[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.50,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.50,0:00:04.59,Default,,0000,0000,0000,,PROFESSOR: You have learned\Na lot in Calculus 2.
Dialogue: 0,0:00:04.59,0:00:12.18,Default,,0000,0000,0000,,Whether you took Calculus\Nrecently or long time ago,
Dialogue: 0,0:00:12.18,0:00:20.96,Default,,0000,0000,0000,,Chapter 9 is about vectors\Nin r3 and eventually
Dialogue: 0,0:00:20.96,0:00:25.49,Default,,0000,0000,0000,,[? a plane ?] and\Noperations with such vectors
Dialogue: 0,0:00:25.49,0:00:28.51,Default,,0000,0000,0000,,and the implications\Nof the vectors
Dialogue: 0,0:00:28.51,0:00:32.88,Default,,0000,0000,0000,,in the equations of a\Nline in space or plane
Dialogue: 0,0:00:32.88,0:00:35.18,Default,,0000,0000,0000,,in space-- stuff like that.
Dialogue: 0,0:00:35.18,0:00:42.82,Default,,0000,0000,0000,,Now, 9.1 to 9.5 was considered\Nto be covered completely
Dialogue: 0,0:00:42.82,0:00:46.97,Default,,0000,0000,0000,,in Calc 2 here at Tech.
Dialogue: 0,0:00:46.97,0:00:52.40,Default,,0000,0000,0000,,However, lots of students\Ncome from South Plains College
Dialogue: 0,0:00:52.40,0:00:54.88,Default,,0000,0000,0000,,and [? Rio ?] College,\Nlots of colleges
Dialogue: 0,0:00:54.88,0:00:59.22,Default,,0000,0000,0000,,where by the nature of\Nthe course Calculus 2,
Dialogue: 0,0:00:59.22,0:01:02.44,Default,,0000,0000,0000,,vectors in r3 are not covered.
Dialogue: 0,0:01:02.44,0:01:08.73,Default,,0000,0000,0000,,Therefore, I'd like to make an\Nattempt to review 9.1 and 9.5
Dialogue: 0,0:01:08.73,0:01:12.82,Default,,0000,0000,0000,,quickly with the knowledge\Nyou have now as grown-ups
Dialogue: 0,0:01:12.82,0:01:15.26,Default,,0000,0000,0000,,in the area of vectors in r2.
Dialogue: 0,0:01:15.26,0:01:17.13,Default,,0000,0000,0000,,So again, what are vectors?
Dialogue: 0,0:01:17.13,0:01:20.70,Default,,0000,0000,0000,,They are oriented segments.
Dialogue: 0,0:01:20.70,0:01:22.94,Default,,0000,0000,0000,,Not only that they\Nare oriented segments,
Dialogue: 0,0:01:22.94,0:01:29.84,Default,,0000,0000,0000,,but we make the distinction\Nbetween a vector that is fixed
Dialogue: 0,0:01:29.84,0:01:33.96,Default,,0000,0000,0000,,in the sense that his origin\Nis fixed-- we cannot move him--
Dialogue: 0,0:01:33.96,0:01:36.87,Default,,0000,0000,0000,,and a free vector who is\Nnot married to the origin.
Dialogue: 0,0:01:36.87,0:01:40.83,Default,,0000,0000,0000,,He can shift by parallelism\Nanywhere in space.
Dialogue: 0,0:01:40.83,0:01:43.02,Default,,0000,0000,0000,,And we call that a free vector.
Dialogue: 0,0:01:43.02,0:01:48.03,Default,,0000,0000,0000,,The distinction between those\Nvectors would be vr of v bar.
Dialogue: 0,0:01:48.03,0:01:52.92,Default,,0000,0000,0000,,As you remember, v bar was\Nthe free guy, free vector,
Dialogue: 0,0:01:52.92,0:02:00.67,Default,,0000,0000,0000,,which is the-- actually,\Nit's an equivalence class
Dialogue: 0,0:02:00.67,0:02:14.72,Default,,0000,0000,0000,,of all vectors that can be\Nobtained from the generic v
Dialogue: 0,0:02:14.72,0:02:15.22,Default,,0000,0000,0000,,bounded.
Dialogue: 0,0:02:15.22,0:02:20.97,Default,,0000,0000,0000,,So I'm going to have to\Npoint by translation.
Dialogue: 0,0:02:20.97,0:02:25.09,Default,,0000,0000,0000,,So you have this kind\Nof-- same magnitude
Dialogue: 0,0:02:25.09,0:02:29.41,Default,,0000,0000,0000,,for all vectors, same\Nmagnitude, same orientation,
Dialogue: 0,0:02:29.41,0:02:33.21,Default,,0000,0000,0000,,and parallel directions,\Nparallel lines.
Dialogue: 0,0:02:33.21,0:02:35.38,Default,,0000,0000,0000,,What have we done\Nto such a vector?
Dialogue: 0,0:02:35.38,0:02:38.69,Default,,0000,0000,0000,,As you remember very\Nwell, we decomposed him,
Dialogue: 0,0:02:38.69,0:02:44.94,Default,,0000,0000,0000,,being on the standard canonical\Nbasis, which for most of you
Dialogue: 0,0:02:44.94,0:02:50.14,Default,,0000,0000,0000,,engineers and engineering\Nmajors is denoted as ijk where
Dialogue: 0,0:02:50.14,0:02:55.85,Default,,0000,0000,0000,,ijk is an orthonormal\Nframe with respect
Dialogue: 0,0:02:55.85,0:02:57.11,Default,,0000,0000,0000,,to the Cartesian coordinates.
Dialogue: 0,0:02:57.11,0:03:03.38,Default,,0000,0000,0000,,So i, j, and k will\Nbe their unit vectors
Dialogue: 0,0:03:03.38,0:03:08.78,Default,,0000,0000,0000,,on the x, y, z axes of\Ncoordinates, Cartesian axes
Dialogue: 0,0:03:08.78,0:03:09.73,Default,,0000,0000,0000,,of coordinates.
Dialogue: 0,0:03:09.73,0:03:14.53,Default,,0000,0000,0000,,So remember always that ijk\Nare orthogonal to one another.
Dialogue: 0,0:03:14.53,0:03:17.46,Default,,0000,0000,0000,,Since this is review, I'd\Nlike to attract your attention
Dialogue: 0,0:03:17.46,0:03:25.32,Default,,0000,0000,0000,,to the fact that k is plus j.
Dialogue: 0,0:03:25.32,0:03:29.63,Default,,0000,0000,0000,,Think about it-- what\Nhappens you bring i over j.
Dialogue: 0,0:03:29.63,0:03:32.20,Default,,0000,0000,0000,,And you get k\Nbecause you move up.
Dialogue: 0,0:03:32.20,0:03:35.22,Default,,0000,0000,0000,,Because it's like you\Nare turning [INAUDIBLE]
Dialogue: 0,0:03:35.22,0:03:40.25,Default,,0000,0000,0000,,connection and the screw\Nor whatever from the faucet
Dialogue: 0,0:03:40.25,0:03:41.69,Default,,0000,0000,0000,,is pointing upwards.
Dialogue: 0,0:03:41.69,0:03:43.53,Default,,0000,0000,0000,,It's like the right hand rule.
Dialogue: 0,0:03:43.53,0:03:48.56,Default,,0000,0000,0000,,If you would do the other way\Naround, if you do j cross i,
Dialogue: 0,0:03:48.56,0:03:50.00,Default,,0000,0000,0000,,what are you going to have?
Dialogue: 0,0:03:50.00,0:03:55.33,Default,,0000,0000,0000,,Minus k-- so the properties\Nof the cross product being
Dialogue: 0,0:03:55.33,0:03:57.74,Default,,0000,0000,0000,,antisymmetric are\Nsupposed to be,
Dialogue: 0,0:03:57.74,0:04:02.19,Default,,0000,0000,0000,,no, pay attention to the signs\Nin all the exams that you have.
Dialogue: 0,0:04:02.19,0:04:05.15,Default,,0000,0000,0000,,What do we know about\Ntheir respective products
Dialogue: 0,0:04:05.15,0:04:08.75,Default,,0000,0000,0000,,for vectors in\Nspace or in plain?
Dialogue: 0,0:04:08.75,0:04:13.53,Default,,0000,0000,0000,,If you have two vectors\Nin their standard basis,
Dialogue: 0,0:04:13.53,0:04:17.68,Default,,0000,0000,0000,,you want i plus\Nu2j plus u3k where
Dialogue: 0,0:04:17.68,0:04:24.48,Default,,0000,0000,0000,,ui is a real number and\Ne1i plus v2j plus v3k where
Dialogue: 0,0:04:24.48,0:04:31.60,Default,,0000,0000,0000,,vi are [INAUDIBLE] real numbers\Nthe dot product or the scalar
Dialogue: 0,0:04:31.60,0:04:35.53,Default,,0000,0000,0000,,product-- now, I saw that in\Nall your engineering and physics
Dialogue: 0,0:04:35.53,0:04:38.13,Default,,0000,0000,0000,,classes, you will\Nuse this notation.
Dialogue: 0,0:04:38.13,0:04:40.63,Default,,0000,0000,0000,,Mathematicians\Nsometimes say, no, I'm
Dialogue: 0,0:04:40.63,0:04:42.87,Default,,0000,0000,0000,,going to use angular\Nbrackets because it's
Dialogue: 0,0:04:42.87,0:04:46.02,Default,,0000,0000,0000,,a scalar product in r3\Nor the scalar product
Dialogue: 0,0:04:46.02,0:04:47.91,Default,,0000,0000,0000,,and the dot product\Nis the same thing,
Dialogue: 0,0:04:47.91,0:04:49.79,Default,,0000,0000,0000,,being that's the\Nstandard one here.
Dialogue: 0,0:04:49.79,0:04:54.67,Default,,0000,0000,0000,,You want v1 plus u2v2 plus u3v3.
Dialogue: 0,0:04:54.67,0:04:58.03,Default,,0000,0000,0000,,So what do you to\Nremember what you do?
Dialogue: 0,0:04:58.03,0:04:59.96,Default,,0000,0000,0000,,First component\Nplus first component
Dialogue: 0,0:04:59.96,0:05:02.36,Default,,0000,0000,0000,,times second component\Ntimes second component
Dialogue: 0,0:05:02.36,0:05:05.95,Default,,0000,0000,0000,,plus third component\Ntimes third component, OK?
Dialogue: 0,0:05:05.95,0:05:10.42,Default,,0000,0000,0000,,If you are in\Ncomputer science, I
Dialogue: 0,0:05:10.42,0:05:13.15,Default,,0000,0000,0000,,saw that you use this notation.
Dialogue: 0,0:05:13.15,0:05:15.49,Default,,0000,0000,0000,,I was very happy to see that.
Dialogue: 0,0:05:15.49,0:05:16.57,Default,,0000,0000,0000,,the summation notation.
Dialogue: 0,0:05:16.57,0:05:20.24,Default,,0000,0000,0000,,But you don't have to\Nuse that in our class.
Dialogue: 0,0:05:20.24,0:05:24.41,Default,,0000,0000,0000,,Now, above the [? fresh ?]\Nproduct of two vectors,
Dialogue: 0,0:05:24.41,0:05:29.10,Default,,0000,0000,0000,,you have the definition\Nijk the first row.
Dialogue: 0,0:05:29.10,0:05:31.80,Default,,0000,0000,0000,,So what you get is\Ngoing to be a vector.
Dialogue: 0,0:05:31.80,0:05:36.23,Default,,0000,0000,0000,,Here, what you get is\Na scalar as a result.
Dialogue: 0,0:05:36.23,0:05:40.51,Default,,0000,0000,0000,,Here's what you get as\Na vector, as the result
Dialogue: 0,0:05:40.51,0:05:42.62,Default,,0000,0000,0000,,of the first\Nproduct is a vector.
Dialogue: 0,0:05:42.62,0:05:48.03,Default,,0000,0000,0000,,So you have u1,\Nu2, u3, v1, v2, v3.
Dialogue: 0,0:05:48.03,0:05:49.32,Default,,0000,0000,0000,,These are all friends of yours.
Dialogue: 0,0:05:49.32,0:05:56.84,Default,,0000,0000,0000,,I'm just reminding you\Nthe lucrative definitions.
Dialogue: 0,0:05:56.84,0:05:59.99,Default,,0000,0000,0000,,Now, some people\Nsaid, yes, but I'd
Dialogue: 0,0:05:59.99,0:06:03.25,Default,,0000,0000,0000,,like to see the lucrative\Ndefinitions that
Dialogue: 0,0:06:03.25,0:06:05.06,Default,,0000,0000,0000,,have to do with trig as well.
Dialogue: 0,0:06:05.06,0:06:06.13,Default,,0000,0000,0000,,OK, let's see.
Dialogue: 0,0:06:06.13,0:06:09.12,Default,,0000,0000,0000,,For those of you who asked me\Nto remind you what they were,
Dialogue: 0,0:06:09.12,0:06:13.17,Default,,0000,0000,0000,,I will remind you\Nwhat they were.
Dialogue: 0,0:06:13.17,0:06:20.48,Default,,0000,0000,0000,,For u.v, you get the same\Nthing as writing magnitude u
Dialogue: 0,0:06:20.48,0:06:25.17,Default,,0000,0000,0000,,magnitude v and cosine\Nof the angle between them
Dialogue: 0,0:06:25.17,0:06:27.38,Default,,0000,0000,0000,,no matter in which\Ndirection you take it
Dialogue: 0,0:06:27.38,0:06:29.37,Default,,0000,0000,0000,,because the cosine is the same.
Dialogue: 0,0:06:29.37,0:06:32.33,Default,,0000,0000,0000,,Cosine of pi is equal to\Ncosine of negative phi
Dialogue: 0,0:06:32.33,0:06:34.50,Default,,0000,0000,0000,,or theta [INAUDIBLE].
Dialogue: 0,0:06:34.50,0:06:35.93,Default,,0000,0000,0000,,How about the other one?
Dialogue: 0,0:06:35.93,0:06:38.48,Default,,0000,0000,0000,,Here's where one of\Nyou had a little bit
Dialogue: 0,0:06:38.48,0:06:41.59,Default,,0000,0000,0000,,of a misunderstanding.
Dialogue: 0,0:06:41.59,0:06:45.21,Default,,0000,0000,0000,,And I saw that happen in\Ntwo finals, unfortunately.
Dialogue: 0,0:06:45.21,0:06:51.54,Default,,0000,0000,0000,,This is not the scalar\Nvector that I'm right here.
Dialogue: 0,0:06:51.54,0:06:52.62,Default,,0000,0000,0000,,It's a vector.
Dialogue: 0,0:06:52.62,0:06:53.65,Default,,0000,0000,0000,,So what's missing?
Dialogue: 0,0:06:53.65,0:06:55.83,Default,,0000,0000,0000,,This is the scalar part.
Dialogue: 0,0:06:55.83,0:07:00.90,Default,,0000,0000,0000,,And then you have times e\Nwhere e is the unit vector
Dialogue: 0,0:07:00.90,0:07:10.05,Default,,0000,0000,0000,,of the direction of the\Nvector, the direction of u.v.
Dialogue: 0,0:07:10.05,0:07:12.48,Default,,0000,0000,0000,,Why I cannot use another notion?
Dialogue: 0,0:07:12.48,0:07:13.98,Default,,0000,0000,0000,,Because u is already taken.
Dialogue: 0,0:07:13.98,0:07:17.68,Default,,0000,0000,0000,,But e in itself\Nshould suggest to you
Dialogue: 0,0:07:17.68,0:07:22.09,Default,,0000,0000,0000,,that you have a unit vector,\N[? length of ?] one vector, OK?
Dialogue: 0,0:07:22.09,0:07:26.64,Default,,0000,0000,0000,,All right, what is the--\Nlet's review a little
Dialogue: 0,0:07:26.64,0:07:29.28,Default,,0000,0000,0000,,bit the absolute value.
Dialogue: 0,0:07:29.28,0:07:31.61,Default,,0000,0000,0000,,Well, the absolute\Nvalue is a scalar.
Dialogue: 0,0:07:31.61,0:07:34.43,Default,,0000,0000,0000,,So that scalar will be\Nmagnitude of your magnitude
Dialogue: 0,0:07:34.43,0:07:36.77,Default,,0000,0000,0000,,of [INAUDIBLE]\Nsine of the angle.
Dialogue: 0,0:07:36.77,0:07:41.01,Default,,0000,0000,0000,,And do you guys remember\Nthe geometric interpretation
Dialogue: 0,0:07:41.01,0:07:42.56,Default,,0000,0000,0000,,of that?
Dialogue: 0,0:07:42.56,0:07:43.47,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:07:43.47,0:07:45.14,Default,,0000,0000,0000,,PROFESSOR: The area\Nof the parallelogram
Dialogue: 0,0:07:45.14,0:07:48.38,Default,,0000,0000,0000,,based on the two vectors--\Nvery good [INAUDIBLE].
Dialogue: 0,0:07:48.38,0:07:50.87,Default,,0000,0000,0000,,U plus b is the area\Nof the parallelogram
Dialogue: 0,0:07:50.87,0:07:56.98,Default,,0000,0000,0000,,that you would draw based\Non those two vectors.
Dialogue: 0,0:07:56.98,0:07:58.34,Default,,0000,0000,0000,,All right, good.
Dialogue: 0,0:07:58.34,0:08:00.73,Default,,0000,0000,0000,,Now, say goodbye, vectors.
Dialogue: 0,0:08:00.73,0:08:04.46,Default,,0000,0000,0000,,We've seen-- you've seen\Nthem through 9.4, 9.5.
Dialogue: 0,0:08:04.46,0:08:08.19,Default,,0000,0000,0000,,What was important\Nto remember was
Dialogue: 0,0:08:08.19,0:08:10.85,Default,,0000,0000,0000,,that these vectors\Nwere the building
Dialogue: 0,0:08:10.85,0:08:14.63,Default,,0000,0000,0000,,blocks, the foundations,\Nof the equations
Dialogue: 0,0:08:14.63,0:08:16.84,Default,,0000,0000,0000,,of the lines in space.
Dialogue: 0,0:08:16.84,0:08:18.86,Default,,0000,0000,0000,,That's your work [INAUDIBLE].
Dialogue: 0,0:08:18.86,0:08:21.37,Default,,0000,0000,0000,,So what did we work with?
Dialogue: 0,0:08:21.37,0:08:26.11,Default,,0000,0000,0000,,Lines in space-- lines in space\Ncan be given in many ways.
Dialogue: 0,0:08:26.11,0:08:28.03,Default,,0000,0000,0000,,But now that you\Nremember them, I'm
Dialogue: 0,0:08:28.03,0:08:35.60,Default,,0000,0000,0000,,going to give you the symmetric\Nequation of a line in space.
Dialogue: 0,0:08:35.60,0:08:38.91,Default,,0000,0000,0000,,
Dialogue: 0,0:08:38.91,0:08:42.64,Default,,0000,0000,0000,,OK, [INAUDIBLE] this\Ncan see [INAUDIBLE]
Dialogue: 0,0:08:42.64,0:08:44.35,Default,,0000,0000,0000,,included on the final.
Dialogue: 0,0:08:44.35,0:08:46.76,Default,,0000,0000,0000,,You are expected to know it.
Dialogue: 0,0:08:46.76,0:08:55.52,Default,,0000,0000,0000,,
Dialogue: 0,0:08:55.52,0:08:59.19,Default,,0000,0000,0000,,So that is the\Nsymmetric equation,
Dialogue: 0,0:08:59.19,0:09:03.41,Default,,0000,0000,0000,,meaning the equation of a what?
Dialogue: 0,0:09:03.41,0:09:11.28,Default,,0000,0000,0000,,Of a line in space passing\Nthrough or containing
Dialogue: 0,0:09:11.28,0:09:21.47,Default,,0000,0000,0000,,the point of p, not of\Ncoordinates x0, y0, [? z0, ?]
Dialogue: 0,0:09:21.47,0:09:29.82,Default,,0000,0000,0000,,and of direction [INAUDIBLE]\Nin the sense of a vector.
Dialogue: 0,0:09:29.82,0:09:32.95,Default,,0000,0000,0000,,Now, if I were to\Ndraw such a line,
Dialogue: 0,0:09:32.95,0:09:35.34,Default,,0000,0000,0000,,I'm going to have\Nthe line over here.
Dialogue: 0,0:09:35.34,0:09:39.05,Default,,0000,0000,0000,,Going to have a vector for\Nthe point p0 on the line.
Dialogue: 0,0:09:39.05,0:09:41.97,Default,,0000,0000,0000,,I can put this free\Nvector because he's free.
Dialogue: 0,0:09:41.97,0:09:43.72,Default,,0000,0000,0000,,He says, I'm a free guy.
Dialogue: 0,0:09:43.72,0:09:47.66,Default,,0000,0000,0000,,I can slide any way I want.
Dialogue: 0,0:09:47.66,0:09:52.19,Default,,0000,0000,0000,,So I'm going to have\Nli plus mj plus mk.
Dialogue: 0,0:09:52.19,0:09:56.60,Default,,0000,0000,0000,,
Dialogue: 0,0:09:56.60,0:09:58.07,Default,,0000,0000,0000,,This is the blue vector.
Dialogue: 0,0:09:58.07,0:10:01.57,Default,,0000,0000,0000,,Now, you don't have blue\Nmarkers or blue pens,
Dialogue: 0,0:10:01.57,0:10:05.94,Default,,0000,0000,0000,,but you can still do a\Ngood job taking notes.
Dialogue: 0,0:10:05.94,0:10:09.57,Default,,0000,0000,0000,,Now somebody asked\Nme just a week ago,
Dialogue: 0,0:10:09.57,0:10:11.97,Default,,0000,0000,0000,,saying that I've started\Ndoing review already
Dialogue: 0,0:10:11.97,0:10:16.34,Default,,0000,0000,0000,,and I don't understand\Nwhat the difference is
Dialogue: 0,0:10:16.34,0:10:20.40,Default,,0000,0000,0000,,between the symmetric\Nequation of a line in space
Dialogue: 0,0:10:20.40,0:10:24.84,Default,,0000,0000,0000,,and the parametric equations\Nof a line in space.
Dialogue: 0,0:10:24.84,0:10:27.83,Default,,0000,0000,0000,,This is no essential difference.
Dialogue: 0,0:10:27.83,0:10:28.77,Default,,0000,0000,0000,,So what do we do?
Dialogue: 0,0:10:28.77,0:10:31.95,Default,,0000,0000,0000,,We denote this whole\Nanimal by t, a real number.
Dialogue: 0,0:10:31.95,0:10:35.22,Default,,0000,0000,0000,,
Dialogue: 0,0:10:35.22,0:10:37.64,Default,,0000,0000,0000,,And then we erase the board.
Dialogue: 0,0:10:37.64,0:10:43.31,Default,,0000,0000,0000,,And then we write\Nthe three equations
Dialogue: 0,0:10:43.31,0:10:47.33,Default,,0000,0000,0000,,that govern-- I'm going\Nto put if and only
Dialogue: 0,0:10:47.33,0:10:51.59,Default,,0000,0000,0000,,if xyz satisfy the following.
Dialogue: 0,0:10:51.59,0:10:53.36,Default,,0000,0000,0000,,So I'm going to have, what?
Dialogue: 0,0:10:53.36,0:11:07.88,Default,,0000,0000,0000,,X equals lt plus x0, y equals\Nmt plus y0, n equals nt plus z0.
Dialogue: 0,0:11:07.88,0:11:11.60,Default,,0000,0000,0000,,Well, of course, we\Nunderstand-- we know the meaning
Dialogue: 0,0:11:11.60,0:11:16.49,Default,,0000,0000,0000,,that lmn are like what\N[INAUDIBLE] physics direction
Dialogue: 0,0:11:16.49,0:11:18.69,Default,,0000,0000,0000,,cosines were\Ntelling me about it.
Dialogue: 0,0:11:18.69,0:11:21.96,Default,,0000,0000,0000,,And then x0, y0,\Nz0 is a fixed point
Dialogue: 0,0:11:21.96,0:11:24.58,Default,,0000,0000,0000,,that belongs to that line.
Dialogue: 0,0:11:24.58,0:11:28.65,Default,,0000,0000,0000,,Now, since you know a\Nlittle bit more than
Dialogue: 0,0:11:28.65,0:11:30.99,Default,,0000,0000,0000,,you knew in Calculus\N2 when you saw
Dialogue: 0,0:11:30.99,0:11:35.58,Default,,0000,0000,0000,,that for the first time, what\Nis the typical notation that we
Dialogue: 0,0:11:35.58,0:11:40.04,Default,,0000,0000,0000,,use all through Calc 3,\Nall through the chapters?
Dialogue: 0,0:11:40.04,0:11:44.92,Default,,0000,0000,0000,,The position vector-- the\Nposition vector of the point
Dialogue: 0,0:11:44.92,0:11:53.93,Default,,0000,0000,0000,,on the line that is\Nrelated to, what?
Dialogue: 0,0:11:53.93,0:11:57.28,Default,,0000,0000,0000,,So practically you\Nhave the origin here.
Dialogue: 0,0:11:57.28,0:12:04.20,Default,,0000,0000,0000,,[? Op0 ?] represent the\Nvector x0i plus y0j plus e0k.
Dialogue: 0,0:12:04.20,0:12:06.88,Default,,0000,0000,0000,,So now you have a little bit\Nof a different understanding
Dialogue: 0,0:12:06.88,0:12:08.91,Default,,0000,0000,0000,,of what's going on.
Dialogue: 0,0:12:08.91,0:12:15.30,Default,,0000,0000,0000,,And then after, let's say, t\Nequals 1 hour, what do you do?
Dialogue: 0,0:12:15.30,0:12:18.10,Default,,0000,0000,0000,,You are adding the\Nblue vector here.
Dialogue: 0,0:12:18.10,0:12:20.80,Default,,0000,0000,0000,,
Dialogue: 0,0:12:20.80,0:12:23.95,Default,,0000,0000,0000,,Let's say at t equals\N1, you are here.
Dialogue: 0,0:12:23.95,0:12:25.69,Default,,0000,0000,0000,,You are here at p1.
Dialogue: 0,0:12:25.69,0:12:29.92,Default,,0000,0000,0000,,So to get to p1, you have to\Nadd two vectors, right guys?
Dialogue: 0,0:12:29.92,0:12:34.03,Default,,0000,0000,0000,,This is the addition between the\Nblue vector and the red vector.
Dialogue: 0,0:12:34.03,0:12:36.29,Default,,0000,0000,0000,,So what you get is your result.
Dialogue: 0,0:12:36.29,0:12:41.12,Default,,0000,0000,0000,,So if I am smart enough to\Nunderstand my concepts are all
Dialogue: 0,0:12:41.12,0:12:44.11,Default,,0000,0000,0000,,connected, the\Nposition in this case
Dialogue: 0,0:12:44.11,0:12:49.86,Default,,0000,0000,0000,,will be r of t, which is--\NI hate angular brackets,
Dialogue: 0,0:12:49.86,0:12:52.65,Default,,0000,0000,0000,,but just because\Nyou like them, I'm
Dialogue: 0,0:12:52.65,0:12:55.76,Default,,0000,0000,0000,,going to use them--\Nx of ty tz of tm
Dialogue: 0,0:12:55.76,0:12:58.14,Default,,0000,0000,0000,,to be consistent with the book.
Dialogue: 0,0:12:58.14,0:13:04.44,Default,,0000,0000,0000,,This is the same as\Nxi plux yj plus zk.
Dialogue: 0,0:13:04.44,0:13:08.52,Default,,0000,0000,0000,,And what is this by\Nthe actual notations
Dialogue: 0,0:13:08.52,0:13:10.51,Default,,0000,0000,0000,,from the parametric equation?
Dialogue: 0,0:13:10.51,0:13:15.57,Default,,0000,0000,0000,,This is nothing\Nbut a certain lmn
Dialogue: 0,0:13:15.57,0:13:20.08,Default,,0000,0000,0000,,vector that is the vector\Nli plus mj plus nk written
Dialogue: 0,0:13:20.08,0:13:21.75,Default,,0000,0000,0000,,with angular brackets\Nbecause I know
Dialogue: 0,0:13:21.75,0:13:28.35,Default,,0000,0000,0000,,you like that times the time\Nt plus the fixed vector x0,
Dialogue: 0,0:13:28.35,0:13:29.30,Default,,0000,0000,0000,,y0, z0.
Dialogue: 0,0:13:29.30,0:13:31.89,Default,,0000,0000,0000,,You can say, yeah, I\Nthought it was a point.
Dialogue: 0,0:13:31.89,0:13:33.41,Default,,0000,0000,0000,,It is a point and a vector.
Dialogue: 0,0:13:33.41,0:13:41.96,Default,,0000,0000,0000,,You identified the point p0 with\Nthe position of the point p0
Dialogue: 0,0:13:41.96,0:13:44.66,Default,,0000,0000,0000,,starting with respect\Nto the origin.
Dialogue: 0,0:13:44.66,0:13:47.24,Default,,0000,0000,0000,,So whether you're\Ntalking about mister p0
Dialogue: 0,0:13:47.24,0:13:51.90,Default,,0000,0000,0000,,being a point in\Nspace-- x0, y0, z0.
Dialogue: 0,0:13:51.90,0:13:54.81,Default,,0000,0000,0000,,Or you're talking about\Nthe [INAUDIBLE] position
Dialogue: 0,0:13:54.81,0:13:58.19,Default,,0000,0000,0000,,vector that [INAUDIBLE]\Nis practically
Dialogue: 0,0:13:58.19,0:14:00.00,Default,,0000,0000,0000,,the same after identification.
Dialogue: 0,0:14:00.00,0:14:02.67,Default,,0000,0000,0000,,So you have something very nice.
Dialogue: 0,0:14:02.67,0:14:05.97,Default,,0000,0000,0000,,And if I asked you with\Nthe mind and the knowledge
Dialogue: 0,0:14:05.97,0:14:13.43,Default,,0000,0000,0000,,you have now what that\Ndoes is mean-- r prime of t
Dialogue: 0,0:14:13.43,0:14:16.19,Default,,0000,0000,0000,,equals what?
Dialogue: 0,0:14:16.19,0:14:19.18,Default,,0000,0000,0000,,It's the velocity vector.
Dialogue: 0,0:14:19.18,0:14:22.30,Default,,0000,0000,0000,,And what is that as a vector?
Dialogue: 0,0:14:22.30,0:14:24.54,Default,,0000,0000,0000,,Do the differentiation.
Dialogue: 0,0:14:24.54,0:14:27.18,Default,,0000,0000,0000,,What do we get in terms\Nof velocity vector?
Dialogue: 0,0:14:27.18,0:14:29.48,Default,,0000,0000,0000,,Prime with respect\Nto t-- what do I get?
Dialogue: 0,0:14:29.48,0:14:30.15,Default,,0000,0000,0000,,STUDENT: Lmn.
Dialogue: 0,0:14:30.15,0:14:31.27,Default,,0000,0000,0000,,PROFESSOR: Lmn as a vector.
Dialogue: 0,0:14:31.27,0:14:33.86,Default,,0000,0000,0000,,But of course, as I\Nhate angular notations,
Dialogue: 0,0:14:33.86,0:14:38.47,Default,,0000,0000,0000,,I will rewrite it--\Nli plus mj plus nk.
Dialogue: 0,0:14:38.47,0:14:40.19,Default,,0000,0000,0000,,So this is your velocity.
Dialogue: 0,0:14:40.19,0:14:44.16,Default,,0000,0000,0000,,What can you say about\Nthis type of motion?
Dialogue: 0,0:14:44.16,0:14:44.89,Default,,0000,0000,0000,,This is a--
Dialogue: 0,0:14:44.89,0:14:46.51,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Nconstant velocity.
Dialogue: 0,0:14:46.51,0:14:48.84,Default,,0000,0000,0000,,PROFESSOR: Yeah, you\Nhave a constant velocity
Dialogue: 0,0:14:48.84,0:14:51.23,Default,,0000,0000,0000,,for this motion.
Dialogue: 0,0:14:51.23,0:14:56.36,Default,,0000,0000,0000,,If somebody would ask you\Nyou have-- 10 years from now,
Dialogue: 0,0:14:56.36,0:15:00.37,Default,,0000,0000,0000,,you have a boy who\Nsaid, dad-- or a girl.
Dialogue: 0,0:15:00.37,0:15:01.51,Default,,0000,0000,0000,,let's not be biased.
Dialogue: 0,0:15:01.51,0:15:05.69,Default,,0000,0000,0000,,So he learns, math, good at\Nmath or physics, and says,
Dialogue: 0,0:15:05.69,0:15:08.37,Default,,0000,0000,0000,,what is the difference\Nbetween velocity and speed?
Dialogue: 0,0:15:08.37,0:15:11.17,Default,,0000,0000,0000,,Well, most parents will\Nsay it's the same thing.
Dialogue: 0,0:15:11.17,0:15:14.41,Default,,0000,0000,0000,,Well, you're not most parents.
Dialogue: 0,0:15:14.41,0:15:16.29,Default,,0000,0000,0000,,You are educated parents.
Dialogue: 0,0:15:16.29,0:15:19.34,Default,,0000,0000,0000,,So this is-- don't tell\Nyour kid about vectors,
Dialogue: 0,0:15:19.34,0:15:22.67,Default,,0000,0000,0000,,but you can show them you\Nhave an oriented segment.
Dialogue: 0,0:15:22.67,0:15:26.39,Default,,0000,0000,0000,,So make your child run around\Naround in circles and say,
Dialogue: 0,0:15:26.39,0:15:30.66,Default,,0000,0000,0000,,this is the velocity that's\Nalways tangent to the circle
Dialogue: 0,0:15:30.66,0:15:32.42,Default,,0000,0000,0000,,that you are running on.
Dialogue: 0,0:15:32.42,0:15:34.86,Default,,0000,0000,0000,,That's a velocity.
Dialogue: 0,0:15:34.86,0:15:38.89,Default,,0000,0000,0000,,And if they ask,\Nwell, they will catch
Dialogue: 0,0:15:38.89,0:15:41.99,Default,,0000,0000,0000,,the notions of acceleration\Nand force faster than you
Dialogue: 0,0:15:41.99,0:15:44.02,Default,,0000,0000,0000,,because they see\Nall these cartoons.
Dialogue: 0,0:15:44.02,0:15:48.36,Default,,0000,0000,0000,,And my son was telling me\Nthe other thing-- he's 10
Dialogue: 0,0:15:48.36,0:15:50.57,Default,,0000,0000,0000,,and I asked him, what\Nthe heck is that?
Dialogue: 0,0:15:50.57,0:15:53.91,Default,,0000,0000,0000,,It looked like an\Nelectromagnetic field
Dialogue: 0,0:15:53.91,0:15:55.78,Default,,0000,0000,0000,,surrounding some hero.
Dialogue: 0,0:15:55.78,0:15:59.70,Default,,0000,0000,0000,,And he said, mom, that's\Nthe force field of course.
Dialogue: 0,0:15:59.70,0:16:02.16,Default,,0000,0000,0000,,And I was thinking, force field?
Dialogue: 0,0:16:02.16,0:16:04.77,Default,,0000,0000,0000,,This is what I taught\Nthe other day when
Dialogue: 0,0:16:04.77,0:16:08.05,Default,,0000,0000,0000,,I was talking about [? crux. ?]\NDouble integral of f.n
Dialogue: 0,0:16:08.05,0:16:10.01,Default,,0000,0000,0000,,[INAUDIBLE] f was\Nthe force field.
Dialogue: 0,0:16:10.01,0:16:13.95,Default,,0000,0000,0000,,So he was, like, talking\Nabout something very normal
Dialogue: 0,0:16:13.95,0:16:15.67,Default,,0000,0000,0000,,that you see every day.
Dialogue: 0,0:16:15.67,0:16:21.25,Default,,0000,0000,0000,,So do not underestimate your\Nnephews, nieces, children.
Dialogue: 0,0:16:21.25,0:16:23.10,Default,,0000,0000,0000,,They will catch\Nup on these things
Dialogue: 0,0:16:23.10,0:16:25.54,Default,,0000,0000,0000,,faster than you, which is good.
Dialogue: 0,0:16:25.54,0:16:30.82,Default,,0000,0000,0000,,Now, the speed in this\Ncase will be, what?
Dialogue: 0,0:16:30.82,0:16:33.17,Default,,0000,0000,0000,,What is the speed\Nof this-- the speed
Dialogue: 0,0:16:33.17,0:16:36.34,Default,,0000,0000,0000,,of this motion, linear motion?
Dialogue: 0,0:16:36.34,0:16:38.13,Default,,0000,0000,0000,,STUDENT: Square\Nroot of l squared.
Dialogue: 0,0:16:38.13,0:16:42.88,Default,,0000,0000,0000,,PROFESSOR: Square root of l\Nsquared plus m squared plus n
Dialogue: 0,0:16:42.88,0:16:45.53,Default,,0000,0000,0000,,squared, which again is\Ndifferent from velocity.
Dialogue: 0,0:16:45.53,0:16:48.58,Default,,0000,0000,0000,,Velocity is a vector,\Nspeed is a scalar.
Dialogue: 0,0:16:48.58,0:16:50.64,Default,,0000,0000,0000,,Velocity is a vector,\Nspeed is a scalar.
Dialogue: 0,0:16:50.64,0:16:52.59,Default,,0000,0000,0000,,In general, doesn't\Nhave to be constant,
Dialogue: 0,0:16:52.59,0:16:54.51,Default,,0000,0000,0000,,but this is the\Nblessing because lmn
Dialogue: 0,0:16:54.51,0:16:57.41,Default,,0000,0000,0000,,are given constants.\N[INAUDIBLE] in this case,
Dialogue: 0,0:16:57.41,0:16:59.66,Default,,0000,0000,0000,,you are on cruise control.
Dialogue: 0,0:16:59.66,0:17:02.24,Default,,0000,0000,0000,,You are moving on\Na line directly
Dialogue: 0,0:17:02.24,0:17:05.20,Default,,0000,0000,0000,,in your motion on\Ncruise control driving
Dialogue: 0,0:17:05.20,0:17:07.84,Default,,0000,0000,0000,,to Amarillo at 60 miles\Nan hour because you
Dialogue: 0,0:17:07.84,0:17:09.26,Default,,0000,0000,0000,,are afraid of the cops.
Dialogue: 0,0:17:09.26,0:17:11.06,Default,,0000,0000,0000,,And you are doing\Nthe right thing
Dialogue: 0,0:17:11.06,0:17:12.52,Default,,0000,0000,0000,,because don't mess with Texas.
Dialogue: 0,0:17:12.52,0:17:18.70,Default,,0000,0000,0000,,I have friends who came here\Nto visit-- Texas, New Mexico,
Dialogue: 0,0:17:18.70,0:17:22.41,Default,,0000,0000,0000,,go to Santa Fe, go\Nto Carlsbad Caverns.
Dialogue: 0,0:17:22.41,0:17:24.19,Default,,0000,0000,0000,,Many of them got caught.
Dialogue: 0,0:17:24.19,0:17:27.09,Default,,0000,0000,0000,,Many of them got tickets.
Dialogue: 0,0:17:27.09,0:17:29.27,Default,,0000,0000,0000,,So it's really serious.
Dialogue: 0,0:17:29.27,0:17:36.58,Default,,0000,0000,0000,,OK, that's go further\Nand see what we
Dialogue: 0,0:17:36.58,0:17:40.75,Default,,0000,0000,0000,,remember about planes in space.
Dialogue: 0,0:17:40.75,0:17:44.12,Default,,0000,0000,0000,,Because planes in\Nspace are magic?
Dialogue: 0,0:17:44.12,0:17:44.98,Default,,0000,0000,0000,,No.
Dialogue: 0,0:17:44.98,0:17:47.64,Default,,0000,0000,0000,,Planes in space\Nare very important.
Dialogue: 0,0:17:47.64,0:17:55.36,Default,,0000,0000,0000,,Planes in space are\Ntwo dimensional objects
Dialogue: 0,0:17:55.36,0:18:00.75,Default,,0000,0000,0000,,embedded three dimensional\N[? area ?] spaces.
Dialogue: 0,0:18:00.75,0:18:02.55,Default,,0000,0000,0000,,This is what we're\Ntalking about.
Dialogue: 0,0:18:02.55,0:18:04.93,Default,,0000,0000,0000,,But even if you lived\Nin a four dimensional
Dialogue: 0,0:18:04.93,0:18:08.22,Default,,0000,0000,0000,,space, five dimensional\Nspace, n dimensional space,
Dialogue: 0,0:18:08.22,0:18:10.34,Default,,0000,0000,0000,,in the space of\Nyour imagination,
Dialogue: 0,0:18:10.34,0:18:13.38,Default,,0000,0000,0000,,if you have this two\Ndimensional object,
Dialogue: 0,0:18:13.38,0:18:16.42,Default,,0000,0000,0000,,it would still be\Ncalled a plane.
Dialogue: 0,0:18:16.42,0:18:20.03,Default,,0000,0000,0000,,All right, so how about planes?
Dialogue: 0,0:18:20.03,0:18:22.88,Default,,0000,0000,0000,,What is their equation?
Dialogue: 0,0:18:22.88,0:18:29.98,Default,,0000,0000,0000,,In your case ax plus by plus cz\Nplus d is the general equation.
Dialogue: 0,0:18:29.98,0:18:36.57,Default,,0000,0000,0000,,We now have a plane in r3.
Dialogue: 0,0:18:36.57,0:18:38.75,Default,,0000,0000,0000,,You should not forget about it.
Dialogue: 0,0:18:38.75,0:18:41.78,Default,,0000,0000,0000,,It's going to haunt\Nyou in the final
Dialogue: 0,0:18:41.78,0:18:45.25,Default,,0000,0000,0000,,and in other exams in your\Nlife through at least two
Dialogue: 0,0:18:45.25,0:18:47.54,Default,,0000,0000,0000,,or three different exercises.
Dialogue: 0,0:18:47.54,0:18:53.40,Default,,0000,0000,0000,,Now I'm going to ask you\Nto do a simple exercise.
Dialogue: 0,0:18:53.40,0:19:07.83,Default,,0000,0000,0000,,What is the equation of the\Nplane normal to the given line?
Dialogue: 0,0:19:07.83,0:19:09.35,Default,,0000,0000,0000,,And this is the given line.
Dialogue: 0,0:19:09.35,0:19:11.35,Default,,0000,0000,0000,,Look at it, how\Nbeautiful [INAUDIBLE].
Dialogue: 0,0:19:11.35,0:19:25.85,Default,,0000,0000,0000,,And passing through-- that\Npasses through the point
Dialogue: 0,0:19:25.85,0:19:32.41,Default,,0000,0000,0000,,another point-- x1, y1,\Nz1-- that I give you.
Dialogue: 0,0:19:32.41,0:19:35.27,Default,,0000,0000,0000,,How do you solve solution?
Dialogue: 0,0:19:35.27,0:19:37.78,Default,,0000,0000,0000,,How do you solve this quickly?
Dialogue: 0,0:19:37.78,0:19:41.49,Default,,0000,0000,0000,,You should just remember\Nwhat you learned
Dialogue: 0,0:19:41.49,0:19:43.66,Default,,0000,0000,0000,,and write that as\Nsoon as possible.
Dialogue: 0,0:19:43.66,0:19:46.77,Default,,0000,0000,0000,,Because, OK, this\Nmay be a little piece
Dialogue: 0,0:19:46.77,0:19:51.71,Default,,0000,0000,0000,,of a bigger problem in my exam.
Dialogue: 0,0:19:51.71,0:19:52.71,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:19:52.71,0:19:54.96,Default,,0000,0000,0000,,[? PROFESSOR: Who is ?] a?
Dialogue: 0,0:19:54.96,0:19:57.73,Default,,0000,0000,0000,,If this is normal to the line--
Dialogue: 0,0:19:57.73,0:19:59.42,Default,,0000,0000,0000,,STUDENT: A is 1.
Dialogue: 0,0:19:59.42,0:20:04.48,Default,,0000,0000,0000,,PROFESSOR: You\Npick up abc exactly
Dialogue: 0,0:20:04.48,0:20:09.34,Default,,0000,0000,0000,,from the lmn of the line.
Dialogue: 0,0:20:09.34,0:20:12.00,Default,,0000,0000,0000,,Remember this was an essential\Npiece of information.
Dialogue: 0,0:20:12.00,0:20:17.27,Default,,0000,0000,0000,,So the relationship between\Na line and its normal plane
Dialogue: 0,0:20:17.27,0:20:22.76,Default,,0000,0000,0000,,is that the direction\Nof that line lmn
Dialogue: 0,0:20:22.76,0:20:27.90,Default,,0000,0000,0000,,gives the coefficients abc\Nof the plane, all right?
Dialogue: 0,0:20:27.90,0:20:31.57,Default,,0000,0000,0000,,Don't forget that because you're\Ngoing to stumble right into it
Dialogue: 0,0:20:31.57,0:20:35.99,Default,,0000,0000,0000,,in the exams [? lx ?] in the\Ncoming up-- in the one that's
Dialogue: 0,0:20:35.99,0:20:36.72,Default,,0000,0000,0000,,coming up.
Dialogue: 0,0:20:36.72,0:20:39.01,Default,,0000,0000,0000,,And c, is this good?
Dialogue: 0,0:20:39.01,0:20:41.94,Default,,0000,0000,0000,,No, I cannot say d\Nand then look for d.
Dialogue: 0,0:20:41.94,0:20:44.71,Default,,0000,0000,0000,,I could-- I could [INAUDIBLE].
Dialogue: 0,0:20:44.71,0:20:46.82,Default,,0000,0000,0000,,Whatever you want.
Dialogue: 0,0:20:46.82,0:20:48.57,Default,,0000,0000,0000,,But then it's more work for me.
Dialogue: 0,0:20:48.57,0:20:52.13,Default,,0000,0000,0000,,Look, I don't know-- suppose\NI don't know who d is.
Dialogue: 0,0:20:52.13,0:20:55.82,Default,,0000,0000,0000,,I have to make the\Nplane satisfy--
Dialogue: 0,0:20:55.82,0:20:59.09,Default,,0000,0000,0000,,make the point\Nx1, y1, z1 satisfy
Dialogue: 0,0:20:59.09,0:21:02.07,Default,,0000,0000,0000,,the equation of the plane.
Dialogue: 0,0:21:02.07,0:21:03.97,Default,,0000,0000,0000,,And that is more work.
Dialogue: 0,0:21:03.97,0:21:06.55,Default,,0000,0000,0000,,I can do that if I forget.
Dialogue: 0,0:21:06.55,0:21:09.06,Default,,0000,0000,0000,,If I forget the theory,\NI can always do that.
Dialogue: 0,0:21:09.06,0:21:13.37,Default,,0000,0000,0000,,Subtract the two lines, subtract\Nthe second out of the first.
Dialogue: 0,0:21:13.37,0:21:15.33,Default,,0000,0000,0000,,I get something\Nmagic that I should
Dialogue: 0,0:21:15.33,0:21:19.58,Default,,0000,0000,0000,,have known from my\Nprevious knowledge,
Dialogue: 0,0:21:19.58,0:21:21.80,Default,,0000,0000,0000,,from a previous life-- no.
Dialogue: 0,0:21:21.80,0:21:29.32,Default,,0000,0000,0000,,L times x minus x1 plus m\Ntimes y minus y1 plus z times
Dialogue: 0,0:21:29.32,0:21:30.56,Default,,0000,0000,0000,,z minus 1.
Dialogue: 0,0:21:30.56,0:21:34.83,Default,,0000,0000,0000,,And I notice that most of\Nyou-- you prove me on exams,
Dialogue: 0,0:21:34.83,0:21:38.49,Default,,0000,0000,0000,,you prove me on homework--\Nknow that if you have
Dialogue: 0,0:21:38.49,0:21:44.29,Default,,0000,0000,0000,,the coefficients and you\Nalso have the point that
Dialogue: 0,0:21:44.29,0:21:48.12,Default,,0000,0000,0000,,is containing the plane,\Nyou can go ahead and write
Dialogue: 0,0:21:48.12,0:21:49.95,Default,,0000,0000,0000,,this equation from the start.
Dialogue: 0,0:21:49.95,0:21:54.05,Default,,0000,0000,0000,,So you know very well\Nthat x1, y1, z1 satisfies
Dialogue: 0,0:21:54.05,0:21:57.87,Default,,0000,0000,0000,,your [INAUDIBLE] the plane Then\Nyou can go ahead and write it.
Dialogue: 0,0:21:57.87,0:22:02.10,Default,,0000,0000,0000,,Save time on that\Nexam Don't waste time.
Dialogue: 0,0:22:02.10,0:22:05.45,Default,,0000,0000,0000,,It's like a star test\Nthat's a four hour test.
Dialogue: 0,0:22:05.45,0:22:07.78,Default,,0000,0000,0000,,No, ours is only two\Nhours and a half.
Dialogue: 0,0:22:07.78,0:22:11.37,Default,,0000,0000,0000,,But still, the pressure\Nis about the same.
Dialogue: 0,0:22:11.37,0:22:15.72,Default,,0000,0000,0000,,So we have to remember\Nthese notions.
Dialogue: 0,0:22:15.72,0:22:18.81,Default,,0000,0000,0000,,We cannot survive without them.
Dialogue: 0,0:22:18.81,0:22:20.33,Default,,0000,0000,0000,,Let's move on.
Dialogue: 0,0:22:20.33,0:22:26.21,Default,,0000,0000,0000,,And one of you asked me.
Dialogue: 0,0:22:26.21,0:22:31.31,Default,,0000,0000,0000,,Do I need to know by\Nheart the formula that
Dialogue: 0,0:22:31.31,0:22:35.34,Default,,0000,0000,0000,,give-- a formula that will give\Nthe distance between a point
Dialogue: 0,0:22:35.34,0:22:37.55,Default,,0000,0000,0000,,in space and a line in space?
Dialogue: 0,0:22:37.55,0:22:38.94,Default,,0000,0000,0000,,No, that is not assumed.
Dialogue: 0,0:22:38.94,0:22:41.29,Default,,0000,0000,0000,,You can build up to that one.
Dialogue: 0,0:22:41.29,0:22:42.35,Default,,0000,0000,0000,,It's not so immediate.
Dialogue: 0,0:22:42.35,0:22:44.44,Default,,0000,0000,0000,,It takes about 15 minutes.
Dialogue: 0,0:22:44.44,0:22:45.84,Default,,0000,0000,0000,,That's not a problem.
Dialogue: 0,0:22:45.84,0:22:47.89,Default,,0000,0000,0000,,What you are\Nsupposed to remember,
Dialogue: 0,0:22:47.89,0:22:53.55,Default,,0000,0000,0000,,though, is that the formula\Nfor distance between a given
Dialogue: 0,0:22:53.55,0:23:00.91,Default,,0000,0000,0000,,point in plane and a\Npoint in space and a given
Dialogue: 0,0:23:00.91,0:23:06.37,Default,,0000,0000,0000,,plane in space-- that was a long\Ntime ago that you knew that,
Dialogue: 0,0:23:06.37,0:23:09.46,Default,,0000,0000,0000,,but I said you should\Nnever for get it
Dialogue: 0,0:23:09.46,0:23:13.92,Default,,0000,0000,0000,,because it's similar\Nto the formula
Dialogue: 0,0:23:13.92,0:23:21.21,Default,,0000,0000,0000,,for the distance between a point\Nin plane and a line in plane.
Dialogue: 0,0:23:21.21,0:23:25.80,Default,,0000,0000,0000,,I'm not testing you, but I\Nwill-- I hope-- maybe I do.
Dialogue: 0,0:23:25.80,0:23:31.05,Default,,0000,0000,0000,,I hope that you remember how\Nto write this as a fraction.
Dialogue: 0,0:23:31.05,0:23:34.09,Default,,0000,0000,0000,,I'm already giving you hits.
Dialogue: 0,0:23:34.09,0:23:34.59,Default,,0000,0000,0000,,What is--
Dialogue: 0,0:23:34.59,0:23:35.97,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:23:35.97,0:23:38.15,Default,,0000,0000,0000,,PROFESSOR: Absolute value\Nbecause it's a distance.
Dialogue: 0,0:23:38.15,0:23:39.09,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:23:39.09,0:23:40.30,Default,,0000,0000,0000,,PROFESSOR: Of what?
Dialogue: 0,0:23:40.30,0:23:40.80,Default,,0000,0000,0000,,STUDENT: Ax.
Dialogue: 0,0:23:40.80,0:23:41.56,Default,,0000,0000,0000,,
Dialogue: 0,0:23:41.56,0:23:47.17,Default,,0000,0000,0000,,PROFESSOR: Ax0\Nplus by0 plus cz0--
Dialogue: 0,0:23:47.17,0:23:48.13,Default,,0000,0000,0000,,STUDENT: Plus b.
Dialogue: 0,0:23:48.13,0:23:50.05,Default,,0000,0000,0000,,PROFESSOR: Plus b, O. Good.
Dialogue: 0,0:23:50.05,0:23:51.80,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:23:51.80,0:23:52.93,Default,,0000,0000,0000,,PROFESSOR: Square root of--
Dialogue: 0,0:23:52.93,0:23:53.72,Default,,0000,0000,0000,,STUDENT: A squared.
Dialogue: 0,0:23:53.72,0:23:56.71,Default,,0000,0000,0000,,PROFESSOR: A squared plus\Nb squared plus c squared.
Dialogue: 0,0:23:56.71,0:24:01.42,Default,,0000,0000,0000,,Right, so it's a generalization\Nof the formula of the--
Dialogue: 0,0:24:01.42,0:24:04.84,Default,,0000,0000,0000,,in plane if you have a\Npoint and a line that
Dialogue: 0,0:24:04.84,0:24:11.19,Default,,0000,0000,0000,,doesn't contain the point, you\Nhave a similar type of formula.
Dialogue: 0,0:24:11.19,0:24:16.56,Default,,0000,0000,0000,,Good, let's remember\Nthe basics of conics.
Dialogue: 0,0:24:16.56,0:24:20.23,Default,,0000,0000,0000,,Because I'm afraid that\Nyou forgot them from Calc 2
Dialogue: 0,0:24:20.23,0:24:24.86,Default,,0000,0000,0000,,and from analytic or\Ntrigonometry class.
Dialogue: 0,0:24:24.86,0:24:30.52,Default,,0000,0000,0000,,What were the standard conics\Nthat were used in this class
Dialogue: 0,0:24:30.52,0:24:34.42,Default,,0000,0000,0000,,and I would like\Nyou to never forget?
Dialogue: 0,0:24:34.42,0:24:38.78,Default,,0000,0000,0000,,Well, when you are\Nin an exam, you
Dialogue: 0,0:24:38.78,0:24:42.86,Default,,0000,0000,0000,,may be asked the [INAUDIBLE]\Ninside of an ellipse.
Dialogue: 0,0:24:42.86,0:24:44.66,Default,,0000,0000,0000,,But if you don't know\Nthe standard equation
Dialogue: 0,0:24:44.66,0:24:46.34,Default,,0000,0000,0000,,of an ellipse, that's bad.
Dialogue: 0,0:24:46.34,0:24:47.79,Default,,0000,0000,0000,,So you should.
Dialogue: 0,0:24:47.79,0:24:48.83,Default,,0000,0000,0000,,What is that?
Dialogue: 0,0:24:48.83,0:24:52.38,Default,,0000,0000,0000,,Ab are semi-axis.
Dialogue: 0,0:24:52.38,0:24:53.97,Default,,0000,0000,0000,,STUDENT: X squared\Nover a squared.
Dialogue: 0,0:24:53.97,0:24:56.79,Default,,0000,0000,0000,,PROFESSOR: X squared over\Na squared plus y squared
Dialogue: 0,0:24:56.79,0:25:01.21,Default,,0000,0000,0000,,over b squared equals 1.
Dialogue: 0,0:25:01.21,0:25:05.92,Default,,0000,0000,0000,,Excellent, and what\Nif I have-- I'm
Dialogue: 0,0:25:05.92,0:25:11.41,Default,,0000,0000,0000,,going to draw a\Nrectangle with these kind
Dialogue: 0,0:25:11.41,0:25:14.16,Default,,0000,0000,0000,,of semi axes a and b.
Dialogue: 0,0:25:14.16,0:25:19.65,Default,,0000,0000,0000,,And I'm going to draw the\Ndiagonals-- the diagonals.
Dialogue: 0,0:25:19.65,0:25:23.36,Default,,0000,0000,0000,,And I'm going to draw\Na [INAUDIBLE] something
Dialogue: 0,0:25:23.36,0:25:27.58,Default,,0000,0000,0000,,that is touching, kissing\Nat this point tangent to it.
Dialogue: 0,0:25:27.58,0:25:32.35,Default,,0000,0000,0000,,And it's asymptotic to\Nthe blue asymptotes.
Dialogue: 0,0:25:32.35,0:25:34.02,Default,,0000,0000,0000,,What is this animal?
Dialogue: 0,0:25:34.02,0:25:35.88,Default,,0000,0000,0000,,STUDENT: Hyperbola.
Dialogue: 0,0:25:35.88,0:25:38.94,Default,,0000,0000,0000,,PROFESSOR: The\Nstandard hyperbola?
Dialogue: 0,0:25:38.94,0:25:41.69,Default,,0000,0000,0000,,Tell me what-- it\Nhas these branches.
Dialogue: 0,0:25:41.69,0:25:42.72,Default,,0000,0000,0000,,The equation is what?
Dialogue: 0,0:25:42.72,0:25:43.55,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:25:43.55,0:25:46.56,Default,,0000,0000,0000,,PROFESSOR: X squared over\Na squared minus y squared
Dialogue: 0,0:25:46.56,0:25:49.14,Default,,0000,0000,0000,,over b squared equals 1.
Dialogue: 0,0:25:49.14,0:25:59.31,Default,,0000,0000,0000,,If I were to draw\Nits brother-- oh--
Dialogue: 0,0:25:59.31,0:26:05.89,Default,,0000,0000,0000,,that brother would\Nbe the conjugate, OK?
Dialogue: 0,0:26:05.89,0:26:11.49,Default,,0000,0000,0000,,And you would have to swap\Nthe sides of plus minus.
Dialogue: 0,0:26:11.49,0:26:14.49,Default,,0000,0000,0000,,And you'll get the conjugate.
Dialogue: 0,0:26:14.49,0:26:18.42,Default,,0000,0000,0000,,Quadrics-- OK, the parabola, I\Ndon't remind you the parabola
Dialogue: 0,0:26:18.42,0:26:20.85,Default,,0000,0000,0000,,because you see it everywhere.
Dialogue: 0,0:26:20.85,0:26:24.74,Default,,0000,0000,0000,,I'm going to review it when\NI work with some quadrics.
Dialogue: 0,0:26:24.74,0:26:30.74,Default,,0000,0000,0000,,So the [INAUDIBLE]\Nquadrics-- and I really
Dialogue: 0,0:26:30.74,0:26:37.30,Default,,0000,0000,0000,,would like you to, if you feel\Nthe need to remind yourself
Dialogue: 0,0:26:37.30,0:26:43.22,Default,,0000,0000,0000,,when quadrics are, go to the\Nso-called gallery of quadrics.
Dialogue: 0,0:26:43.22,0:26:48.73,Default,,0000,0000,0000,,Type these magic words\Nas keywords in Google.
Dialogue: 0,0:26:48.73,0:26:52.58,Default,,0000,0000,0000,,And it's going to send\Nyou to a beautiful website
Dialogue: 0,0:26:52.58,0:26:57.16,Default,,0000,0000,0000,,from University of Minnesota\Nthat has a gallery of quadrics
Dialogue: 0,0:26:57.16,0:27:00.88,Default,,0000,0000,0000,,where not only do you see\Nthe most important quadrics
Dialogue: 0,0:27:00.88,0:27:06.84,Default,,0000,0000,0000,,in standard forms, but you also\Nsee the cross sections that you
Dialogue: 0,0:27:06.84,0:27:11.57,Default,,0000,0000,0000,,have when you curve those\Nquardics with horizontal planes
Dialogue: 0,0:27:11.57,0:27:14.82,Default,,0000,0000,0000,,or other planes parallel to\Nthe planes of coordinates.
Dialogue: 0,0:27:14.82,0:27:18.04,Default,,0000,0000,0000,,So I don't know in which\Norder to present them to you.
Dialogue: 0,0:27:18.04,0:27:22.94,Default,,0000,0000,0000,,But how about I present\Nthem to you in the order
Dialogue: 0,0:27:22.94,0:27:28.48,Default,,0000,0000,0000,,that they were mostly\Nfrequently used
Dialogue: 0,0:27:28.48,0:27:36.94,Default,,0000,0000,0000,,rather than starting with--\Nso ellipsoid and respectively
Dialogue: 0,0:27:36.94,0:27:39.69,Default,,0000,0000,0000,,a sphere.
Dialogue: 0,0:27:39.69,0:27:44.22,Default,,0000,0000,0000,,Depends if you like football--\NAmerican football or soccer.
Dialogue: 0,0:27:44.22,0:27:48.72,Default,,0000,0000,0000,,Well, let's see what\Nthe equations were.
Dialogue: 0,0:27:48.72,0:27:52.39,Default,,0000,0000,0000,,X squared over a\Nsquared plus y squared
Dialogue: 0,0:27:52.39,0:27:56.14,Default,,0000,0000,0000,,over b squared plus z\Nsquared over c squared
Dialogue: 0,0:27:56.14,0:27:58.76,Default,,0000,0000,0000,,equals 1 for the ellipsoids.
Dialogue: 0,0:27:58.76,0:28:06.23,Default,,0000,0000,0000,,If abc are equal and\Nequal to r, what is that?
Dialogue: 0,0:28:06.23,0:28:12.46,Default,,0000,0000,0000,,That's a sphere of center\Norigin-- standard sphere--
Dialogue: 0,0:28:12.46,0:28:13.77,Default,,0000,0000,0000,,in radius .
Dialogue: 0,0:28:13.77,0:28:18.80,Default,,0000,0000,0000,,R These are your friends.
Dialogue: 0,0:28:18.80,0:28:20.61,Default,,0000,0000,0000,,Don't forget about them.
Dialogue: 0,0:28:20.61,0:28:28.22,Default,,0000,0000,0000,,When you draw the\Nellipsoid, remember
Dialogue: 0,0:28:28.22,0:28:32.22,Default,,0000,0000,0000,,that the first line,\Nthe dotted one,
Dialogue: 0,0:28:32.22,0:28:36.70,Default,,0000,0000,0000,,is an ellipse on the\Nother behind the board.
Dialogue: 0,0:28:36.70,0:28:38.62,Default,,0000,0000,0000,,And that is obtained\Nas x squared
Dialogue: 0,0:28:38.62,0:28:41.30,Default,,0000,0000,0000,,over a squared plus y squared\Nover b squared equals 1.
Dialogue: 0,0:28:41.30,0:28:47.57,Default,,0000,0000,0000,,So it's going to be an\Nintersection with z equals 0
Dialogue: 0,0:28:47.57,0:28:53.15,Default,,0000,0000,0000,,And similarly, you can take\Nthe plain that's x equals 0.
Dialogue: 0,0:28:53.15,0:28:56.27,Default,,0000,0000,0000,,And you get this ellipse,\Nthe plane that is y equals 0.
Dialogue: 0,0:28:56.27,0:28:57.94,Default,,0000,0000,0000,,And you get this ellipse.
Dialogue: 0,0:28:57.94,0:29:01.11,Default,,0000,0000,0000,,So those are all\Nfriends of yours.
Dialogue: 0,0:29:01.11,0:29:03.16,Default,,0000,0000,0000,,Remember that all\Nthe cross sections
Dialogue: 0,0:29:03.16,0:29:10.47,Default,,0000,0000,0000,,you have cutting with planes,\Nthe football, you have, what?
Dialogue: 0,0:29:10.47,0:29:12.14,Default,,0000,0000,0000,,Ellipses.
Dialogue: 0,0:29:12.14,0:29:15.24,Default,,0000,0000,0000,,That is easy and\Nbeautiful and it's not
Dialogue: 0,0:29:15.24,0:29:18.15,Default,,0000,0000,0000,,something you need a\Nlot of thinking about.
Dialogue: 0,0:29:18.15,0:29:23.47,Default,,0000,0000,0000,,But let's move on some other\Nguys that I'm afraid you forgot
Dialogue: 0,0:29:23.47,0:29:27.94,Default,,0000,0000,0000,,and you should not\Nforget in any case.
Dialogue: 0,0:29:27.94,0:29:33.83,Default,,0000,0000,0000,,And the hyperboloids--\Nhyperboloids,
Dialogue: 0,0:29:33.83,0:29:42.83,Default,,0000,0000,0000,,the most standard ones,\Nthe classification
Dialogue: 0,0:29:42.83,0:29:48.54,Default,,0000,0000,0000,,that we had in the classroom\Nwas based on putting everybody
Dialogue: 0,0:29:48.54,0:29:49.88,Default,,0000,0000,0000,,to the left hand side.
Dialogue: 0,0:29:49.88,0:29:53.70,Default,,0000,0000,0000,,How many pluses, how many\Nminuses you have had?
Dialogue: 0,0:29:53.70,0:29:57.30,Default,,0000,0000,0000,,If you have plus, plus, plus,\Nminus or minus, minus, minus,
Dialogue: 0,0:29:57.30,0:30:01.35,Default,,0000,0000,0000,,plus, you have an uneven\Nnumber of pluses and minus.
Dialogue: 0,0:30:01.35,0:30:03.52,Default,,0000,0000,0000,,That was the\Ntwo-sheeted hyperbola.
Dialogue: 0,0:30:03.52,0:30:07.02,Default,,0000,0000,0000,,If you had an even number\Nof pluses and minuses,
Dialogue: 0,0:30:07.02,0:30:09.97,Default,,0000,0000,0000,,that's the one sheet hyperbola.
Dialogue: 0,0:30:09.97,0:30:14.04,Default,,0000,0000,0000,,So let us remember\Nhow that went.
Dialogue: 0,0:30:14.04,0:30:21.41,Default,,0000,0000,0000,,Assuming that I\Nlove this one, this
Dialogue: 0,0:30:21.41,0:30:25.54,Default,,0000,0000,0000,,is the first one--\Nthe first kind which
Dialogue: 0,0:30:25.54,0:30:31.15,Default,,0000,0000,0000,,is the one-sheeted hyperboloid.
Dialogue: 0,0:30:31.15,0:30:33.24,Default,,0000,0000,0000,,What is the symmetry axis?
Dialogue: 0,0:30:33.24,0:30:41.30,Default,,0000,0000,0000,,The surface of\Nrevolution-- What axis?
Dialogue: 0,0:30:41.30,0:30:44.60,Default,,0000,0000,0000,,Of axis 0x.
Dialogue: 0,0:30:44.60,0:30:47.96,Default,,0000,0000,0000,,So I'm going to go\Nahead and draw that.
Dialogue: 0,0:30:47.96,0:30:50.93,Default,,0000,0000,0000,,I'm going to draw\Nas well as I can.
Dialogue: 0,0:30:50.93,0:30:53.32,Default,,0000,0000,0000,,I cannot draw very well today.
Dialogue: 0,0:30:53.32,0:30:56.10,Default,,0000,0000,0000,,Although I had three cups\Nof coffee, doesn't matter.
Dialogue: 0,0:30:56.10,0:31:00.19,Default,,0000,0000,0000,,I'm still shaking when\Nit comes to drawing.
Dialogue: 0,0:31:00.19,0:31:03.53,Default,,0000,0000,0000,,So in order to get the cross\Nsection, the first cross
Dialogue: 0,0:31:03.53,0:31:06.76,Default,,0000,0000,0000,,section, the red one,\Nwhat do you guys do?
Dialogue: 0,0:31:06.76,0:31:08.11,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:31:08.11,0:31:09.67,Default,,0000,0000,0000,,PROFESSOR: It's a-- what?
Dialogue: 0,0:31:09.67,0:31:11.08,Default,,0000,0000,0000,,It's an ellipse\Nbecause you said z
Dialogue: 0,0:31:11.08,0:31:13.65,Default,,0000,0000,0000,,equal to 0 just as you said now.
Dialogue: 0,0:31:13.65,0:31:18.11,Default,,0000,0000,0000,,So I get the ellipse\Nof semi axis a and b.
Dialogue: 0,0:31:18.11,0:31:19.33,Default,,0000,0000,0000,,This is the x-axis.
Dialogue: 0,0:31:19.33,0:31:20.82,Default,,0000,0000,0000,,This is a.
Dialogue: 0,0:31:20.82,0:31:22.18,Default,,0000,0000,0000,,This is b.
Dialogue: 0,0:31:22.18,0:31:25.50,Default,,0000,0000,0000,,Well, it looks\Nlike horrible in b.
Dialogue: 0,0:31:25.50,0:31:28.87,Default,,0000,0000,0000,,And that's the\N[INAUDIBLE] we have.
Dialogue: 0,0:31:28.87,0:31:31.31,Default,,0000,0000,0000,,But now you say,\Nbut wait a minute.
Dialogue: 0,0:31:31.31,0:31:37.21,Default,,0000,0000,0000,,I would like to draw the cross\Nsection that corresponds to x
Dialogue: 0,0:31:37.21,0:31:38.04,Default,,0000,0000,0000,,equals 0.
Dialogue: 0,0:31:38.04,0:31:41.66,Default,,0000,0000,0000,,And that should be in\Nthe plane of the board.
Dialogue: 0,0:31:41.66,0:31:50.70,Default,,0000,0000,0000,,So if you set x to be 0, then\Nyou have the standard hyperbola
Dialogue: 0,0:31:50.70,0:31:53.45,Default,,0000,0000,0000,,based on semi axes b and c.
Dialogue: 0,0:31:53.45,0:31:55.60,Default,,0000,0000,0000,,Now, b, you believe me.
Dialogue: 0,0:31:55.60,0:31:59.76,Default,,0000,0000,0000,,But c, you don't believe me\Nat all because you cannot see.
Dialogue: 0,0:31:59.76,0:32:05.13,Default,,0000,0000,0000,,So if I were to be proactive--\Nwhich right now I'm
Dialogue: 0,0:32:05.13,0:32:07.96,Default,,0000,0000,0000,,not very proactive,\Nbut I'll try--
Dialogue: 0,0:32:07.96,0:32:12.61,Default,,0000,0000,0000,,I'm going to have\Nto draw-- look,
Dialogue: 0,0:32:12.61,0:32:16.88,Default,,0000,0000,0000,,I'm not done even if I\Ndidn't have enough coffee.
Dialogue: 0,0:32:16.88,0:32:20.88,Default,,0000,0000,0000,,So the rectangle--\Nyou see b and c here?
Dialogue: 0,0:32:20.88,0:32:22.37,Default,,0000,0000,0000,,OK, you see the asymptote?
Dialogue: 0,0:32:22.37,0:32:25.22,Default,,0000,0000,0000,,It was not a bad guess\Nof the asymptote.
Dialogue: 0,0:32:25.22,0:32:28.70,Default,,0000,0000,0000,,This branch of the cross\Nsection looks like, really,
Dialogue: 0,0:32:28.70,0:32:30.51,Default,,0000,0000,0000,,a good branch for the asymptote.
Dialogue: 0,0:32:30.51,0:32:33.14,Default,,0000,0000,0000,,Good, and the other\None in a similar way,
Dialogue: 0,0:32:33.14,0:32:35.30,Default,,0000,0000,0000,,you can find the\Nother cross section,
Dialogue: 0,0:32:35.30,0:32:37.50,Default,,0000,0000,0000,,which is also a hyperbola.
Dialogue: 0,0:32:37.50,0:32:43.20,Default,,0000,0000,0000,,So your old friend which\Nis one-sheeted hyperboloid,
Dialogue: 0,0:32:43.20,0:32:50.40,Default,,0000,0000,0000,,hyperboloid-- it sounds\Nlike a monster-- what
Dialogue: 0,0:32:50.40,0:32:53.73,Default,,0000,0000,0000,,was special about him?
Dialogue: 0,0:32:53.73,0:32:54.100,Default,,0000,0000,0000,,You have some extra credit.
Dialogue: 0,0:32:54.100,0:32:56.40,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:32:56.40,0:32:58.57,Default,,0000,0000,0000,,PROFESSOR: It's a\N[? ruled ?] surface generated
Dialogue: 0,0:32:58.57,0:33:01.29,Default,,0000,0000,0000,,by two families of lines.
Dialogue: 0,0:33:01.29,0:33:03.31,Default,,0000,0000,0000,,And thanks again for the model.
Dialogue: 0,0:33:03.31,0:33:05.63,Default,,0000,0000,0000,,I will keep it for\Nthe rest of my life.
Dialogue: 0,0:33:05.63,0:33:08.09,Default,,0000,0000,0000,,You got five bonus\Npoints because of that.
Dialogue: 0,0:33:08.09,0:33:10.69,Default,,0000,0000,0000,,I'm just-- well,\Nthis is something
Dialogue: 0,0:33:10.69,0:33:12.69,Default,,0000,0000,0000,,I will always remember.
Dialogue: 0,0:33:12.69,0:33:19.35,Default,,0000,0000,0000,,Number two, how do I write\Nthat two-sheeted hyperboloid
Dialogue: 0,0:33:19.35,0:33:23.26,Default,,0000,0000,0000,,if I wanted me to have\Nthe same axis of symmetry?
Dialogue: 0,0:33:23.26,0:33:25.32,Default,,0000,0000,0000,,It should be a\Nsurface of revolution
Dialogue: 0,0:33:25.32,0:33:29.53,Default,,0000,0000,0000,,consisting of two parts, two.
Dialogue: 0,0:33:29.53,0:33:30.97,Default,,0000,0000,0000,,They are disconnected, right?
Dialogue: 0,0:33:30.97,0:33:34.46,Default,,0000,0000,0000,,You have two sheets,\Ntwo somethings,
Dialogue: 0,0:33:34.46,0:33:35.71,Default,,0000,0000,0000,,two connected components.
Dialogue: 0,0:33:35.71,0:33:38.69,Default,,0000,0000,0000,,
Dialogue: 0,0:33:38.69,0:33:40.32,Default,,0000,0000,0000,,It's not hard at all.
Dialogue: 0,0:33:40.32,0:33:41.66,Default,,0000,0000,0000,,What do I need to do?
Dialogue: 0,0:33:41.66,0:33:44.55,Default,,0000,0000,0000,,
Dialogue: 0,0:33:44.55,0:33:49.74,Default,,0000,0000,0000,,The same thing as here-- just\Nchange the minus to a plus.
Dialogue: 0,0:33:49.74,0:33:52.51,Default,,0000,0000,0000,,All righty, x squared over\Na squared plus y squared
Dialogue: 0,0:33:52.51,0:33:59.66,Default,,0000,0000,0000,,over b squared minus z squared\Nover c squared plus 1 equals 0.
Dialogue: 0,0:33:59.66,0:34:04.28,Default,,0000,0000,0000,,Great, so I can go ahead and\Nreminds you what that was.
Dialogue: 0,0:34:04.28,0:34:06.67,Default,,0000,0000,0000,,You didn't like\Nit when you first,
Dialogue: 0,0:34:06.67,0:34:09.19,Default,,0000,0000,0000,,but maybe now you\Nlike it better.
Dialogue: 0,0:34:09.19,0:34:12.94,Default,,0000,0000,0000,,This is always yz.
Dialogue: 0,0:34:12.94,0:34:20.00,Default,,0000,0000,0000,,And I'm going to\Ndraw the two sheets.
Dialogue: 0,0:34:20.00,0:34:21.82,Default,,0000,0000,0000,,And I'm going to\Nask you eventually,
Dialogue: 0,0:34:21.82,0:34:26.31,Default,,0000,0000,0000,,because I am mean, how\Nfar apart they are.
Dialogue: 0,0:34:26.31,0:34:28.12,Default,,0000,0000,0000,,It's the surface of revolution.
Dialogue: 0,0:34:28.12,0:34:29.58,Default,,0000,0000,0000,,These two guys\Nshould be symmetric.
Dialogue: 0,0:34:29.58,0:34:32.20,Default,,0000,0000,0000,,
Dialogue: 0,0:34:32.20,0:34:40.00,Default,,0000,0000,0000,,Well, so when I were-- if\NI were to take z equals 0,
Dialogue: 0,0:34:40.00,0:34:44.01,Default,,0000,0000,0000,,I would get no solution\Nbecause this is impossible.
Dialogue: 0,0:34:44.01,0:34:48.61,Default,,0000,0000,0000,,I have a sum of\Nsquares equal 0, right?
Dialogue: 0,0:34:48.61,0:34:52.11,Default,,0000,0000,0000,,It's impossible\Nto get 0 this way.
Dialogue: 0,0:34:52.11,0:34:57.94,Default,,0000,0000,0000,,When would I get 0 on\Nthe axis of rotation?
Dialogue: 0,0:34:57.94,0:35:01.57,Default,,0000,0000,0000,,Well, axis of rotation\Nmeans forget about x and y.
Dialogue: 0,0:35:01.57,0:35:03.43,Default,,0000,0000,0000,,X is 0, y is 0.
Dialogue: 0,0:35:03.43,0:35:05.89,Default,,0000,0000,0000,,Z would be how much?
Dialogue: 0,0:35:05.89,0:35:06.70,Default,,0000,0000,0000,,STUDENT: C.
Dialogue: 0,0:35:06.70,0:35:07.70,Default,,0000,0000,0000,,PROFESSOR: Plus minus c.
Dialogue: 0,0:35:07.70,0:35:08.66,Default,,0000,0000,0000,,Plus minus-- very good.
Dialogue: 0,0:35:08.66,0:35:13.67,Default,,0000,0000,0000,,C, practically c, if c is\Npositive, and minus c here.
Dialogue: 0,0:35:13.67,0:35:19.20,Default,,0000,0000,0000,,So I know how far apart they\Nare, these two-- [INAUDIBLE]
Dialogue: 0,0:35:19.20,0:35:21.54,Default,,0000,0000,0000,,this is not [? x ?] [INAUDIBLE]\Nminimum and the maximum
Dialogue: 0,0:35:21.54,0:35:23.09,Default,,0000,0000,0000,,over here.
Dialogue: 0,0:35:23.09,0:35:24.94,Default,,0000,0000,0000,,Now, one last question.
Dialogue: 0,0:35:24.94,0:35:28.16,Default,,0000,0000,0000,,Well-- OK, no.
Dialogue: 0,0:35:28.16,0:35:34.65,Default,,0000,0000,0000,,More questions-- when I were\Nto intersect with, let's
Dialogue: 0,0:35:34.65,0:35:41.25,Default,,0000,0000,0000,,say, a z that is bigger\Nthan c, a z plane that
Dialogue: 0,0:35:41.25,0:35:46.81,Default,,0000,0000,0000,,is bigger than c over here,\Nwhat am I going to get?
Dialogue: 0,0:35:46.81,0:35:47.55,Default,,0000,0000,0000,,No--
Dialogue: 0,0:35:47.55,0:35:48.38,Default,,0000,0000,0000,,STUDENT: An ellipse.
Dialogue: 0,0:35:48.38,0:35:49.75,Default,,0000,0000,0000,,PROFESSOR: An\Nelipse-- excellent.
Dialogue: 0,0:35:49.75,0:35:52.34,Default,,0000,0000,0000,,An ellipse here, an\Nellipse there everything
Dialogue: 0,0:35:52.34,0:35:53.31,Default,,0000,0000,0000,,is symmetrical.
Dialogue: 0,0:35:53.31,0:35:57.35,Default,,0000,0000,0000,,And finally, what\Nif I take x to be 0?
Dialogue: 0,0:35:57.35,0:35:59.06,Default,,0000,0000,0000,,I'm in the plane of the board.
Dialogue: 0,0:35:59.06,0:36:00.79,Default,,0000,0000,0000,,I hide the x.
Dialogue: 0,0:36:00.79,0:36:02.22,Default,,0000,0000,0000,,I get this.
Dialogue: 0,0:36:02.22,0:36:04.64,Default,,0000,0000,0000,,What is this?
Dialogue: 0,0:36:04.64,0:36:11.48,Default,,0000,0000,0000,,A hyperbola in the plane\Nof the board, which is yz.
Dialogue: 0,0:36:11.48,0:36:14.64,Default,,0000,0000,0000,,Y is going this\Nway, z is going up.
Dialogue: 0,0:36:14.64,0:36:17.37,Default,,0000,0000,0000,,X doesn't exist anymore.
Dialogue: 0,0:36:17.37,0:36:19.70,Default,,0000,0000,0000,,So what kind of\Nhyperbola is this?
Dialogue: 0,0:36:19.70,0:36:23.02,Default,,0000,0000,0000,,Do you like it?
Dialogue: 0,0:36:23.02,0:36:23.52,Default,,0000,0000,0000,,So--
Dialogue: 0,0:36:23.52,0:36:26.43,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:36:26.43,0:36:28.79,Default,,0000,0000,0000,,PROFESSOR: Right, mean smart.
Dialogue: 0,0:36:28.79,0:36:31.67,Default,,0000,0000,0000,,go ahead and multiply by\Nnegative-- who said that?
Dialogue: 0,0:36:31.67,0:36:34.79,Default,,0000,0000,0000,,Zander, you got two extra\Npoints, extra [INAUDIBLE].
Dialogue: 0,0:36:34.79,0:36:38.45,Default,,0000,0000,0000,,Minus y squared over\Nb squared equals 1.
Dialogue: 0,0:36:38.45,0:36:39.62,Default,,0000,0000,0000,,What did he notice?
Dialogue: 0,0:36:39.62,0:36:41.54,Default,,0000,0000,0000,,What did he-- he gets my mind.
Dialogue: 0,0:36:41.54,0:36:45.91,Default,,0000,0000,0000,,I'm trying to say you have\Nno hyperbola like that.
Dialogue: 0,0:36:45.91,0:36:48.63,Default,,0000,0000,0000,,So Zander said, I\Nknow what which ones.
Dialogue: 0,0:36:48.63,0:36:53.25,Default,,0000,0000,0000,,She wants these two branches\Nto be the hyperbola.
Dialogue: 0,0:36:53.25,0:36:56.19,Default,,0000,0000,0000,,But that's a\Nconjugate hyperbola.
Dialogue: 0,0:36:56.19,0:36:58.48,Default,,0000,0000,0000,,That is a conjugate\Nhyperbola because you
Dialogue: 0,0:36:58.48,0:37:04.13,Default,,0000,0000,0000,,don't have y and z with minus\Nbetween the squares and a y.
Dialogue: 0,0:37:04.13,0:37:10.16,Default,,0000,0000,0000,,So this is the conjugate\Nhyperbola-- hyperbola--
Dialogue: 0,0:37:10.16,0:37:13.61,Default,,0000,0000,0000,,that I'm going to draw.
Dialogue: 0,0:37:13.61,0:37:14.94,Default,,0000,0000,0000,,In what color?
Dialogue: 0,0:37:14.94,0:37:16.03,Default,,0000,0000,0000,,That's the question.
Dialogue: 0,0:37:16.03,0:37:18.95,Default,,0000,0000,0000,,It's really essential what\Ncolor I'm going to use.
Dialogue: 0,0:37:18.95,0:37:23.32,Default,,0000,0000,0000,,So I'm going to use--\NI'm going to use green.
Dialogue: 0,0:37:23.32,0:37:26.79,Default,,0000,0000,0000,,And this is the hyperbola\Nwe are talking about.
Dialogue: 0,0:37:26.79,0:37:32.25,Default,,0000,0000,0000,,It's a conjugate one drawn\Nin the plane of the board.
Dialogue: 0,0:37:32.25,0:37:33.56,Default,,0000,0000,0000,,OK, all right.
Dialogue: 0,0:37:33.56,0:37:36.22,Default,,0000,0000,0000,,So if I wanted to\Ndrop those asymptotes,
Dialogue: 0,0:37:36.22,0:37:38.50,Default,,0000,0000,0000,,they will look very ugly.
Dialogue: 0,0:37:38.50,0:37:42.63,Default,,0000,0000,0000,,And I cannot do better,\Nbut that's [INAUDIBLE].
Dialogue: 0,0:37:42.63,0:37:50.07,Default,,0000,0000,0000,,So we have reviewed the\Nmost awful quadrics.
Dialogue: 0,0:37:50.07,0:37:54.27,Default,,0000,0000,0000,,A friend of yours that\Nby now all of you love
Dialogue: 0,0:37:54.27,0:37:57.02,Default,,0000,0000,0000,,is mister paraboloid.
Dialogue: 0,0:37:57.02,0:38:01.53,Default,,0000,0000,0000,,You have used that in\Nall sorts of examples.
Dialogue: 0,0:38:01.53,0:38:06.04,Default,,0000,0000,0000,,I'm going to remind you\Nwhat the standard one was
Dialogue: 0,0:38:06.04,0:38:08.98,Default,,0000,0000,0000,,that we used before.
Dialogue: 0,0:38:08.98,0:38:19.21,Default,,0000,0000,0000,,So [INAUDIBLE] paraboloids,\Nelliptic paraboloid.
Dialogue: 0,0:38:19.21,0:38:22.65,Default,,0000,0000,0000,,
Dialogue: 0,0:38:22.65,0:38:25.87,Default,,0000,0000,0000,,Circular paraboloid is\Njust the particular case.
Dialogue: 0,0:38:25.87,0:38:29.02,Default,,0000,0000,0000,,
Dialogue: 0,0:38:29.02,0:38:32.98,Default,,0000,0000,0000,,The elliptic paraboloid\Nthat you're used to
Dialogue: 0,0:38:32.98,0:38:37.46,Default,,0000,0000,0000,,is the following-- z equals\Nx squared over a squared
Dialogue: 0,0:38:37.46,0:38:41.81,Default,,0000,0000,0000,,plus y squared over b squared.
Dialogue: 0,0:38:41.81,0:38:44.64,Default,,0000,0000,0000,,They may be positive\Nif you want.
Dialogue: 0,0:38:44.64,0:38:49.48,Default,,0000,0000,0000,,They don't-- in general,\Nthey are not equal.
Dialogue: 0,0:38:49.48,0:38:53.90,Default,,0000,0000,0000,,The circular paraboloid--\Nwell, you simply
Dialogue: 0,0:38:53.90,0:38:58.58,Default,,0000,0000,0000,,assume that a and b are equal.
Dialogue: 0,0:38:58.58,0:39:04.23,Default,,0000,0000,0000,,And then you put-- you want\Na c squared or an r squared.
Dialogue: 0,0:39:04.23,0:39:06.66,Default,,0000,0000,0000,,Let's put an r squared on top.
Dialogue: 0,0:39:06.66,0:39:10.42,Default,,0000,0000,0000,,It really doesn't matter\Nwhat you're putting there.
Dialogue: 0,0:39:10.42,0:39:11.63,Default,,0000,0000,0000,,Can I draw?
Dialogue: 0,0:39:11.63,0:39:15.93,Default,,0000,0000,0000,,Hopefully, hopefully,\Nhopefully I can draw.
Dialogue: 0,0:39:15.93,0:39:18.71,Default,,0000,0000,0000,,It looks like a\Nvalley whose minimum
Dialogue: 0,0:39:18.71,0:39:22.16,Default,,0000,0000,0000,,is at the origin\NI'm going to draw
Dialogue: 0,0:39:22.16,0:39:29.68,Default,,0000,0000,0000,,so that the intersection\Nwith the horizontal plane
Dialogue: 0,0:39:29.68,0:39:32.17,Default,,0000,0000,0000,,will be visible to you.
Dialogue: 0,0:39:32.17,0:39:36.82,Default,,0000,0000,0000,,And I take this\Nz greater than 0.
Dialogue: 0,0:39:36.82,0:39:39.41,Default,,0000,0000,0000,,And then I'm going to\Nhave some sort of ellipse.
Dialogue: 0,0:39:39.41,0:39:42.18,Default,,0000,0000,0000,,
Dialogue: 0,0:39:42.18,0:39:44.54,Default,,0000,0000,0000,,Under that, there is nothing.
Dialogue: 0,0:39:44.54,0:39:46.11,Default,,0000,0000,0000,,Under the origin,\Nthere is nothing
Dialogue: 0,0:39:46.11,0:39:50.21,Default,,0000,0000,0000,,because z is going to be\Npositive at x equals 0,
Dialogue: 0,0:39:50.21,0:39:53.17,Default,,0000,0000,0000,,y equals 0, and passing\Nthrough the origin-- very
Dialogue: 0,0:39:53.17,0:39:56.28,Default,,0000,0000,0000,,nice and [? sassy ?] Quadric.
Dialogue: 0,0:39:56.28,0:40:01.79,Default,,0000,0000,0000,,There is one that occurred in\Nmany examples like a nightmare.
Dialogue: 0,0:40:01.79,0:40:04.07,Default,,0000,0000,0000,,And it was based on that one.
Dialogue: 0,0:40:04.07,0:40:05.91,Default,,0000,0000,0000,,And I'm going to\Ndraw-- no, no, no.
Dialogue: 0,0:40:05.91,0:40:07.73,Default,,0000,0000,0000,,I'm going to write\Nit and you draw it
Dialogue: 0,0:40:07.73,0:40:09.56,Default,,0000,0000,0000,,with the eyes of\Nyour imagination
Dialogue: 0,0:40:09.56,0:40:11.76,Default,,0000,0000,0000,,and see what that is.
Dialogue: 0,0:40:11.76,0:40:16.67,Default,,0000,0000,0000,,Because you are, again, going\Nto bump into it into the exam.
Dialogue: 0,0:40:16.67,0:40:20.76,Default,,0000,0000,0000,,We had all sorts\Nof patches of that.
Dialogue: 0,0:40:20.76,0:40:22.45,Default,,0000,0000,0000,,Look at the areas of the patch.
Dialogue: 0,0:40:22.45,0:40:25.86,Default,,0000,0000,0000,,And you cannot get rid of that.
Dialogue: 0,0:40:25.86,0:40:29.02,Default,,0000,0000,0000,,It's haunting your dreams.
Dialogue: 0,0:40:29.02,0:40:29.73,Default,,0000,0000,0000,,What is this?
Dialogue: 0,0:40:29.73,0:40:31.35,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:40:31.35,0:40:34.39,Default,,0000,0000,0000,,PROFESSOR: Upside\Ndown paraboloid--
Dialogue: 0,0:40:34.39,0:40:35.87,Default,,0000,0000,0000,,what is the vertex?
Dialogue: 0,0:40:35.87,0:40:37.50,Default,,0000,0000,0000,,Where is the vertex at?
Dialogue: 0,0:40:37.50,0:40:38.40,Default,,0000,0000,0000,,STUDENT: 0, 0, 1.
Dialogue: 0,0:40:38.40,0:40:41.87,Default,,0000,0000,0000,,PROFESSOR: 0, 0, 1-- very good.
Dialogue: 0,0:40:41.87,0:40:45.97,Default,,0000,0000,0000,,What's special about it?
Dialogue: 0,0:40:45.97,0:40:49.49,Default,,0000,0000,0000,,So assume that I\Nwould draw the-- I
Dialogue: 0,0:40:49.49,0:40:56.26,Default,,0000,0000,0000,,would draw it to compute\Nthe normal to the surface.
Dialogue: 0,0:40:56.26,0:40:58.23,Default,,0000,0000,0000,,How would I do that?
Dialogue: 0,0:40:58.23,0:40:59.72,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:40:59.72,0:41:00.70,Default,,0000,0000,0000,,PROFESSOR: Uh, yeah.
Dialogue: 0,0:41:00.70,0:41:02.41,Default,,0000,0000,0000,,Well, it's a little\Nbit more complicated.
Dialogue: 0,0:41:02.41,0:41:04.90,Default,,0000,0000,0000,,I would have to shift\Neverybody to once side,
Dialogue: 0,0:41:04.90,0:41:07.96,Default,,0000,0000,0000,,the side that I have a certain\Nincrease in form [? than to ?]
Dialogue: 0,0:41:07.96,0:41:09.89,Default,,0000,0000,0000,,the gradient\N[? to stuff ?] like that.
Dialogue: 0,0:41:09.89,0:41:15.78,Default,,0000,0000,0000,,So don't forget about this type\Nof project is an essential one.
Dialogue: 0,0:41:15.78,0:41:18.75,Default,,0000,0000,0000,,
Dialogue: 0,0:41:18.75,0:41:21.30,Default,,0000,0000,0000,,Am I missing anybody important?
Dialogue: 0,0:41:21.30,0:41:22.60,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:41:22.60,0:41:23.91,Default,,0000,0000,0000,,We live in tests.
Dialogue: 0,0:41:23.91,0:41:27.76,Default,,0000,0000,0000,,We cannot say goodbye to the\Nlast section of the chapter
Dialogue: 0,0:41:27.76,0:41:32.66,Default,,0000,0000,0000,,nine, which is 9.7, without\Nmeeting again our friend
Dialogue: 0,0:41:32.66,0:41:35.22,Default,,0000,0000,0000,,the saddle, right?
Dialogue: 0,0:41:35.22,0:41:39.05,Default,,0000,0000,0000,,The saddle is-- this\Nis elliptic paraboloid.
Dialogue: 0,0:41:39.05,0:41:42.20,Default,,0000,0000,0000,,And the last very\Nimportant quadric
Dialogue: 0,0:41:42.20,0:41:46.89,Default,,0000,0000,0000,,that I wanted to talk\Nabout today is the--
Dialogue: 0,0:41:46.89,0:41:50.85,Default,,0000,0000,0000,,
Dialogue: 0,0:41:50.85,0:41:52.20,Default,,0000,0000,0000,,STUDENT: What about a cone?
Dialogue: 0,0:41:52.20,0:41:52.83,Default,,0000,0000,0000,,PROFESSOR: Huh?
Dialogue: 0,0:41:52.83,0:41:53.82,Default,,0000,0000,0000,,STUDENT: How about a cone?
Dialogue: 0,0:41:53.82,0:41:55.32,Default,,0000,0000,0000,,PROFESSOR: Oh, a\Ncone is too easy.
Dialogue: 0,0:41:55.32,0:41:58.72,Default,,0000,0000,0000,,But yeah, let's talk\Nabout the cone as well.
Dialogue: 0,0:41:58.72,0:42:01.60,Default,,0000,0000,0000,,Give me an example\Nof the standard cone.
Dialogue: 0,0:42:01.60,0:42:04.24,Default,,0000,0000,0000,,Thank you, [INAUDIBLE].
Dialogue: 0,0:42:04.24,0:42:06.51,Default,,0000,0000,0000,,X squared-- well--
Dialogue: 0,0:42:06.51,0:42:08.01,Default,,0000,0000,0000,,STUDENT: T squared\Nequals x squared.
Dialogue: 0,0:42:08.01,0:42:11.45,Default,,0000,0000,0000,,PROFESSOR: I'm going\Nto draw it first
Dialogue: 0,0:42:11.45,0:42:14.39,Default,,0000,0000,0000,,so that you know what I want.
Dialogue: 0,0:42:14.39,0:42:16.25,Default,,0000,0000,0000,,Unless I draw it, how\Nwould you know what
Dialogue: 0,0:42:16.25,0:42:19.07,Default,,0000,0000,0000,,to invent or to come up with?
Dialogue: 0,0:42:19.07,0:42:23.20,Default,,0000,0000,0000,,It's not an ice cream\Ncone-- it's a double cone.
Dialogue: 0,0:42:23.20,0:42:27.01,Default,,0000,0000,0000,,So I can have a positive\Nz and a negative z-- two
Dialogue: 0,0:42:27.01,0:42:29.77,Default,,0000,0000,0000,,different sheets that are\Nsymmetric with one another.
Dialogue: 0,0:42:29.77,0:42:33.57,Default,,0000,0000,0000,,
Dialogue: 0,0:42:33.57,0:42:35.74,Default,,0000,0000,0000,,So how do I write that?
Dialogue: 0,0:42:35.74,0:42:36.72,Default,,0000,0000,0000,,STUDENT: T squared.
Dialogue: 0,0:42:36.72,0:42:37.68,Default,,0000,0000,0000,,PROFESSOR: Yes, equals?
Dialogue: 0,0:42:37.68,0:42:38.51,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:42:38.51,0:42:41.30,Default,,0000,0000,0000,,
Dialogue: 0,0:42:41.30,0:42:45.93,Default,,0000,0000,0000,,PROFESSOR: Well, would you like\Nit to be like most of those
Dialogue: 0,0:42:45.93,0:42:50.77,Default,,0000,0000,0000,,that we see in the examples\Nin the book, right?
Dialogue: 0,0:42:50.77,0:42:54.20,Default,,0000,0000,0000,,But it doesn't have\Nto be like that.
Dialogue: 0,0:42:54.20,0:42:56.64,Default,,0000,0000,0000,,Of course, if it's\Nlike that, of course
Dialogue: 0,0:42:56.64,0:42:59.24,Default,,0000,0000,0000,,you realize z-- set the z.
Dialogue: 0,0:42:59.24,0:43:00.89,Default,,0000,0000,0000,,Set the plane and altitude.
Dialogue: 0,0:43:00.89,0:43:03.32,Default,,0000,0000,0000,,Then you're going to have\Ncircle, circle, circle,
Dialogue: 0,0:43:03.32,0:43:07.34,Default,,0000,0000,0000,,circle-- circle after\Ncircle of different radii
Dialogue: 0,0:43:07.34,0:43:08.81,Default,,0000,0000,0000,,as cross sections.
Dialogue: 0,0:43:08.81,0:43:10.45,Default,,0000,0000,0000,,Z could also be negative.
Dialogue: 0,0:43:10.45,0:43:13.96,Default,,0000,0000,0000,,Except for the case of the\Norigin, where you have 0, 0, 0.
Dialogue: 0,0:43:13.96,0:43:17.89,Default,,0000,0000,0000,,
Dialogue: 0,0:43:17.89,0:43:21.72,Default,,0000,0000,0000,,Now, if you were to set\Nx equals 0, of course
Dialogue: 0,0:43:21.72,0:43:24.26,Default,,0000,0000,0000,,you would get y\Nequals plus minus
Dialogue: 0,0:43:24.26,0:43:30.05,Default,,0000,0000,0000,,z, which are exactly these\Nlines, the red lines that I'm
Dialogue: 0,0:43:30.05,0:43:31.05,Default,,0000,0000,0000,,drawing in this picture.
Dialogue: 0,0:43:31.05,0:43:34.33,Default,,0000,0000,0000,,
Dialogue: 0,0:43:34.33,0:43:35.91,Default,,0000,0000,0000,,So practically, this is, what?
Dialogue: 0,0:43:35.91,0:43:37.24,Default,,0000,0000,0000,,Called a what?
Dialogue: 0,0:43:37.24,0:43:39.45,Default,,0000,0000,0000,,A circular cone.
Dialogue: 0,0:43:39.45,0:43:42.09,Default,,0000,0000,0000,,If I wanted to make\Nit more interesting,
Dialogue: 0,0:43:42.09,0:43:45.56,Default,,0000,0000,0000,,I would put a squared\Nand b squared.
Dialogue: 0,0:43:45.56,0:43:48.36,Default,,0000,0000,0000,,And it would be\Nan elliptic cone.
Dialogue: 0,0:43:48.36,0:43:51.74,Default,,0000,0000,0000,,And we stayed away from\Nthat as much as we could.
Dialogue: 0,0:43:51.74,0:43:55.16,Default,,0000,0000,0000,,We brought it up now because\NZander asked about it.
Dialogue: 0,0:43:55.16,0:43:57.95,Default,,0000,0000,0000,,So how about the number four,\Nnumber five, whatever it
Dialogue: 0,0:43:57.95,0:43:59.07,Default,,0000,0000,0000,,is-- number four?
Dialogue: 0,0:43:59.07,0:44:03.09,Default,,0000,0000,0000,,The [INAUDIBLE] what\Nwas the typical equation
Dialogue: 0,0:44:03.09,0:44:15.72,Default,,0000,0000,0000,,of the hyperbolic paraboloid\Nthat I had in mind?
Dialogue: 0,0:44:15.72,0:44:16.55,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:44:16.55,0:44:19.99,Default,,0000,0000,0000,,
Dialogue: 0,0:44:19.99,0:44:31.97,Default,,0000,0000,0000,,PROFESSOR: Z equals x squared\Nminus y squared, very good.
Dialogue: 0,0:44:31.97,0:44:34.53,Default,,0000,0000,0000,,So I will try again and draw it.
Dialogue: 0,0:44:34.53,0:44:36.77,Default,,0000,0000,0000,,It's not so easy to draw.
Dialogue: 0,0:44:36.77,0:44:40.45,Default,,0000,0000,0000,,
Dialogue: 0,0:44:40.45,0:44:47.20,Default,,0000,0000,0000,,If I were to choose x\Nto be 0 and draw exactly
Dialogue: 0,0:44:47.20,0:44:51.14,Default,,0000,0000,0000,,in the plane of the board,\Nz equals minus y squared
Dialogue: 0,0:44:51.14,0:44:59.55,Default,,0000,0000,0000,,would be some\Ncoordinate line, right?
Dialogue: 0,0:44:59.55,0:45:02.00,Default,,0000,0000,0000,,This is what we\Ncall such a thing.
Dialogue: 0,0:45:02.00,0:45:06.22,Default,,0000,0000,0000,,If we fix the x to be x0,\Nwe get a coordinate line.
Dialogue: 0,0:45:06.22,0:45:09.80,Default,,0000,0000,0000,,If we fix the y to be y0, we\Nget another coordinate line.
Dialogue: 0,0:45:09.80,0:45:12.36,Default,,0000,0000,0000,,There are two families of lines.
Dialogue: 0,0:45:12.36,0:45:17.16,Default,,0000,0000,0000,,Why is z equals minus y\Nsquared drawn in this board,
Dialogue: 0,0:45:17.16,0:45:19.32,Default,,0000,0000,0000,,on the board, in this plane?
Dialogue: 0,0:45:19.32,0:45:23.53,Default,,0000,0000,0000,,It's a parabola that\Nopens upside down.
Dialogue: 0,0:45:23.53,0:45:34.03,Default,,0000,0000,0000,,OK, so you have something like\Nthis which you are drawing,
Dialogue: 0,0:45:34.03,0:45:36.05,Default,,0000,0000,0000,,right?
Dialogue: 0,0:45:36.05,0:45:39.88,Default,,0000,0000,0000,,And then what if y would be 0?
Dialogue: 0,0:45:39.88,0:45:43.47,Default,,0000,0000,0000,,Then you get z equals x0.
Dialogue: 0,0:45:43.47,0:45:48.40,Default,,0000,0000,0000,,So it's going to a\Nparabola that opens up.
Dialogue: 0,0:45:48.40,0:45:53.72,Default,,0000,0000,0000,,Then I have to locate myself\Nand draw it on that wall.
Dialogue: 0,0:45:53.72,0:45:54.72,Default,,0000,0000,0000,,But I can't.
Dialogue: 0,0:45:54.72,0:45:57.70,Default,,0000,0000,0000,,Because if I do that, I'm\Ngoing to get in trouble.
Dialogue: 0,0:45:57.70,0:46:01.74,Default,,0000,0000,0000,,So I better draw it like\Nthis in perspective.
Dialogue: 0,0:46:01.74,0:46:04.72,Default,,0000,0000,0000,,And you guys should\Nimagine what we have.
Dialogue: 0,0:46:04.72,0:46:12.08,Default,,0000,0000,0000,,So if we were to cut\Ndown with a knife,
Dialogue: 0,0:46:12.08,0:46:15.57,Default,,0000,0000,0000,,we would get-- we will\Nstill get these parabolas
Dialogue: 0,0:46:15.57,0:46:18.07,Default,,0000,0000,0000,,that all point down.
Dialogue: 0,0:46:18.07,0:46:22.16,Default,,0000,0000,0000,,And in those directions, these\Nare just the highest parts
Dialogue: 0,0:46:22.16,0:46:23.59,Default,,0000,0000,0000,,of the saddle.
Dialogue: 0,0:46:23.59,0:46:27.03,Default,,0000,0000,0000,,And let's say this would be\Nthe lowest part of the saddle.
Dialogue: 0,0:46:27.03,0:46:32.28,Default,,0000,0000,0000,,Where-- where is the\Npart of the rider?
Dialogue: 0,0:46:32.28,0:46:34.99,Default,,0000,0000,0000,,A guy's butt is here.
Dialogue: 0,0:46:34.99,0:46:38.76,Default,,0000,0000,0000,,And his leg is following the\Nshape of the saddle going down.
Dialogue: 0,0:46:38.76,0:46:43.32,Default,,0000,0000,0000,,That's the cowboy boot, OK?
Dialogue: 0,0:46:43.32,0:46:46.53,Default,,0000,0000,0000,,And he is-- hold on.
Dialogue: 0,0:46:46.53,0:46:52.66,Default,,0000,0000,0000,,I don't know how-- what's the\Nattitude of the [INAUDIBLE]?
Dialogue: 0,0:46:52.66,0:46:54.62,Default,,0000,0000,0000,,Well, it doesn't look\Nlike a cowboy hat.
Dialogue: 0,0:46:54.62,0:46:57.55,Default,,0000,0000,0000,,But anyway, I'm sorry.
Dialogue: 0,0:46:57.55,0:46:59.50,Default,,0000,0000,0000,,He looks a little\Nbit Vietnamese.
Dialogue: 0,0:46:59.50,0:47:03.42,Default,,0000,0000,0000,,That was not the intention.
Dialogue: 0,0:47:03.42,0:47:05.86,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:47:05.86,0:47:07.84,Default,,0000,0000,0000,,PROFESSOR: Then\Nlet him be Mexican.
Dialogue: 0,0:47:07.84,0:47:11.16,Default,,0000,0000,0000,,Half of the population\Nin this town are Mexican.
Dialogue: 0,0:47:11.16,0:47:14.50,Default,,0000,0000,0000,,So this is his leg\Nthat goes down.
Dialogue: 0,0:47:14.50,0:47:18.91,Default,,0000,0000,0000,,
Dialogue: 0,0:47:18.91,0:47:20.38,Default,,0000,0000,0000,,OK, very good.
Dialogue: 0,0:47:20.38,0:47:23.81,Default,,0000,0000,0000,,Look-- he even has a--\Nwhat do you call that?
Dialogue: 0,0:47:23.81,0:47:24.79,Default,,0000,0000,0000,,That's so beautiful.
Dialogue: 0,0:47:24.79,0:47:30.62,Default,,0000,0000,0000,,In the Mexican culture, they\Nmake those embroidered by hand
Dialogue: 0,0:47:30.62,0:47:33.05,Default,,0000,0000,0000,,with many colors belts.
Dialogue: 0,0:47:33.05,0:47:35.64,Default,,0000,0000,0000,,But there are some\Nspecial belts.
Dialogue: 0,0:47:35.64,0:47:39.45,Default,,0000,0000,0000,,OK-- depends on the area\Nof Mexico You visit.
Dialogue: 0,0:47:39.45,0:47:41.26,Default,,0000,0000,0000,,I liked several of them.
Dialogue: 0,0:47:41.26,0:47:42.80,Default,,0000,0000,0000,,They're so beautiful.
Dialogue: 0,0:47:42.80,0:47:45.58,Default,,0000,0000,0000,,But my favorite\None is, of course,
Dialogue: 0,0:47:45.58,0:47:50.54,Default,,0000,0000,0000,,the Rivera Maya, which is where\Nyou go to the Chichen Itza,
Dialogue: 0,0:47:50.54,0:47:54.64,Default,,0000,0000,0000,,to the mystic areas, to the\Nsea, and eat the good food
Dialogue: 0,0:47:54.64,0:48:00.20,Default,,0000,0000,0000,,and go to Cozumel and forget\Nabout school for a week.
Dialogue: 0,0:48:00.20,0:48:02.68,Default,,0000,0000,0000,,That is paradise for me.
Dialogue: 0,0:48:02.68,0:48:05.16,Default,,0000,0000,0000,,But [INAUDIBLE]\Nis not bad either.
Dialogue: 0,0:48:05.16,0:48:08.12,Default,,0000,0000,0000,,If I were to choose where\Nto live and I had money,
Dialogue: 0,0:48:08.12,0:48:11.11,Default,,0000,0000,0000,,I would live in Cozumel\Nfor the rest of my life.
Dialogue: 0,0:48:11.11,0:48:15.61,Default,,0000,0000,0000,,OK, so this is the\Nsaddle that is oriented
Dialogue: 0,0:48:15.61,0:48:18.45,Default,,0000,0000,0000,,so that you have a parabola\Ngoing in this direction,
Dialogue: 0,0:48:18.45,0:48:23.16,Default,,0000,0000,0000,,going up, a parabola going\Ndown in this direction.
Dialogue: 0,0:48:23.16,0:48:28.09,Default,,0000,0000,0000,,What is magic about\Na saddle point?
Dialogue: 0,0:48:28.09,0:48:29.48,Default,,0000,0000,0000,,Do you remember?
Dialogue: 0,0:48:29.48,0:48:33.16,Default,,0000,0000,0000,,STUDENT: It was [INAUDIBLE]
Dialogue: 0,0:48:33.16,0:48:36.32,Default,,0000,0000,0000,,PROFESSOR: It's-- one direction\Nis like a max and one direction
Dialogue: 0,0:48:36.32,0:48:37.45,Default,,0000,0000,0000,,is like a min.
Dialogue: 0,0:48:37.45,0:48:41.34,Default,,0000,0000,0000,,So when you compute that\Ndiscriminant, you get negative.
Dialogue: 0,0:48:41.34,0:48:44.67,Default,,0000,0000,0000,,You get like the product\Nbetween the curvatures.
Dialogue: 0,0:48:44.67,0:48:46.71,Default,,0000,0000,0000,,One curvature in one\Ndirection would be positive.
Dialogue: 0,0:48:46.71,0:48:48.39,Default,,0000,0000,0000,,The other one would be negative.
Dialogue: 0,0:48:48.39,0:48:52.88,Default,,0000,0000,0000,,So it's like getting the product\Nplus minus equals y, [? yes. ?]
Dialogue: 0,0:48:52.88,0:48:55.62,Default,,0000,0000,0000,,But again, we will talk\Nabout this sometimes later.
Dialogue: 0,0:48:55.62,0:48:58.84,Default,,0000,0000,0000,,You don't even\Nhave to know that.
Dialogue: 0,0:48:58.84,0:49:01.84,Default,,0000,0000,0000,,Shall we say goodbye to This?
Dialogue: 0,0:49:01.84,0:49:03.78,Default,,0000,0000,0000,,I guess it's time.
Dialogue: 0,0:49:03.78,0:49:08.27,Default,,0000,0000,0000,,And chapter nine is now\Nfresh in your memory.
Dialogue: 0,0:49:08.27,0:49:11.87,Default,,0000,0000,0000,,You would be really\Nto start chapter 10.
Dialogue: 0,0:49:11.87,0:49:14.80,Default,,0000,0000,0000,,
Dialogue: 0,0:49:14.80,0:49:19.51,Default,,0000,0000,0000,,What I want to do, I want\Nto-- without being recorded.
Dialogue: 0,0:49:19.51,0:49:20.54,Default,,0000,0000,0000,,