, same magnitude, same orientation, 00:02:29.411 --> 00:02:33.210 and parallel directions, parallel lines. 00:02:33.210 --> 00:02:35.380 What have we done to such a vector? 00:02:35.380 --> 00:02:38.690 As you remember very well, we decomposed him, 00:02:38.690 --> 00:02:44.940 being on the standard canonical basis, which for most of you 00:02:44.940 --> 00:02:50.140 engineers and engineering majors is denoted as ijk where 00:02:50.140 --> 00:02:55.850 ijk is an orthonormal frame with respect 00:02:55.850 --> 00:02:57.110 to the Cartesian coordinates. 00:02:57.110 --> 00:03:03.380 So i, j, and k will be their unit vectors 00:03:03.380 --> 00:03:08.775 on the x, y, z axes of coordinates, Cartesian axes 00:03:08.775 --> 00:03:09.730 of coordinates. 00:03:09.730 --> 00:03:14.530 So remember always that ijk are orthogonal to one another. 00:03:14.530 --> 00:03:17.460 Since this is review, I'd like to attract your attention 00:03:17.460 --> 00:03:25.320 to the fact that k is plus j. 00:03:25.320 --> 00:03:29.630 Think about it-- what happens you bring i over j. 00:03:29.630 --> 00:03:32.205 And you get k because you move up. 00:03:32.205 --> 00:03:35.219 Because it's like you are turning [INAUDIBLE] 00:03:35.219 --> 00:03:40.250 connection and the screw or whatever from the faucet 00:03:40.250 --> 00:03:41.690 is pointing upwards. 00:03:41.690 --> 00:03:43.530 It's like the right hand rule. 00:03:43.530 --> 00:03:48.560 If you would do the other way around, if you do j cross i, 00:03:48.560 --> 00:03:50.000 what are you going to have? 00:03:50.000 --> 00:03:55.330 Minus k-- so the properties of the cross product being 00:03:55.330 --> 00:03:57.740 antisymmetric are supposed to be, 00:03:57.740 --> 00:04:02.190 no, pay attention to the signs in all the exams that you have. 00:04:02.190 --> 00:04:05.150 What do we know about their respective products 00:04:05.150 --> 00:04:08.750 for vectors in space or in plain? 00:04:08.750 --> 00:04:13.530 If you have two vectors in their standard basis, 00:04:13.530 --> 00:04:17.680 you want i plus u2j plus u3k where 00:04:17.680 --> 00:04:24.480 ui is a real number and e1i plus v2j plus v3k where 00:04:24.480 --> 00:04:31.603 vi are [INAUDIBLE] real numbers the dot product or the scalar 00:04:31.603 --> 00:04:35.530 product-- now, I saw that in all your engineering and physics 00:04:35.530 --> 00:04:38.130 classes, you will use this notation. 00:04:38.130 --> 00:04:40.630 Mathematicians sometimes say, no, I'm 00:04:40.630 --> 00:04:42.870 going to use angular brackets because it's 00:04:42.870 --> 00:04:46.020 a scalar product in r3 or the scalar product 00:04:46.020 --> 00:04:47.910 and the dot product is the same thing, 00:04:47.910 --> 00:04:49.790 being that's the standard one here. 00:04:49.790 --> 00:04:54.670 You want v1 plus u2v2 plus u3v3. 00:04:54.670 --> 00:04:58.031 So what do you to remember what you do? 00:04:58.031 --> 00:04:59.955 First component plus first component 00:04:59.955 --> 00:05:02.365 times second component times second component 00:05:02.365 --> 00:05:05.950 plus third component times third component, OK? 00:05:05.950 --> 00:05:10.420 If you are in computer science, I 00:05:10.420 --> 00:05:13.150 saw that you use this notation. 00:05:13.150 --> 00:05:15.490 I was very happy to see that. 00:05:15.490 --> 00:05:16.570 the summation notation. 00:05:16.570 --> 00:05:20.240 But you don't have to use that in our class. 00:05:20.240 --> 00:05:24.410 Now, above the [? fresh ?] product of two vectors, 00:05:24.410 --> 00:05:29.100 you have the definition ijk the first row. 00:05:29.100 --> 00:05:31.800 So what you get is going to be a vector. 00:05:31.800 --> 00:05:36.230 Here, what you get is a scalar as a result. 00:05:36.230 --> 00:05:40.510 Here's what you get as a vector, as the result 00:05:40.510 --> 00:05:42.625 of the first product is a vector. 00:05:42.625 --> 00:05:48.029 So you have u1, u2, u3, v1, v2, v3. 00:05:48.029 --> 00:05:49.320 These are all friends of yours. 00:05:49.320 --> 00:05:56.840 I'm just reminding you the lucrative definitions. 00:05:56.840 --> 00:05:59.992 Now, some people said, yes, but I'd 00:05:59.992 --> 00:06:03.250 like to see the lucrative definitions that 00:06:03.250 --> 00:06:05.060 have to do with trig as well. 00:06:05.060 --> 00:06:06.130 OK, let's see. 00:06:06.130 --> 00:06:09.120 For those of you who asked me to remind you what they were, 00:06:09.120 --> 00:06:13.170 I will remind you what they were. 00:06:13.170 --> 00:06:20.480 For u.v, you get the same thing as writing magnitude u 00:06:20.480 --> 00:06:25.170 magnitude v and cosine of the angle between them 00:06:25.170 --> 00:06:27.380 no matter in which direction you take it 00:06:27.380 --> 00:06:29.370 because the cosine is the same. 00:06:29.370 --> 00:06:32.330 Cosine of pi is equal to cosine of negative phi 00:06:32.330 --> 00:06:34.500 or theta [INAUDIBLE]. 00:06:34.500 --> 00:06:35.930 How about the other one? 00:06:35.930 --> 00:06:38.482 Here's where one of you had a little bit 00:06:38.482 --> 00:06:41.590 of a misunderstanding. 00:06:41.590 --> 00:06:45.210 And I saw that happen in two finals, unfortunately. 00:06:45.210 --> 00:06:51.540 This is not the scalar vector that I'm right here. 00:06:51.540 --> 00:06:52.620 It's a vector. 00:06:52.620 --> 00:06:53.650 So what's missing? 00:06:53.650 --> 00:06:55.830 This is the scalar part. 00:06:55.830 --> 00:07:00.896 And then you have times e where e is the unit vector 00:07:00.896 --> 00:07:10.050 of the direction of the vector, the direction of u.v. 00:07:10.050 --> 00:07:12.480 Why I cannot use another notion? 00:07:12.480 --> 00:07:13.980 Because u is already taken. 00:07:13.980 --> 00:07:17.676 But e in itself should suggest to you 00:07:17.676 --> 00:07:22.090 that you have a unit vector, [? length of ?] one vector, OK? 00:07:22.090 --> 00:07:26.640 All right, what is the-- let's review a little 00:07:26.640 --> 00:07:29.280 bit the absolute value. 00:07:29.280 --> 00:07:31.610 Well, the absolute value is a scalar. 00:07:31.610 --> 00:07:34.430 So that scalar will be magnitude of your magnitude 00:07:34.430 --> 00:07:36.770 of [INAUDIBLE] sine of the angle. 00:07:36.770 --> 00:07:41.010 And do you guys remember the geometric interpretation 00:07:41.010 --> 00:07:42.556 of that? 00:07:42.556 --> 00:07:43.472 STUDENT: [INAUDIBLE] 00:07:43.472 --> 00:07:45.138 PROFESSOR: The area of the parallelogram 00:07:45.138 --> 00:07:48.380 based on the two vectors-- very good [INAUDIBLE]. 00:07:48.380 --> 00:07:50.870 U plus b is the area of the parallelogram 00:07:50.870 --> 00:07:56.980 that you would draw based on those two vectors. 00:07:56.980 --> 00:07:58.340 All right, good. 00:07:58.340 --> 00:08:00.730 Now, say goodbye, vectors. 00:08:00.730 --> 00:08:04.464 We've seen-- you've seen them through 9.4, 9.5. 00:08:04.464 --> 00:08:08.190 What was important to remember was 00:08:08.190 --> 00:08:10.850 that these vectors were the building 00:08:10.850 --> 00:08:14.630 blocks, the foundations, of the equations 00:08:14.630 --> 00:08:16.845 of the lines in space. 00:08:16.845 --> 00:08:18.860 That's your work [INAUDIBLE]. 00:08:18.860 --> 00:08:21.370 So what did we work with? 00:08:21.370 --> 00:08:26.110 Lines in space-- lines in space can be given in many ways. 00:08:26.110 --> 00:08:28.030 But now that you remember them, I'm 00:08:28.030 --> 00:08:35.600 going to give you the symmetric equation of a line in space. 00:08:35.600 --> 00:08:38.908 00:08:38.908 --> 00:08:42.640 OK, [INAUDIBLE] this can see [INAUDIBLE] 00:08:42.640 --> 00:08:44.350 included on the final. 00:08:44.350 --> 00:08:46.760 You are expected to know it. 00:08:46.760 --> 00:08:55.520 00:08:55.520 --> 00:08:59.190 So that is the symmetric equation, 00:08:59.190 --> 00:09:03.410 meaning the equation of a what? 00:09:03.410 --> 00:09:11.280 Of a line in space passing through or containing 00:09:11.280 --> 00:09:21.470 the point of p, not of coordinates x0, y0, [? z0, ?] 00:09:21.470 --> 00:09:29.820 and of direction [INAUDIBLE] in the sense of a vector. 00:09:29.820 --> 00:09:32.949 Now, if I were to draw such a line, 00:09:32.949 --> 00:09:35.344 I'm going to have the line over here. 00:09:35.344 --> 00:09:39.050 Going to have a vector for the point p0 on the line. 00:09:39.050 --> 00:09:41.970 I can put this free vector because he's free. 00:09:41.970 --> 00:09:43.720 He says, I'm a free guy. 00:09:43.720 --> 00:09:47.660 I can slide any way I want. 00:09:47.660 --> 00:09:52.190 So I'm going to have li plus mj plus mk. 00:09:52.190 --> 00:09:56.600 00:09:56.600 --> 00:09:58.070 This is the blue vector. 00:09:58.070 --> 00:10:01.570 Now, you don't have blue markers or blue pens, 00:10:01.570 --> 00:10:05.940 but you can still do a good job taking notes. 00:10:05.940 --> 00:10:09.570 Now somebody asked me just a week ago, 00:10:09.570 --> 00:10:11.970 saying that I've started doing review already 00:10:11.970 --> 00:10:16.340 and I don't understand what the difference is 00:10:16.340 --> 00:10:20.400 between the symmetric equation of a line in space 00:10:20.400 --> 00:10:24.840 and the parametric equations of a line in space. 00:10:24.840 --> 00:10:27.830 This is no essential difference. 00:10:27.830 --> 00:10:28.770 So what do we do? 00:10:28.770 --> 00:10:31.947 We denote this whole animal by t, a real number. 00:10:31.947 --> 00:10:35.220 00:10:35.220 --> 00:10:37.640 And then we erase the board. 00:10:37.640 --> 00:10:43.310 And then we write the three equations 00:10:43.310 --> 00:10:47.326 that govern-- I'm going to put if and only 00:10:47.326 --> 00:10:51.590 if xyz satisfy the following. 00:10:51.590 --> 00:10:53.360 So I'm going to have, what? 00:10:53.360 --> 00:11:07.881 X equals lt plus x0, y equals mt plus y0, n equals nt plus z0. 00:11:07.881 --> 00:11:11.600 Well, of course, we understand-- we know the meaning 00:11:11.600 --> 00:11:16.490 that lmn are like what [INAUDIBLE] physics direction 00:11:16.490 --> 00:11:18.690 cosines were telling me about it. 00:11:18.690 --> 00:11:21.960 And then x0, y0, z0 is a fixed point 00:11:21.960 --> 00:11:24.580 that belongs to that line. 00:11:24.580 --> 00:11:28.650 Now, since you know a little bit more than 00:11:28.650 --> 00:11:30.990 you knew in Calculus 2 when you saw 00:11:30.990 --> 00:11:35.580 that for the first time, what is the typical notation that we 00:11:35.580 --> 00:11:40.040 use all through Calc 3, all through the chapters? 00:11:40.040 --> 00:11:44.920 The position vector-- the position vector of the point 00:11:44.920 --> 00:11:53.930 on the line that is related to, what? 00:11:53.930 --> 00:11:57.280 So practically you have the origin here. 00:11:57.280 --> 00:12:04.200 [? Op0 ?] represent the vector x0i plus y0j plus e0k. 00:12:04.200 --> 00:12:06.880 So now you have a little bit of a different understanding 00:12:06.880 --> 00:12:08.910 of what's going on. 00:12:08.910 --> 00:12:15.300 And then after, let's say, t equals 1 hour, what do you do? 00:12:15.300 --> 00:12:18.100 You are adding the blue vector here. 00:12:18.100 --> 00:12:20.800 00:12:20.800 --> 00:12:23.950 Let's say at t equals 1, you are here. 00:12:23.950 --> 00:12:25.690 You are here at p1. 00:12:25.690 --> 00:12:29.920 So to get to p1, you have to add two vectors, right guys? 00:12:29.920 --> 00:12:34.030 This is the addition between the blue vector and the red vector. 00:12:34.030 --> 00:12:36.290 So what you get is your result. 00:12:36.290 --> 00:12:41.120 So if I am smart enough to understand my concepts are all 00:12:41.120 --> 00:12:44.110 connected, the position in this case 00:12:44.110 --> 00:12:49.860 will be r of t, which is-- I hate angular brackets, 00:12:49.860 --> 00:12:52.650 but just because you like them, I'm 00:12:52.650 --> 00:12:55.760 going to use them-- x of ty tz of tm 00:12:55.760 --> 00:12:58.140 to be consistent with the book. 00:12:58.140 --> 00:13:04.440 This is the same as xi plux yj plus zk. 00:13:04.440 --> 00:13:08.518 And what is this by the actual notations 00:13:08.518 --> 00:13:10.510 from the parametric equation? 00:13:10.510 --> 00:13:15.570 This is nothing but a certain lmn 00:13:15.570 --> 00:13:20.080 vector that is the vector li plus mj plus nk written 00:13:20.080 --> 00:13:21.750 with angular brackets because I know 00:13:21.750 --> 00:13:28.350 you like that times the time t plus the fixed vector x0, 00:13:28.350 --> 00:13:29.300 y0, z0. 00:13:29.300 --> 00:13:31.890 You can say, yeah, I thought it was a point. 00:13:31.890 --> 00:13:33.410 It is a point and a vector. 00:13:33.410 --> 00:13:41.960 You identified the point p0 with the position of the point p0 00:13:41.960 --> 00:13:44.660 starting with respect to the origin. 00:13:44.660 --> 00:13:47.240 So whether you're talking about mister p0 00:13:47.240 --> 00:13:51.900 being a point in space-- x0, y0, z0. 00:13:51.900 --> 00:13:54.810 Or you're talking about the [INAUDIBLE] position 00:13:54.810 --> 00:13:58.190 vector that [INAUDIBLE] is practically 00:13:58.190 --> 00:14:00.000 the same after identification. 00:14:00.000 --> 00:14:02.670 So you have something very nice. 00:14:02.670 --> 00:14:05.970 And if I asked you with the mind and the knowledge 00:14:05.970 --> 00:14:13.430 you have now what that does is mean-- r prime of t 00:14:13.430 --> 00:14:16.190 equals what? 00:14:16.190 --> 00:14:19.180 It's the velocity vector. 00:14:19.180 --> 00:14:22.295 And what is that as a vector? 00:14:22.295 --> 00:14:24.540 Do the differentiation. 00:14:24.540 --> 00:14:27.180 What do we get in terms of velocity vector? 00:14:27.180 --> 00:14:29.480 Prime with respect to t-- what do I get? 00:14:29.480 --> 00:14:30.146 STUDENT: Lmn. 00:14:30.146 --> 00:14:31.270 PROFESSOR: Lmn as a vector. 00:14:31.270 --> 00:14:33.860 But of course, as I hate angular notations, 00:14:33.860 --> 00:14:38.470 I will rewrite it-- li plus mj plus nk. 00:14:38.470 --> 00:14:40.190 So this is your velocity. 00:14:40.190 --> 00:14:44.155 What can you say about this type of motion? 00:14:44.155 --> 00:14:44.886 This is a-- 00:14:44.886 --> 00:14:46.510 STUDENT: [INAUDIBLE] constant velocity. 00:14:46.510 --> 00:14:48.835 PROFESSOR: Yeah, you have a constant velocity 00:14:48.835 --> 00:14:51.232 for this motion. 00:14:51.232 --> 00:14:56.356 If somebody would ask you you have-- 10 years from now, 00:14:56.356 --> 00:15:00.370 you have a boy who said, dad-- or a girl. 00:15:00.370 --> 00:15:01.513 let's not be biased. 00:15:01.513 --> 00:15:05.690 So he learns, math, good at math or physics, and says, 00:15:05.690 --> 00:15:08.370 what is the difference between velocity and speed? 00:15:08.370 --> 00:15:11.172 Well, most parents will say it's the same thing. 00:15:11.172 --> 00:15:14.406 Well, you're not most parents. 00:15:14.406 --> 00:15:16.290 You are educated parents. 00:15:16.290 --> 00:15:19.340 So this is-- don't tell your kid about vectors, 00:15:19.340 --> 00:15:22.670 but you can show them you have an oriented segment. 00:15:22.670 --> 00:15:26.390 So make your child run around around in circles and say, 00:15:26.390 --> 00:15:30.660 this is the velocity that's always tangent to the circle 00:15:30.660 --> 00:15:32.420 that you are running on. 00:15:32.420 --> 00:15:34.860 That's a velocity. 00:15:34.860 --> 00:15:38.890 And if they ask, well, they will catch 00:15:38.890 --> 00:15:41.990 the notions of acceleration and force faster than you 00:15:41.990 --> 00:15:44.020 because they see all these cartoons. 00:15:44.020 --> 00:15:48.360 And my son was telling me the other thing-- he's 10 00:15:48.360 --> 00:15:50.570 and I asked him, what the heck is that? 00:15:50.570 --> 00:15:53.910 It looked like an electromagnetic field 00:15:53.910 --> 00:15:55.780 surrounding some hero. 00:15:55.780 --> 00:15:59.700 And he said, mom, that's the force field of course. 00:15:59.700 --> 00:16:02.155 And I was thinking, force field? 00:16:02.155 --> 00:16:04.773 This is what I taught the other day when 00:16:04.773 --> 00:16:08.047 I was talking about [? crux. ?] Double integral of f.n 00:16:08.047 --> 00:16:10.011 [INAUDIBLE] f was the force field. 00:16:10.011 --> 00:16:13.950 So he was, like, talking about something very normal 00:16:13.950 --> 00:16:15.670 that you see every day. 00:16:15.670 --> 00:16:21.250 So do not underestimate your nephews, nieces, children. 00:16:21.250 --> 00:16:23.100 They will catch up on these things 00:16:23.100 --> 00:16:25.540 faster than you, which is good. 00:16:25.540 --> 00:16:30.820 Now, the speed in this case will be, what? 00:16:30.820 --> 00:16:33.170 What is the speed of this-- the speed 00:16:33.170 --> 00:16:36.345 of this motion, linear motion? 00:16:36.345 --> 00:16:38.132 STUDENT: Square root of l squared. 00:16:38.132 --> 00:16:42.880 PROFESSOR: Square root of l squared plus m squared plus n 00:16:42.880 --> 00:16:45.530 squared, which again is different from velocity. 00:16:45.530 --> 00:16:48.580 Velocity is a vector, speed is a scalar. 00:16:48.580 --> 00:16:50.640 Velocity is a vector, speed is a scalar. 00:16:50.640 --> 00:16:52.590 In general, doesn't have to be constant, 00:16:52.590 --> 00:16:54.510 but this is the blessing because lmn 00:16:54.510 --> 00:16:57.410 are given constants. [INAUDIBLE] in this case, 00:16:57.410 --> 00:16:59.660 you are on cruise control. 00:16:59.660 --> 00:17:02.240 You are moving on a line directly 00:17:02.240 --> 00:17:05.200 in your motion on cruise control driving 00:17:05.200 --> 00:17:07.835 to Amarillo at 60 miles an hour because you 00:17:07.835 --> 00:17:09.257 are afraid of the cops. 00:17:09.257 --> 00:17:11.060 And you are doing the right thing 00:17:11.060 --> 00:17:12.519 because don't mess with Texas. 00:17:12.519 --> 00:17:18.700 I have friends who came here to visit-- Texas, New Mexico, 00:17:18.700 --> 00:17:22.410 go to Santa Fe, go to Carlsbad Caverns. 00:17:22.410 --> 00:17:24.189 Many of them got caught. 00:17:24.189 --> 00:17:27.089 Many of them got tickets. 00:17:27.089 --> 00:17:29.270 So it's really serious. 00:17:29.270 --> 00:17:36.580 OK, that's go further and see what we 00:17:36.580 --> 00:17:40.750 remember about planes in space. 00:17:40.750 --> 00:17:44.120 Because planes in space are magic? 00:17:44.120 --> 00:17:44.980 No. 00:17:44.980 --> 00:17:47.640 Planes in space are very important. 00:17:47.640 --> 00:17:55.360 Planes in space are two dimensional objects 00:17:55.360 --> 00:18:00.750 embedded three dimensional [? area ?] spaces. 00:18:00.750 --> 00:18:02.550 This is what we're talking about. 00:18:02.550 --> 00:18:04.930 But even if you lived in a four dimensional 00:18:04.930 --> 00:18:08.220 space, five dimensional space, n dimensional space, 00:18:08.220 --> 00:18:10.340 in the space of your imagination, 00:18:10.340 --> 00:18:13.375 if you have this two dimensional object, 00:18:13.375 --> 00:18:16.420 it would still be called a plane. 00:18:16.420 --> 00:18:20.030 All right, so how about planes? 00:18:20.030 --> 00:18:22.880 What is their equation? 00:18:22.880 --> 00:18:29.980 In your case ax plus by plus cz plus d is the general equation. 00:18:29.980 --> 00:18:36.570 We now have a plane in r3. 00:18:36.570 --> 00:18:38.750 You should not forget about it. 00:18:38.750 --> 00:18:41.780 It's going to haunt you in the final 00:18:41.780 --> 00:18:45.250 and in other exams in your life through at least two 00:18:45.250 --> 00:18:47.540 or three different exercises. 00:18:47.540 --> 00:18:53.400 Now I'm going to ask you to do a simple exercise. 00:18:53.400 --> 00:19:07.831 What is the equation of the plane normal to the given line? 00:19:07.831 --> 00:19:09.350 And this is the given line. 00:19:09.350 --> 00:19:11.350 Look at it, how beautiful [INAUDIBLE]. 00:19:11.350 --> 00:19:25.850 And passing through-- that passes through the point 00:19:25.850 --> 00:19:32.410 another point-- x1, y1, z1-- that I give you. 00:19:32.410 --> 00:19:35.270 How do you solve solution? 00:19:35.270 --> 00:19:37.776 How do you solve this quickly? 00:19:37.776 --> 00:19:41.490 You should just remember what you learned 00:19:41.490 --> 00:19:43.660 and write that as soon as possible. 00:19:43.660 --> 00:19:46.772 Because, OK, this may be a little piece 00:19:46.772 --> 00:19:51.712 of a bigger problem in my exam. 00:19:51.712 --> 00:19:52.710 STUDENT: [INAUDIBLE] 00:19:52.710 --> 00:19:54.960 [? PROFESSOR: Who is ?] a? 00:19:54.960 --> 00:19:57.730 If this is normal to the line-- 00:19:57.730 --> 00:19:59.420 STUDENT: A is 1. 00:19:59.420 --> 00:20:04.480 PROFESSOR: You pick up abc exactly 00:20:04.480 --> 00:20:09.340 from the lmn of the line. 00:20:09.340 --> 00:20:12.000 Remember this was an essential piece of information. 00:20:12.000 --> 00:20:17.270 So the relationship between a line and its normal plane 00:20:17.270 --> 00:20:22.760 is that the direction of that line lmn 00:20:22.760 --> 00:20:27.900 gives the coefficients abc of the plane, all right? 00:20:27.900 --> 00:20:31.570 Don't forget that because you're going to stumble right into it 00:20:31.570 --> 00:20:35.990 in the exams [? lx ?] in the coming up-- in the one that's 00:20:35.990 --> 00:20:36.720 coming up. 00:20:36.720 --> 00:20:39.010 And c, is this good? 00:20:39.010 --> 00:20:41.940 No, I cannot say d and then look for d. 00:20:41.940 --> 00:20:44.710 I could-- I could [INAUDIBLE]. 00:20:44.710 --> 00:20:46.820 Whatever you want. 00:20:46.820 --> 00:20:48.570 But then it's more work for me. 00:20:48.570 --> 00:20:52.130 Look, I don't know-- suppose I don't know who d is. 00:20:52.130 --> 00:20:55.820 I have to make the plane satisfy-- 00:20:55.820 --> 00:20:59.092 make the point x1, y1, z1 satisfy 00:20:59.092 --> 00:21:02.070 the equation of the plane. 00:21:02.070 --> 00:21:03.970 And that is more work. 00:21:03.970 --> 00:21:06.550 I can do that if I forget. 00:21:06.550 --> 00:21:09.060 If I forget the theory, I can always do that. 00:21:09.060 --> 00:21:13.370 Subtract the two lines, subtract the second out of the first. 00:21:13.370 --> 00:21:15.330 I get something magic that I should 00:21:15.330 --> 00:21:19.585 have known from my previous knowledge, 00:21:19.585 --> 00:21:21.800 from a previous life-- no. 00:21:21.800 --> 00:21:29.320 L times x minus x1 plus m times y minus y1 plus z times 00:21:29.320 --> 00:21:30.560 z minus 1. 00:21:30.560 --> 00:21:34.830 And I notice that most of you-- you prove me on exams, 00:21:34.830 --> 00:21:38.490 you prove me on homework-- know that if you have 00:21:38.490 --> 00:21:44.290 the coefficients and you also have the point that 00:21:44.290 --> 00:21:48.120 is containing the plane, you can go ahead and write 00:21:48.120 --> 00:21:49.950 this equation from the start. 00:21:49.950 --> 00:21:54.050 So you know very well that x1, y1, z1 satisfies 00:21:54.050 --> 00:21:57.870 your [INAUDIBLE] the plane Then you can go ahead and write it. 00:21:57.870 --> 00:22:02.095 Save time on that exam Don't waste time. 00:22:02.095 --> 00:22:05.453 It's like a star test that's a four hour test. 00:22:05.453 --> 00:22:07.780 No, ours is only two hours and a half. 00:22:07.780 --> 00:22:11.370 But still, the pressure is about the same. 00:22:11.370 --> 00:22:15.720 So we have to remember these notions. 00:22:15.720 --> 00:22:18.810 We cannot survive without them. 00:22:18.810 --> 00:22:20.330 Let's move on. 00:22:20.330 --> 00:22:26.210 And one of you asked me. 00:22:26.210 --> 00:22:31.310 Do I need to know by heart the formula that 00:22:31.310 --> 00:22:35.340 give-- a formula that will give the distance between a point 00:22:35.340 --> 00:22:37.550 in space and a line in space? 00:22:37.550 --> 00:22:38.940 No, that is not assumed. 00:22:38.940 --> 00:22:41.290 You can build up to that one. 00:22:41.290 --> 00:22:42.350 It's not so immediate. 00:22:42.350 --> 00:22:44.435 It takes about 15 minutes. 00:22:44.435 --> 00:22:45.840 That's not a problem. 00:22:45.840 --> 00:22:47.890 What you are supposed to remember, 00:22:47.890 --> 00:22:53.550 though, is that the formula for distance between a given 00:22:53.550 --> 00:23:00.910 point in plane and a point in space and a given 00:23:00.910 --> 00:23:06.370 plane in space-- that was a long time ago that you knew that, 00:23:06.370 --> 00:23:09.465 but I said you should never for get it 00:23:09.465 --> 00:23:13.920 because it's similar to the formula 00:23:13.920 --> 00:23:21.210 for the distance between a point in plane and a line in plane. 00:23:21.210 --> 00:23:25.800 I'm not testing you, but I will-- I hope-- maybe I do. 00:23:25.800 --> 00:23:31.050 I hope that you remember how to write this as a fraction. 00:23:31.050 --> 00:23:34.090 I'm already giving you hits. 00:23:34.090 --> 00:23:34.590 What is-- 00:23:34.590 --> 00:23:35.970 STUDENT: [INAUDIBLE] 00:23:35.970 --> 00:23:38.150 PROFESSOR: Absolute value because it's a distance. 00:23:38.150 --> 00:23:39.090 STUDENT: [INAUDIBLE] 00:23:39.090 --> 00:23:40.296 PROFESSOR: Of what? 00:23:40.296 --> 00:23:40.796 STUDENT: Ax. 00:23:40.796 --> 00:23:41.555 00:23:41.555 --> 00:23:47.170 PROFESSOR: Ax0 plus by0 plus cz0-- 00:23:47.170 --> 00:23:48.130 STUDENT: Plus b. 00:23:48.130 --> 00:23:50.050 PROFESSOR: Plus b, O. Good. 00:23:50.050 --> 00:23:51.805 STUDENT: [INAUDIBLE] 00:23:51.805 --> 00:23:52.930 PROFESSOR: Square root of-- 00:23:52.930 --> 00:23:53.721 STUDENT: A squared. 00:23:53.721 --> 00:23:56.710 PROFESSOR: A squared plus b squared plus c squared. 00:23:56.710 --> 00:24:01.420 Right, so it's a generalization of the formula of the-- 00:24:01.420 --> 00:24:04.840 in plane if you have a point and a line that 00:24:04.840 --> 00:24:11.190 doesn't contain the point, you have a similar type of formula. 00:24:11.190 --> 00:24:16.560 Good, let's remember the basics of conics. 00:24:16.560 --> 00:24:20.230 Because I'm afraid that you forgot them from Calc 2 00:24:20.230 --> 00:24:24.860 and from analytic or trigonometry class. 00:24:24.860 --> 00:24:30.520 What were the standard conics that were used in this class 00:24:30.520 --> 00:24:34.418 and I would like you to never forget? 00:24:34.418 --> 00:24:38.780 Well, when you are in an exam, you 00:24:38.780 --> 00:24:42.860 may be asked the [INAUDIBLE] inside of an ellipse. 00:24:42.860 --> 00:24:44.655 But if you don't know the standard equation 00:24:44.655 --> 00:24:46.340 of an ellipse, that's bad. 00:24:46.340 --> 00:24:47.790 So you should. 00:24:47.790 --> 00:24:48.830 What is that? 00:24:48.830 --> 00:24:52.383 Ab are semi-axis. 00:24:52.383 --> 00:24:53.970 STUDENT: X squared over a squared. 00:24:53.970 --> 00:24:56.787 PROFESSOR: X squared over a squared plus y squared 00:24:56.787 --> 00:25:01.210 over b squared equals 1. 00:25:01.210 --> 00:25:05.920 Excellent, and what if I have-- I'm 00:25:05.920 --> 00:25:11.412 going to draw a rectangle with these kind 00:25:11.412 --> 00:25:14.160 of semi axes a and b. 00:25:14.160 --> 00:25:19.650 And I'm going to draw the diagonals-- the diagonals. 00:25:19.650 --> 00:25:23.360 And I'm going to draw a [INAUDIBLE] something 00:25:23.360 --> 00:25:27.580 that is touching, kissing at this point tangent to it. 00:25:27.580 --> 00:25:32.350 And it's asymptotic to the blue asymptotes. 00:25:32.350 --> 00:25:34.024 What is this animal? 00:25:34.024 --> 00:25:35.880 STUDENT: Hyperbola. 00:25:35.880 --> 00:25:38.940 PROFESSOR: The standard hyperbola? 00:25:38.940 --> 00:25:41.690 Tell me what-- it has these branches. 00:25:41.690 --> 00:25:42.717 The equation is what? 00:25:42.717 --> 00:25:43.550 STUDENT: [INAUDIBLE] 00:25:43.550 --> 00:25:46.560 PROFESSOR: X squared over a squared minus y squared 00:25:46.560 --> 00:25:49.140 over b squared equals 1. 00:25:49.140 --> 00:25:59.310 If I were to draw its brother-- oh-- 00:25:59.310 --> 00:26:05.890 that brother would be the conjugate, OK? 00:26:05.890 --> 00:26:11.490 And you would have to swap the sides of plus minus. 00:26:11.490 --> 00:26:14.490 And you'll get the conjugate. 00:26:14.490 --> 00:26:18.425 Quadrics-- OK, the parabola, I don't remind you the parabola 00:26:18.425 --> 00:26:20.850 because you see it everywhere. 00:26:20.850 --> 00:26:24.740 I'm going to review it when I work with some quadrics. 00:26:24.740 --> 00:26:30.735 So the [INAUDIBLE] quadrics-- and I really 00:26:30.735 --> 00:26:37.300 would like you to, if you feel the need to remind yourself 00:26:37.300 --> 00:26:43.220 when quadrics are, go to the so-called gallery of quadrics. 00:26:43.220 --> 00:26:48.730 Type these magic words as keywords in Google. 00:26:48.730 --> 00:26:52.580 And it's going to send you to a beautiful website 00:26:52.580 --> 00:26:57.160 from University of Minnesota that has a gallery of quadrics 00:26:57.160 --> 00:27:00.880 where not only do you see the most important quadrics 00:27:00.880 --> 00:27:06.840 in standard forms, but you also see the cross sections that you 00:27:06.840 --> 00:27:11.570 have when you curve those quardics with horizontal planes 00:27:11.570 --> 00:27:14.820 or other planes parallel to the planes of coordinates. 00:27:14.820 --> 00:27:18.040 So I don't know in which order to present them to you. 00:27:18.040 --> 00:27:22.940 But how about I present them to you in the order 00:27:22.940 --> 00:27:28.482 that they were mostly frequently used 00:27:28.482 --> 00:27:36.940 rather than starting with-- so ellipsoid and respectively 00:27:36.940 --> 00:27:39.688 a sphere. 00:27:39.688 --> 00:27:44.220 Depends if you like football-- American football or soccer. 00:27:44.220 --> 00:27:48.720 Well, let's see what the equations were. 00:27:48.720 --> 00:27:52.390 X squared over a squared plus y squared 00:27:52.390 --> 00:27:56.140 over b squared plus z squared over c squared 00:27:56.140 --> 00:27:58.760 equals 1 for the ellipsoids. 00:27:58.760 --> 00:28:06.230 If abc are equal and equal to r, what is that? 00:28:06.230 --> 00:28:12.455 That's a sphere of center origin-- standard sphere-- 00:28:12.455 --> 00:28:13.770 in radius . 00:28:13.770 --> 00:28:18.800 R These are your friends. 00:28:18.800 --> 00:28:20.610 Don't forget about them. 00:28:20.610 --> 00:28:28.220 When you draw the ellipsoid, remember 00:28:28.220 --> 00:28:32.220 that the first line, the dotted one, 00:28:32.220 --> 00:28:36.700 is an ellipse on the other behind the board. 00:28:36.700 --> 00:28:38.625 And that is obtained as x squared 00:28:38.625 --> 00:28:41.305 over a squared plus y squared over b squared equals 1. 00:28:41.305 --> 00:28:47.570 So it's going to be an intersection with z equals 0 00:28:47.570 --> 00:28:53.150 And similarly, you can take the plain that's x equals 0. 00:28:53.150 --> 00:28:56.270 And you get this ellipse, the plane that is y equals 0. 00:28:56.270 --> 00:28:57.940 And you get this ellipse. 00:28:57.940 --> 00:29:01.110 So those are all friends of yours. 00:29:01.110 --> 00:29:03.160 Remember that all the cross sections 00:29:03.160 --> 00:29:10.470 you have cutting with planes, the football, you have, what? 00:29:10.470 --> 00:29:12.140 Ellipses. 00:29:12.140 --> 00:29:15.240 That is easy and beautiful and it's not 00:29:15.240 --> 00:29:18.150 something you need a lot of thinking about. 00:29:18.150 --> 00:29:23.470 But let's move on some other guys that I'm afraid you forgot 00:29:23.470 --> 00:29:27.936 and you should not forget in any case. 00:29:27.936 --> 00:29:33.830 And the hyperboloids-- hyperboloids, 00:29:33.830 --> 00:29:42.830 the most standard ones, the classification 00:29:42.830 --> 00:29:48.540 that we had in the classroom was based on putting everybody 00:29:48.540 --> 00:29:49.880 to the left hand side. 00:29:49.880 --> 00:29:53.700 How many pluses, how many minuses you have had? 00:29:53.700 --> 00:29:57.300 If you have plus, plus, plus, minus or minus, minus, minus, 00:29:57.300 --> 00:30:01.350 plus, you have an uneven number of pluses and minus. 00:30:01.350 --> 00:30:03.516 That was the two-sheeted hyperbola. 00:30:03.516 --> 00:30:07.020 If you had an even number of pluses and minuses, 00:30:07.020 --> 00:30:09.970 that's the one sheet hyperbola. 00:30:09.970 --> 00:30:14.040 So let us remember how that went. 00:30:14.040 --> 00:30:21.410 Assuming that I love this one, this 00:30:21.410 --> 00:30:25.540 is the first one-- the first kind which 00:30:25.540 --> 00:30:31.150 is the one-sheeted hyperboloid. 00:30:31.150 --> 00:30:33.240 What is the symmetry axis? 00:30:33.240 --> 00:30:41.300 The surface of revolution-- What axis? 00:30:41.300 --> 00:30:44.600 Of axis 0x. 00:30:44.600 --> 00:30:47.962 So I'm going to go ahead and draw that. 00:30:47.962 --> 00:30:50.930 I'm going to draw as well as I can. 00:30:50.930 --> 00:30:53.316 I cannot draw very well today. 00:30:53.316 --> 00:30:56.102 Although I had three cups of coffee, doesn't matter. 00:30:56.102 --> 00:31:00.190 I'm still shaking when it comes to drawing. 00:31:00.190 --> 00:31:03.530 So in order to get the cross section, the first cross 00:31:03.530 --> 00:31:06.760 section, the red one, what do you guys do? 00:31:06.760 --> 00:31:08.110 STUDENT: [INAUDIBLE] 00:31:08.110 --> 00:31:09.669 PROFESSOR: It's a-- what? 00:31:09.669 --> 00:31:11.085 It's an ellipse because you said z 00:31:11.085 --> 00:31:13.650 equal to 0 just as you said now. 00:31:13.650 --> 00:31:18.110 So I get the ellipse of semi axis a and b. 00:31:18.110 --> 00:31:19.330 This is the x-axis. 00:31:19.330 --> 00:31:20.820 This is a. 00:31:20.820 --> 00:31:22.180 This is b. 00:31:22.180 --> 00:31:25.500 Well, it looks like horrible in b. 00:31:25.500 --> 00:31:28.870 And that's the [INAUDIBLE] we have. 00:31:28.870 --> 00:31:31.310 But now you say, but wait a minute. 00:31:31.310 --> 00:31:37.210 I would like to draw the cross section that corresponds to x 00:31:37.210 --> 00:31:38.040 equals 0. 00:31:38.040 --> 00:31:41.660 And that should be in the plane of the board. 00:31:41.660 --> 00:31:50.700 So if you set x to be 0, then you have the standard hyperbola 00:31:50.700 --> 00:31:53.450 based on semi axes b and c. 00:31:53.450 --> 00:31:55.605 Now, b, you believe me. 00:31:55.605 --> 00:31:59.762 But c, you don't believe me at all because you cannot see. 00:31:59.762 --> 00:32:05.130 So if I were to be proactive-- which right now I'm 00:32:05.130 --> 00:32:07.960 not very proactive, but I'll try-- 00:32:07.960 --> 00:32:12.614 I'm going to have to draw-- look, 00:32:12.614 --> 00:32:16.884 I'm not done even if I didn't have enough coffee. 00:32:16.884 --> 00:32:20.880 So the rectangle-- you see b and c here? 00:32:20.880 --> 00:32:22.370 OK, you see the asymptote? 00:32:22.370 --> 00:32:25.220 It was not a bad guess of the asymptote. 00:32:25.220 --> 00:32:28.700 This branch of the cross section looks like, really, 00:32:28.700 --> 00:32:30.510 a good branch for the asymptote. 00:32:30.510 --> 00:32:33.145 Good, and the other one in a similar way, 00:32:33.145 --> 00:32:35.300 you can find the other cross section, 00:32:35.300 --> 00:32:37.500 which is also a hyperbola. 00:32:37.500 --> 00:32:43.200 So your old friend which is one-sheeted hyperboloid, 00:32:43.200 --> 00:32:50.395 hyperboloid-- it sounds like a monster-- what 00:32:50.395 --> 00:32:53.730 was special about him? 00:32:53.730 --> 00:32:54.995 You have some extra credit. 00:32:54.995 --> 00:32:56.400 STUDENT: [INAUDIBLE] 00:32:56.400 --> 00:32:58.570 PROFESSOR: It's a [? ruled ?] surface generated 00:32:58.570 --> 00:33:01.292 by two families of lines. 00:33:01.292 --> 00:33:03.310 And thanks again for the model. 00:33:03.310 --> 00:33:05.630 I will keep it for the rest of my life. 00:33:05.630 --> 00:33:08.090 You got five bonus points because of that. 00:33:08.090 --> 00:33:10.690 I'm just-- well, this is something 00:33:10.690 --> 00:33:12.690 I will always remember. 00:33:12.690 --> 00:33:19.350 Number two, how do I write that two-sheeted hyperboloid 00:33:19.350 --> 00:33:23.260 if I wanted me to have the same axis of symmetry? 00:33:23.260 --> 00:33:25.325 It should be a surface of revolution 00:33:25.325 --> 00:33:29.530 consisting of two parts, two. 00:33:29.530 --> 00:33:30.970 They are disconnected, right? 00:33:30.970 --> 00:33:34.456 You have two sheets, two somethings, 00:33:34.456 --> 00:33:35.710 two connected components. 00:33:35.710 --> 00:33:38.690 00:33:38.690 --> 00:33:40.325 It's not hard at all. 00:33:40.325 --> 00:33:41.660 What do I need to do? 00:33:41.660 --> 00:33:44.550 00:33:44.550 --> 00:33:49.740 The same thing as here-- just change the minus to a plus. 00:33:49.740 --> 00:33:52.510 All righty, x squared over a squared plus y squared 00:33:52.510 --> 00:33:59.660 over b squared minus z squared over c squared plus 1 equals 0. 00:33:59.660 --> 00:34:04.276 Great, so I can go ahead and reminds you what that was. 00:34:04.276 --> 00:34:06.670 You didn't like it when you first, 00:34:06.670 --> 00:34:09.190 but maybe now you like it better. 00:34:09.190 --> 00:34:12.940 This is always yz. 00:34:12.940 --> 00:34:20.000 And I'm going to draw the two sheets. 00:34:20.000 --> 00:34:21.820 And I'm going to ask you eventually, 00:34:21.820 --> 00:34:26.310 because I am mean, how far apart they are. 00:34:26.310 --> 00:34:28.121 It's the surface of revolution. 00:34:28.121 --> 00:34:29.579 These two guys should be symmetric. 00:34:29.579 --> 00:34:32.199 00:34:32.199 --> 00:34:40.000 Well, so when I were-- if I were to take z equals 0, 00:34:40.000 --> 00:34:44.010 I would get no solution because this is impossible. 00:34:44.010 --> 00:34:48.614 I have a sum of squares equal 0, right? 00:34:48.614 --> 00:34:52.110 It's impossible to get 0 this way. 00:34:52.110 --> 00:34:57.945 When would I get 0 on the axis of rotation? 00:34:57.945 --> 00:35:01.570 Well, axis of rotation means forget about x and y. 00:35:01.570 --> 00:35:03.430 X is 0, y is 0. 00:35:03.430 --> 00:35:05.890 Z would be how much? 00:35:05.890 --> 00:35:06.700 STUDENT: C. 00:35:06.700 --> 00:35:07.702 PROFESSOR: Plus minus c. 00:35:07.702 --> 00:35:08.660 Plus minus-- very good. 00:35:08.660 --> 00:35:13.670 C, practically c, if c is positive, and minus c here. 00:35:13.670 --> 00:35:19.200 So I know how far apart they are, these two-- [INAUDIBLE] 00:35:19.200 --> 00:35:21.535 this is not [? x ?] [INAUDIBLE] minimum and the maximum 00:35:21.535 --> 00:35:23.090 over here. 00:35:23.090 --> 00:35:24.944 Now, one last question. 00:35:24.944 --> 00:35:28.160 Well-- OK, no. 00:35:28.160 --> 00:35:34.650 More questions-- when I were to intersect with, let's 00:35:34.650 --> 00:35:41.250 say, a z that is bigger than c, a z plane that 00:35:41.250 --> 00:35:46.810 is bigger than c over here, what am I going to get? 00:35:46.810 --> 00:35:47.547 No-- 00:35:47.547 --> 00:35:48.380 STUDENT: An ellipse. 00:35:48.380 --> 00:35:49.754 PROFESSOR: An elipse-- excellent. 00:35:49.754 --> 00:35:52.342 An ellipse here, an ellipse there everything 00:35:52.342 --> 00:35:53.310 is symmetrical. 00:35:53.310 --> 00:35:57.350 And finally, what if I take x to be 0? 00:35:57.350 --> 00:35:59.060 I'm in the plane of the board. 00:35:59.060 --> 00:36:00.790 I hide the x. 00:36:00.790 --> 00:36:02.220 I get this. 00:36:02.220 --> 00:36:04.640 What is this? 00:36:04.640 --> 00:36:11.480 A hyperbola in the plane of the board, which is yz. 00:36:11.480 --> 00:36:14.640 Y is going this way, z is going up. 00:36:14.640 --> 00:36:17.370 X doesn't exist anymore. 00:36:17.370 --> 00:36:19.700 So what kind of hyperbola is this? 00:36:19.700 --> 00:36:23.020 Do you like it? 00:36:23.020 --> 00:36:23.520 So-- 00:36:23.520 --> 00:36:26.430 STUDENT: [INAUDIBLE] 00:36:26.430 --> 00:36:28.790 PROFESSOR: Right, mean smart. 00:36:28.790 --> 00:36:31.670 go ahead and multiply by negative-- who said that? 00:36:31.670 --> 00:36:34.790 Zander, you got two extra points, extra [INAUDIBLE]. 00:36:34.790 --> 00:36:38.450 Minus y squared over b squared equals 1. 00:36:38.450 --> 00:36:39.620 What did he notice? 00:36:39.620 --> 00:36:41.540 What did he-- he gets my mind. 00:36:41.540 --> 00:36:45.910 I'm trying to say you have no hyperbola like that. 00:36:45.910 --> 00:36:48.630 So Zander said, I know what which ones. 00:36:48.630 --> 00:36:53.250 She wants these two branches to be the hyperbola. 00:36:53.250 --> 00:36:56.190 But that's a conjugate hyperbola. 00:36:56.190 --> 00:36:58.484 That is a conjugate hyperbola because you 00:36:58.484 --> 00:37:04.130 don't have y and z with minus between the squares and a y. 00:37:04.130 --> 00:37:10.160 So this is the conjugate hyperbola-- hyperbola-- 00:37:10.160 --> 00:37:13.610 that I'm going to draw. 00:37:13.610 --> 00:37:14.940 In what color? 00:37:14.940 --> 00:37:16.030 That's the question. 00:37:16.030 --> 00:37:18.950 It's really essential what color I'm going to use. 00:37:18.950 --> 00:37:23.320 So I'm going to use-- I'm going to use green. 00:37:23.320 --> 00:37:26.790 And this is the hyperbola we are talking about. 00:37:26.790 --> 00:37:32.250 It's a conjugate one drawn in the plane of the board. 00:37:32.250 --> 00:37:33.560 OK, all right. 00:37:33.560 --> 00:37:36.218 So if I wanted to drop those asymptotes, 00:37:36.218 --> 00:37:38.500 they will look very ugly. 00:37:38.500 --> 00:37:42.630 And I cannot do better, but that's [INAUDIBLE]. 00:37:42.630 --> 00:37:50.070 So we have reviewed the most awful quadrics. 00:37:50.070 --> 00:37:54.270 A friend of yours that by now all of you love 00:37:54.270 --> 00:37:57.020 is mister paraboloid. 00:37:57.020 --> 00:38:01.530 You have used that in all sorts of examples. 00:38:01.530 --> 00:38:06.040 I'm going to remind you what the standard one was 00:38:06.040 --> 00:38:08.980 that we used before. 00:38:08.980 --> 00:38:19.210 So [INAUDIBLE] paraboloids, elliptic paraboloid. 00:38:19.210 --> 00:38:22.650 00:38:22.650 --> 00:38:25.870 Circular paraboloid is just the particular case. 00:38:25.870 --> 00:38:29.020 00:38:29.020 --> 00:38:32.981 The elliptic paraboloid that you're used to 00:38:32.981 --> 00:38:37.460 is the following-- z equals x squared over a squared 00:38:37.460 --> 00:38:41.810 plus y squared over b squared. 00:38:41.810 --> 00:38:44.640 They may be positive if you want. 00:38:44.640 --> 00:38:49.480 They don't-- in general, they are not equal. 00:38:49.480 --> 00:38:53.900 The circular paraboloid-- well, you simply 00:38:53.900 --> 00:38:58.577 assume that a and b are equal. 00:38:58.577 --> 00:39:04.230 And then you put-- you want a c squared or an r squared. 00:39:04.230 --> 00:39:06.660 Let's put an r squared on top. 00:39:06.660 --> 00:39:10.425 It really doesn't matter what you're putting there. 00:39:10.425 --> 00:39:11.630 Can I draw? 00:39:11.630 --> 00:39:15.930 Hopefully, hopefully, hopefully I can draw. 00:39:15.930 --> 00:39:18.710 It looks like a valley whose minimum 00:39:18.710 --> 00:39:22.160 is at the origin I'm going to draw 00:39:22.160 --> 00:39:29.680 so that the intersection with the horizontal plane 00:39:29.680 --> 00:39:32.170 will be visible to you. 00:39:32.170 --> 00:39:36.820 And I take this z greater than 0. 00:39:36.820 --> 00:39:39.407 And then I'm going to have some sort of ellipse. 00:39:39.407 --> 00:39:42.180 00:39:42.180 --> 00:39:44.535 Under that, there is nothing. 00:39:44.535 --> 00:39:46.110 Under the origin, there is nothing 00:39:46.110 --> 00:39:50.210 because z is going to be positive at x equals 0, 00:39:50.210 --> 00:39:53.170 y equals 0, and passing through the origin-- very 00:39:53.170 --> 00:39:56.280 nice and [? sassy ?] Quadric. 00:39:56.280 --> 00:40:01.790 There is one that occurred in many examples like a nightmare. 00:40:01.790 --> 00:40:04.070 And it was based on that one. 00:40:04.070 --> 00:40:05.910 And I'm going to draw-- no, no, no. 00:40:05.910 --> 00:40:07.730 I'm going to write it and you draw it 00:40:07.730 --> 00:40:09.560 with the eyes of your imagination 00:40:09.560 --> 00:40:11.760 and see what that is. 00:40:11.760 --> 00:40:16.670 Because you are, again, going to bump into it into the exam. 00:40:16.670 --> 00:40:20.760 We had all sorts of patches of that. 00:40:20.760 --> 00:40:22.450 Look at the areas of the patch. 00:40:22.450 --> 00:40:25.860 And you cannot get rid of that. 00:40:25.860 --> 00:40:29.025 It's haunting your dreams. 00:40:29.025 --> 00:40:29.730 What is this? 00:40:29.730 --> 00:40:31.350 STUDENT: [INAUDIBLE] 00:40:31.350 --> 00:40:34.390 PROFESSOR: Upside down paraboloid-- 00:40:34.390 --> 00:40:35.870 what is the vertex? 00:40:35.870 --> 00:40:37.500 Where is the vertex at? 00:40:37.500 --> 00:40:38.400 STUDENT: 0, 0, 1. 00:40:38.400 --> 00:40:41.870 PROFESSOR: 0, 0, 1-- very good. 00:40:41.870 --> 00:40:45.970 What's special about it? 00:40:45.970 --> 00:40:49.490 So assume that I would draw the-- I 00:40:49.490 --> 00:40:56.258 would draw it to compute the normal to the surface. 00:40:56.258 --> 00:40:58.234 How would I do that? 00:40:58.234 --> 00:40:59.716 STUDENT: [INAUDIBLE] 00:40:59.716 --> 00:41:00.704 PROFESSOR: Uh, yeah. 00:41:00.704 --> 00:41:02.412 Well, it's a little bit more complicated. 00:41:02.412 --> 00:41:04.900 I would have to shift everybody to once side, 00:41:04.900 --> 00:41:07.962 the side that I have a certain increase in form [? than to ?] 00:41:07.962 --> 00:41:09.890 the gradient [? to stuff ?] like that. 00:41:09.890 --> 00:41:15.776 So don't forget about this type of project is an essential one. 00:41:15.776 --> 00:41:18.752 00:41:18.752 --> 00:41:21.300 Am I missing anybody important? 00:41:21.300 --> 00:41:22.600 Yes. 00:41:22.600 --> 00:41:23.910 We live in tests. 00:41:23.910 --> 00:41:27.763 We cannot say goodbye to the last section of the chapter 00:41:27.763 --> 00:41:32.655 nine, which is 9.7, without meeting again our friend 00:41:32.655 --> 00:41:35.220 the saddle, right? 00:41:35.220 --> 00:41:39.050 The saddle is-- this is elliptic paraboloid. 00:41:39.050 --> 00:41:42.200 And the last very important quadric 00:41:42.200 --> 00:41:46.890 that I wanted to talk about today is the-- 00:41:46.890 --> 00:41:50.850 00:41:50.850 --> 00:41:52.205 STUDENT: What about a cone? 00:41:52.205 --> 00:41:52.830 PROFESSOR: Huh? 00:41:52.830 --> 00:41:53.820 STUDENT: How about a cone? 00:41:53.820 --> 00:41:55.320 PROFESSOR: Oh, a cone is too easy. 00:41:55.320 --> 00:41:58.720 But yeah, let's talk about the cone as well. 00:41:58.720 --> 00:42:01.600 Give me an example of the standard cone. 00:42:01.600 --> 00:42:04.240 Thank you, [INAUDIBLE]. 00:42:04.240 --> 00:42:06.510 X squared-- well-- 00:42:06.510 --> 00:42:08.010 STUDENT: T squared equals x squared. 00:42:08.010 --> 00:42:11.450 PROFESSOR: I'm going to draw it first 00:42:11.450 --> 00:42:14.390 so that you know what I want. 00:42:14.390 --> 00:42:16.250 Unless I draw it, how would you know what 00:42:16.250 --> 00:42:19.070 to invent or to come up with? 00:42:19.070 --> 00:42:23.200 It's not an ice cream cone-- it's a double cone. 00:42:23.200 --> 00:42:27.010 So I can have a positive z and a negative z-- two 00:42:27.010 --> 00:42:29.770 different sheets that are symmetric with one another. 00:42:29.770 --> 00:42:33.570 00:42:33.570 --> 00:42:35.740 So how do I write that? 00:42:35.740 --> 00:42:36.722 STUDENT: T squared. 00:42:36.722 --> 00:42:37.680 PROFESSOR: Yes, equals? 00:42:37.680 --> 00:42:38.513 STUDENT: [INAUDIBLE] 00:42:38.513 --> 00:42:41.300 00:42:41.300 --> 00:42:45.930 PROFESSOR: Well, would you like it to be like most of those 00:42:45.930 --> 00:42:50.772 that we see in the examples in the book, right? 00:42:50.772 --> 00:42:54.200 But it doesn't have to be like that. 00:42:54.200 --> 00:42:56.640 Of course, if it's like that, of course 00:42:56.640 --> 00:42:59.240 you realize z-- set the z. 00:42:59.240 --> 00:43:00.890 Set the plane and altitude. 00:43:00.890 --> 00:43:03.320 Then you're going to have circle, circle, circle, 00:43:03.320 --> 00:43:07.340 circle-- circle after circle of different radii 00:43:07.340 --> 00:43:08.810 as cross sections. 00:43:08.810 --> 00:43:10.450 Z could also be negative. 00:43:10.450 --> 00:43:13.960 Except for the case of the origin, where you have 0, 0, 0. 00:43:13.960 --> 00:43:17.890 00:43:17.890 --> 00:43:21.720 Now, if you were to set x equals 0, of course 00:43:21.720 --> 00:43:24.260 you would get y equals plus minus 00:43:24.260 --> 00:43:30.050 z, which are exactly these lines, the red lines that I'm 00:43:30.050 --> 00:43:31.050 drawing in this picture. 00:43:31.050 --> 00:43:34.332 00:43:34.332 --> 00:43:35.910 So practically, this is, what? 00:43:35.910 --> 00:43:37.240 Called a what? 00:43:37.240 --> 00:43:39.450 A circular cone. 00:43:39.450 --> 00:43:42.090 If I wanted to make it more interesting, 00:43:42.090 --> 00:43:45.560 I would put a squared and b squared. 00:43:45.560 --> 00:43:48.360 And it would be an elliptic cone. 00:43:48.360 --> 00:43:51.740 And we stayed away from that as much as we could. 00:43:51.740 --> 00:43:55.160 We brought it up now because Zander asked about it. 00:43:55.160 --> 00:43:57.950 So how about the number four, number five, whatever it 00:43:57.950 --> 00:43:59.070 is-- number four? 00:43:59.070 --> 00:44:03.090 The [INAUDIBLE] what was the typical equation 00:44:03.090 --> 00:44:15.715 of the hyperbolic paraboloid that I had in mind? 00:44:15.715 --> 00:44:16.548 STUDENT: [INAUDIBLE] 00:44:16.548 --> 00:44:19.992 00:44:19.992 --> 00:44:31.970 PROFESSOR: Z equals x squared minus y squared, very good. 00:44:31.970 --> 00:44:34.530 So I will try again and draw it. 00:44:34.530 --> 00:44:36.770 It's not so easy to draw. 00:44:36.770 --> 00:44:40.450 00:44:40.450 --> 00:44:47.200 If I were to choose x to be 0 and draw exactly 00:44:47.200 --> 00:44:51.140 in the plane of the board, z equals minus y squared 00:44:51.140 --> 00:44:59.550 would be some coordinate line, right? 00:44:59.550 --> 00:45:02.000 This is what we call such a thing. 00:45:02.000 --> 00:45:06.220 If we fix the x to be x0, we get a coordinate line. 00:45:06.220 --> 00:45:09.800 If we fix the y to be y0, we get another coordinate line. 00:45:09.800 --> 00:45:12.360 There are two families of lines. 00:45:12.360 --> 00:45:17.160 Why is z equals minus y squared drawn in this board, 00:45:17.160 --> 00:45:19.315 on the board, in this plane? 00:45:19.315 --> 00:45:23.530 It's a parabola that opens upside down. 00:45:23.530 --> 00:45:34.030 OK, so you have something like this which you are drawing, 00:45:34.030 --> 00:45:36.050 right? 00:45:36.050 --> 00:45:39.880 And then what if y would be 0? 00:45:39.880 --> 00:45:43.470 Then you get z equals x0. 00:45:43.470 --> 00:45:48.402 So it's going to a parabola that opens up. 00:45:48.402 --> 00:45:53.720 Then I have to locate myself and draw it on that wall. 00:45:53.720 --> 00:45:54.720 But I can't. 00:45:54.720 --> 00:45:57.695 Because if I do that, I'm going to get in trouble. 00:45:57.695 --> 00:46:01.740 So I better draw it like this in perspective. 00:46:01.740 --> 00:46:04.720 And you guys should imagine what we have. 00:46:04.720 --> 00:46:12.079 So if we were to cut down with a knife, 00:46:12.079 --> 00:46:15.572 we would get-- we will still get these parabolas 00:46:15.572 --> 00:46:18.070 that all point down. 00:46:18.070 --> 00:46:22.165 And in those directions, these are just the highest parts 00:46:22.165 --> 00:46:23.590 of the saddle. 00:46:23.590 --> 00:46:27.030 And let's say this would be the lowest part of the saddle. 00:46:27.030 --> 00:46:32.280 Where-- where is the part of the rider? 00:46:32.280 --> 00:46:34.990 A guy's butt is here. 00:46:34.990 --> 00:46:38.760 And his leg is following the shape of the saddle going down. 00:46:38.760 --> 00:46:43.322 That's the cowboy boot, OK? 00:46:43.322 --> 00:46:46.530 And he is-- hold on. 00:46:46.530 --> 00:46:52.659 I don't know how-- what's the attitude of the [INAUDIBLE]? 00:46:52.659 --> 00:46:54.615 Well, it doesn't look like a cowboy hat. 00:46:54.615 --> 00:46:57.549 But anyway, I'm sorry. 00:46:57.549 --> 00:46:59.505 He looks a little bit Vietnamese. 00:46:59.505 --> 00:47:03.417 That was not the intention. 00:47:03.417 --> 00:47:05.862 STUDENT: [INAUDIBLE] 00:47:05.862 --> 00:47:07.840 PROFESSOR: Then let him be Mexican. 00:47:07.840 --> 00:47:11.160 Half of the population in this town are Mexican. 00:47:11.160 --> 00:47:14.500 So this is his leg that goes down. 00:47:14.500 --> 00:47:18.910 00:47:18.910 --> 00:47:20.380 OK, very good. 00:47:20.380 --> 00:47:23.810 Look-- he even has a-- what do you call that? 00:47:23.810 --> 00:47:24.790 That's so beautiful. 00:47:24.790 --> 00:47:30.625 In the Mexican culture, they make those embroidered by hand 00:47:30.625 --> 00:47:33.050 with many colors belts. 00:47:33.050 --> 00:47:35.640 But there are some special belts. 00:47:35.640 --> 00:47:39.450 OK-- depends on the area of Mexico You visit. 00:47:39.450 --> 00:47:41.260 I liked several of them. 00:47:41.260 --> 00:47:42.800 They're so beautiful. 00:47:42.800 --> 00:47:45.585 But my favorite one is, of course, 00:47:45.585 --> 00:47:50.535 the Rivera Maya, which is where you go to the Chichen Itza, 00:47:50.535 --> 00:47:54.645 to the mystic areas, to the sea, and eat the good food 00:47:54.645 --> 00:48:00.205 and go to Cozumel and forget about school for a week. 00:48:00.205 --> 00:48:02.680 That is paradise for me. 00:48:02.680 --> 00:48:05.155 But [INAUDIBLE] is not bad either. 00:48:05.155 --> 00:48:08.125 If I were to choose where to live and I had money, 00:48:08.125 --> 00:48:11.110 I would live in Cozumel for the rest of my life. 00:48:11.110 --> 00:48:15.610 OK, so this is the saddle that is oriented 00:48:15.610 --> 00:48:18.450 so that you have a parabola going in this direction, 00:48:18.450 --> 00:48:23.160 going up, a parabola going down in this direction. 00:48:23.160 --> 00:48:28.090 What is magic about a saddle point? 00:48:28.090 --> 00:48:29.479 Do you remember? 00:48:29.479 --> 00:48:33.160 STUDENT: It was [INAUDIBLE] 00:48:33.160 --> 00:48:36.315 PROFESSOR: It's-- one direction is like a max and one direction 00:48:36.315 --> 00:48:37.450 is like a min. 00:48:37.450 --> 00:48:41.340 So when you compute that discriminant, you get negative. 00:48:41.340 --> 00:48:44.672 You get like the product between the curvatures. 00:48:44.672 --> 00:48:46.713 One curvature in one direction would be positive. 00:48:46.713 --> 00:48:48.390 The other one would be negative. 00:48:48.390 --> 00:48:52.880 So it's like getting the product plus minus equals y, [? yes. ?] 00:48:52.880 --> 00:48:55.620 But again, we will talk about this sometimes later. 00:48:55.620 --> 00:48:58.840 You don't even have to know that. 00:48:58.840 --> 00:49:01.840 Shall we say goodbye to This? 00:49:01.840 --> 00:49:03.780 I guess it's time. 00:49:03.780 --> 00:49:08.270 And chapter nine is now fresh in your memory. 00:49:08.270 --> 00:49:11.866 You would be really to start chapter 10. 00:49:11.866 --> 00:49:14.800 00:49:14.800 --> 00:49:19.510 What I want to do, I want to-- without being recorded. 00:49:19.510 --> 00:49:20.543