[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.76,0:00:03.13,Default,,0000,0000,0000,,So far, when I've told you about\Nthe dot and the cross
Dialogue: 0,0:00:03.13,0:00:06.44,Default,,0000,0000,0000,,products, I've given you the\Ndefinition as the magnitude
Dialogue: 0,0:00:06.44,0:00:08.71,Default,,0000,0000,0000,,times either the cosine\Nor the sine of the
Dialogue: 0,0:00:08.71,0:00:09.71,Default,,0000,0000,0000,,angle between them.
Dialogue: 0,0:00:09.71,0:00:12.43,Default,,0000,0000,0000,,But what if you're not given\Nthe vectors visually?
Dialogue: 0,0:00:12.43,0:00:14.21,Default,,0000,0000,0000,,And what if you're not given\Nthe angle between them?
Dialogue: 0,0:00:14.21,0:00:17.24,Default,,0000,0000,0000,,How do you calculate the dot\Nand the cross products?
Dialogue: 0,0:00:17.24,0:00:19.16,Default,,0000,0000,0000,,Well, let me give you\Nthe definition that
Dialogue: 0,0:00:19.16,0:00:20.00,Default,,0000,0000,0000,,I giving you already.
Dialogue: 0,0:00:20.00,0:00:26.71,Default,,0000,0000,0000,,So let's say I have a\Ndot b dot product.
Dialogue: 0,0:00:26.71,0:00:31.61,Default,,0000,0000,0000,,That's the magnitude of a times\Nthe magnitude of b times
Dialogue: 0,0:00:31.61,0:00:34.20,Default,,0000,0000,0000,,cosine of the angle\Nbetween them.
Dialogue: 0,0:00:34.20,0:00:39.73,Default,,0000,0000,0000,,a cross b is equal to the\Nmagnitude of a times the
Dialogue: 0,0:00:39.73,0:00:44.67,Default,,0000,0000,0000,,magnitude of b times sine of\Nthe angle between them-- so
Dialogue: 0,0:00:44.67,0:00:48.36,Default,,0000,0000,0000,,the perpendicular projections\Nof them-- times the normal
Dialogue: 0,0:00:48.36,0:00:50.13,Default,,0000,0000,0000,,vector that's perpendicular\Nto both of them.
Dialogue: 0,0:00:50.13,0:00:53.75,Default,,0000,0000,0000,,The normal unit vector, and you\Nfigure out which of the
Dialogue: 0,0:00:53.75,0:00:55.50,Default,,0000,0000,0000,,two perpendicular vectors\Nit is by using
Dialogue: 0,0:00:55.50,0:00:56.62,Default,,0000,0000,0000,,the right hand rule.
Dialogue: 0,0:00:56.62,0:01:00.17,Default,,0000,0000,0000,,But what if we don't have\Nthe thetas; the
Dialogue: 0,0:01:00.17,0:01:01.32,Default,,0000,0000,0000,,angles between them?
Dialogue: 0,0:01:01.32,0:01:04.76,Default,,0000,0000,0000,,What if, for example, I were\Nto tell you that the vector
Dialogue: 0,0:01:04.76,0:01:09.99,Default,,0000,0000,0000,,a,-- if I were to give it to you\Nin engineering notation.
Dialogue: 0,0:01:09.99,0:01:12.09,Default,,0000,0000,0000,,In engineering notation,\Nyou're essentially just
Dialogue: 0,0:01:12.09,0:01:16.27,Default,,0000,0000,0000,,breaking down the vector into\Nits x, y and z components.
Dialogue: 0,0:01:16.27,0:01:23.58,Default,,0000,0000,0000,,So let's say vector a is 5i--\Ni is just the unit vector in
Dialogue: 0,0:01:23.58,0:01:31.89,Default,,0000,0000,0000,,the x direction, minus\N6j, plus 3k.
Dialogue: 0,0:01:34.74,0:01:37.79,Default,,0000,0000,0000,,i,j and k are just the unit\Nvectors of the x, y and z
Dialogue: 0,0:01:37.79,0:01:38.31,Default,,0000,0000,0000,,directions.
Dialogue: 0,0:01:38.31,0:01:40.70,Default,,0000,0000,0000,,And the 5 is how much it goes\Nin the x direction.
Dialogue: 0,0:01:40.70,0:01:43.40,Default,,0000,0000,0000,,The minus 6 is how much it\Ngoes in the y direction.
Dialogue: 0,0:01:43.40,0:01:45.89,Default,,0000,0000,0000,,And the 3 is how much it goes\Nin the z direction.
Dialogue: 0,0:01:45.89,0:01:47.04,Default,,0000,0000,0000,,You could try to graph it.
Dialogue: 0,0:01:47.04,0:01:48.96,Default,,0000,0000,0000,,And actually, I'm trying to look\Nfor a graphing calculator
Dialogue: 0,0:01:48.96,0:01:51.37,Default,,0000,0000,0000,,that'll do this, so I can show\Nyou it all in videos to give
Dialogue: 0,0:01:51.37,0:01:52.36,Default,,0000,0000,0000,,you more intuition.
Dialogue: 0,0:01:52.36,0:01:53.83,Default,,0000,0000,0000,,But lets say this is\Nall you're given.
Dialogue: 0,0:01:53.83,0:02:00.10,Default,,0000,0000,0000,,And let's say that b-- I'm just\Nmaking these numbers up--
Dialogue: 0,0:02:00.10,0:02:04.17,Default,,0000,0000,0000,,let's say it's minus 2i-- and,\Nof course, we're working in
Dialogue: 0,0:02:04.17,0:02:14.48,Default,,0000,0000,0000,,three dimensions right now--\Nplus 7j, plus 4k.
Dialogue: 0,0:02:14.48,0:02:15.30,Default,,0000,0000,0000,,You could graph it.
Dialogue: 0,0:02:15.30,0:02:19.03,Default,,0000,0000,0000,,But obviously, if you were given\Na problem, and if you
Dialogue: 0,0:02:19.03,0:02:22.27,Default,,0000,0000,0000,,were actually trying to model\Nvectors on a computer
Dialogue: 0,0:02:22.27,0:02:23.51,Default,,0000,0000,0000,,simulation, this is the\Nway you would do it.
Dialogue: 0,0:02:23.51,0:02:25.69,Default,,0000,0000,0000,,You'd break it up into the x, y,\Nand z components because of
Dialogue: 0,0:02:25.69,0:02:26.78,Default,,0000,0000,0000,,the add vectors.
Dialogue: 0,0:02:26.78,0:02:28.60,Default,,0000,0000,0000,,You just have to add the\Nrespective components.
Dialogue: 0,0:02:28.60,0:02:31.21,Default,,0000,0000,0000,,But how do you multiply them\Neither taking the cross or the
Dialogue: 0,0:02:31.21,0:02:32.34,Default,,0000,0000,0000,,dot product?
Dialogue: 0,0:02:32.34,0:02:34.58,Default,,0000,0000,0000,,Well it actually turns out I'm\Nnot going to prove it here but
Dialogue: 0,0:02:34.58,0:02:35.40,Default,,0000,0000,0000,,I'll just show you\Nhow to do it.
Dialogue: 0,0:02:35.40,0:02:38.10,Default,,0000,0000,0000,,The dot product is very\Neasy when you have it
Dialogue: 0,0:02:38.10,0:02:39.33,Default,,0000,0000,0000,,given in this notation.
Dialogue: 0,0:02:39.33,0:02:40.88,Default,,0000,0000,0000,,And actually another way of\Nwriting this notation,
Dialogue: 0,0:02:40.88,0:02:42.36,Default,,0000,0000,0000,,sometimes it's in bracket\Nnotation.
Dialogue: 0,0:02:42.36,0:02:46.96,Default,,0000,0000,0000,,Sometimes they would rewrite\Nthis as 5 minus 6, 3.
Dialogue: 0,0:02:46.96,0:02:49.46,Default,,0000,0000,0000,,Or it's just the magnitudes of\Nthe x,y and z direction.
Dialogue: 0,0:02:49.46,0:02:53.17,Default,,0000,0000,0000,,I just want to make sure you're\Ncomfortable with all of
Dialogue: 0,0:02:53.17,0:02:54.27,Default,,0000,0000,0000,,these various notations.
Dialogue: 0,0:02:54.27,0:02:57.36,Default,,0000,0000,0000,,You could have written b\Nas minus is 2, 7, 4.
Dialogue: 0,0:02:57.36,0:02:58.38,Default,,0000,0000,0000,,These are all the same things.
Dialogue: 0,0:02:58.38,0:03:00.36,Default,,0000,0000,0000,,You shouldn't get daunted if\Nyou see one or the other.
Dialogue: 0,0:03:00.36,0:03:05.43,Default,,0000,0000,0000,,But anyway, so how do\NI take a dot b?
Dialogue: 0,0:03:08.11,0:03:10.67,Default,,0000,0000,0000,,This, I think you'll find\Nfairly pleasant.
Dialogue: 0,0:03:10.67,0:03:15.41,Default,,0000,0000,0000,,All you do is you multiply the\Ni components, add that to the
Dialogue: 0,0:03:15.41,0:03:18.27,Default,,0000,0000,0000,,j components multiplied, and\Nthen add that to the k
Dialogue: 0,0:03:18.27,0:03:20.21,Default,,0000,0000,0000,,components multiplied\Ntogether.
Dialogue: 0,0:03:20.21,0:03:34.35,Default,,0000,0000,0000,,So it would be 5 times minus 2\Nplus minus 6 times 7 plus 3
Dialogue: 0,0:03:34.35,0:03:45.26,Default,,0000,0000,0000,,times 4, so it equals minus\N10 minus 42 plus 12.
Dialogue: 0,0:03:45.26,0:03:52.02,Default,,0000,0000,0000,,So this is minus 52 plus 12,\Nso it equals minus 40.
Dialogue: 0,0:03:52.02,0:03:52.46,Default,,0000,0000,0000,,That's it.
Dialogue: 0,0:03:52.46,0:03:54.84,Default,,0000,0000,0000,,It's just a number.
Dialogue: 0,0:03:54.84,0:03:57.09,Default,,0000,0000,0000,,And I'd actually be curious\Nto graph this on a three
Dialogue: 0,0:03:57.09,0:04:00.98,Default,,0000,0000,0000,,dimensional grapher to see\Nwhy it's minus 40.
Dialogue: 0,0:04:00.98,0:04:03.60,Default,,0000,0000,0000,,They must be going in\Nopposite directions.
Dialogue: 0,0:04:03.60,0:04:05.68,Default,,0000,0000,0000,,And their projections onto each\Nother go into opposite
Dialogue: 0,0:04:05.68,0:04:06.07,Default,,0000,0000,0000,,directions.
Dialogue: 0,0:04:06.07,0:04:07.77,Default,,0000,0000,0000,,And that's why we get\Na minus number.
Dialogue: 0,0:04:11.00,0:04:13.03,Default,,0000,0000,0000,,The purpose of this-- I don't\Nwant to get too much into the
Dialogue: 0,0:04:13.03,0:04:15.05,Default,,0000,0000,0000,,intuition-- this is just how to\Ncalculate, but it's fairly
Dialogue: 0,0:04:15.05,0:04:15.90,Default,,0000,0000,0000,,straightforward.
Dialogue: 0,0:04:15.90,0:04:18.93,Default,,0000,0000,0000,,You just multiply the\Nx components.
Dialogue: 0,0:04:18.93,0:04:22.03,Default,,0000,0000,0000,,Add that to the y components\Nmultiplied and add that to the
Dialogue: 0,0:04:22.03,0:04:23.45,Default,,0000,0000,0000,,z components multiplied.
Dialogue: 0,0:04:23.45,0:04:25.71,Default,,0000,0000,0000,,So whenever I am given something\Nin engineering or
Dialogue: 0,0:04:25.71,0:04:28.47,Default,,0000,0000,0000,,bracket notation and I have to\Nfind the dot product, it's
Dialogue: 0,0:04:28.47,0:04:33.68,Default,,0000,0000,0000,,very, almost soothing, and\Nnot so error prone.
Dialogue: 0,0:04:33.68,0:04:37.39,Default,,0000,0000,0000,,But, as you will see, taking the\Ncross product of these two
Dialogue: 0,0:04:37.39,0:04:40.16,Default,,0000,0000,0000,,vectors when given in this\Nnotation isn't so
Dialogue: 0,0:04:40.16,0:04:41.49,Default,,0000,0000,0000,,straightforward.
Dialogue: 0,0:04:41.49,0:04:43.02,Default,,0000,0000,0000,,And I want you to keep in mind,\Nanother way you could
Dialogue: 0,0:04:43.02,0:04:44.59,Default,,0000,0000,0000,,have done it, you could have\Nfigured out the magnitude of
Dialogue: 0,0:04:44.59,0:04:49.47,Default,,0000,0000,0000,,each of these vectors and then\Nyou could have used some fancy
Dialogue: 0,0:04:49.47,0:04:51.77,Default,,0000,0000,0000,,trigonometry to figure out the\Nthetas and then used this
Dialogue: 0,0:04:51.77,0:04:52.37,Default,,0000,0000,0000,,definition.
Dialogue: 0,0:04:52.37,0:04:56.23,Default,,0000,0000,0000,,But I think you appreciate the\Nfact that this is a much
Dialogue: 0,0:04:56.23,0:04:57.35,Default,,0000,0000,0000,,simpler way of doing it.
Dialogue: 0,0:04:57.35,0:04:59.14,Default,,0000,0000,0000,,So the dot product\Nis a lot of fun.
Dialogue: 0,0:04:59.14,0:05:02.57,Default,,0000,0000,0000,,Now let's see if we could\Ntake the cross product.
Dialogue: 0,0:05:02.57,0:05:04.45,Default,,0000,0000,0000,,And once again, I'm not\Ngoing to prove it.
Dialogue: 0,0:05:04.45,0:05:06.23,Default,,0000,0000,0000,,I'm just going to show\Nyou how to do it.
Dialogue: 0,0:05:06.23,0:05:09.37,Default,,0000,0000,0000,,In a future video, I'm sure I'll\Nget a request to do it
Dialogue: 0,0:05:09.37,0:05:11.71,Default,,0000,0000,0000,,eventually, and I'll prove it.
Dialogue: 0,0:05:11.71,0:05:15.27,Default,,0000,0000,0000,,But the cross product, this\Nis more involved.
Dialogue: 0,0:05:15.27,0:05:18.21,Default,,0000,0000,0000,,And I never look forward to\Ntaking the cross product of
Dialogue: 0,0:05:18.21,0:05:20.29,Default,,0000,0000,0000,,two vectors in engineering\Nnotation.
Dialogue: 0,0:05:20.29,0:05:22.70,Default,,0000,0000,0000,,a cross b.
Dialogue: 0,0:05:22.70,0:05:23.76,Default,,0000,0000,0000,,It equals.
Dialogue: 0,0:05:23.76,0:05:27.53,Default,,0000,0000,0000,,So this is an application\Nof matrices.
Dialogue: 0,0:05:27.53,0:05:31.85,Default,,0000,0000,0000,,So what you do is you take the\Ndeterminant-- I'll draw a big
Dialogue: 0,0:05:31.85,0:05:34.12,Default,,0000,0000,0000,,determinant line-- on the top\Nline of the determinant.
Dialogue: 0,0:05:34.12,0:05:35.19,Default,,0000,0000,0000,,This is really just\Na way to make you
Dialogue: 0,0:05:35.19,0:05:37.09,Default,,0000,0000,0000,,memorize how to do it.
Dialogue: 0,0:05:37.09,0:05:39.24,Default,,0000,0000,0000,,It doesn't give you much\Nintuition, but the intuition
Dialogue: 0,0:05:39.24,0:05:41.69,Default,,0000,0000,0000,,is given by the actual\Ndefinition.
Dialogue: 0,0:05:41.69,0:05:44.01,Default,,0000,0000,0000,,How much of the vectors are\Nperpendicular to each other.
Dialogue: 0,0:05:44.01,0:05:45.05,Default,,0000,0000,0000,,Multiply those magnitudes.
Dialogue: 0,0:05:45.05,0:05:47.21,Default,,0000,0000,0000,,Right hand rule figures\Nout what direction
Dialogue: 0,0:05:47.21,0:05:48.36,Default,,0000,0000,0000,,you're pointing in.
Dialogue: 0,0:05:48.36,0:05:51.38,Default,,0000,0000,0000,,But the way to do it if you're\Ngiven engineering notation,
Dialogue: 0,0:05:51.38,0:05:55.76,Default,,0000,0000,0000,,you write the i, j, k unit\Nvectors the top row.
Dialogue: 0,0:05:55.76,0:06:00.08,Default,,0000,0000,0000,,i, j, k.
Dialogue: 0,0:06:00.08,0:06:02.23,Default,,0000,0000,0000,,Then you write the first vector\Nin the cross product,
Dialogue: 0,0:06:02.23,0:06:03.56,Default,,0000,0000,0000,,because order matters.
Dialogue: 0,0:06:03.56,0:06:09.55,Default,,0000,0000,0000,,So it's 5 minus 6, 3.
Dialogue: 0,0:06:09.55,0:06:12.32,Default,,0000,0000,0000,,Then you take the second vector\Nwhich is b, which is
Dialogue: 0,0:06:12.32,0:06:16.97,Default,,0000,0000,0000,,minus 2, 7, 4.
Dialogue: 0,0:06:16.97,0:06:19.88,Default,,0000,0000,0000,,So you take the determinant\Nof the 3 by 3 matrix,
Dialogue: 0,0:06:19.88,0:06:21.35,Default,,0000,0000,0000,,and how do I do that?
Dialogue: 0,0:06:21.35,0:06:25.93,Default,,0000,0000,0000,,Well that's equal to the\Nsubdeterminant for i.
Dialogue: 0,0:06:25.93,0:06:28.46,Default,,0000,0000,0000,,So the subdeterminant for i, if\Nyou get rid of this column
Dialogue: 0,0:06:28.46,0:06:31.92,Default,,0000,0000,0000,,and this row, the determinant\Nthat's left over, so that's
Dialogue: 0,0:06:31.92,0:06:40.76,Default,,0000,0000,0000,,minus 6, 3, 7, 4 times i--\Nyou might want to review
Dialogue: 0,0:06:40.76,0:06:42.43,Default,,0000,0000,0000,,determinants if you don't\Nremember how to do this, but
Dialogue: 0,0:06:42.43,0:06:47.77,Default,,0000,0000,0000,,maybe me working through it\Nwill just jog your memory.
Dialogue: 0,0:06:47.77,0:06:50.59,Default,,0000,0000,0000,,And then remember, it's\Nplus, minus, plus.
Dialogue: 0,0:06:50.59,0:06:53.55,Default,,0000,0000,0000,,So then minus the subdeterminant\Nfor j.
Dialogue: 0,0:06:53.55,0:06:55.50,Default,,0000,0000,0000,,What's the subdeterminant\Nfor j?
Dialogue: 0,0:06:55.50,0:06:57.47,Default,,0000,0000,0000,,You cross out j's\Nrow and columns.
Dialogue: 0,0:06:57.47,0:07:01.06,Default,,0000,0000,0000,,You have 5, 3, minus 2, 4.
Dialogue: 0,0:07:05.03,0:07:07.65,Default,,0000,0000,0000,,We just crossed j's\Nrow and column.
Dialogue: 0,0:07:07.65,0:07:09.77,Default,,0000,0000,0000,,And whatever's left over, those\Nare the numbers in its
Dialogue: 0,0:07:09.77,0:07:11.47,Default,,0000,0000,0000,,subdeterminant.
Dialogue: 0,0:07:11.47,0:07:13.42,Default,,0000,0000,0000,,That's what I call it.
Dialogue: 0,0:07:13.42,0:07:18.14,Default,,0000,0000,0000,,j plus-- I want to do them all\Non one line because it would
Dialogue: 0,0:07:18.14,0:07:19.87,Default,,0000,0000,0000,,have been a little bit\Nneater-- plus the
Dialogue: 0,0:07:19.87,0:07:20.84,Default,,0000,0000,0000,,subdeterminant for k.
Dialogue: 0,0:07:20.84,0:07:23.29,Default,,0000,0000,0000,,Cross out the row and\Nthe column for k.
Dialogue: 0,0:07:23.29,0:07:35.01,Default,,0000,0000,0000,,We're left with 5 minus 6,\Nminus 2 and 7 times k.
Dialogue: 0,0:07:35.01,0:07:36.98,Default,,0000,0000,0000,,And now let's calculate them.
Dialogue: 0,0:07:36.98,0:07:39.44,Default,,0000,0000,0000,,And let me make some\Nspace, because I've
Dialogue: 0,0:07:39.44,0:07:41.13,Default,,0000,0000,0000,,written this too big.
Dialogue: 0,0:07:41.13,0:07:43.79,Default,,0000,0000,0000,,I don't think we need\Nthis anymore.
Dialogue: 0,0:07:43.79,0:07:46.46,Default,,0000,0000,0000,,So what do we get?
Dialogue: 0,0:07:46.46,0:07:49.40,Default,,0000,0000,0000,,Let's take this up here.
Dialogue: 0,0:07:49.40,0:07:51.09,Default,,0000,0000,0000,,So these 2 by 2 determinants\Nare pretty easy.
Dialogue: 0,0:07:51.09,0:07:58.69,Default,,0000,0000,0000,,This is minus 6 times\N4 minus 7 times 3.
Dialogue: 0,0:07:58.69,0:08:00.18,Default,,0000,0000,0000,,I always make careless\Nmistakes here.
Dialogue: 0,0:08:00.18,0:08:10.77,Default,,0000,0000,0000,,Minus 24 minus 21 times i minus\N5 times 4 is 20, minus
Dialogue: 0,0:08:10.77,0:08:23.27,Default,,0000,0000,0000,,minus 2 times 3, so minus minus\N6 j, plus 5 times 7, 35
Dialogue: 0,0:08:23.27,0:08:25.64,Default,,0000,0000,0000,,minus minus 2 times minus 6.
Dialogue: 0,0:08:25.64,0:08:29.33,Default,,0000,0000,0000,,So it's minus positive 12k.
Dialogue: 0,0:08:29.33,0:08:34.33,Default,,0000,0000,0000,,We could simplify this, which\Nequals minus 24 minus 21.
Dialogue: 0,0:08:34.33,0:08:40.83,Default,,0000,0000,0000,,It is minus 35-- I didn't have\Nto put a parentheses-- i, and
Dialogue: 0,0:08:40.83,0:08:43.72,Default,,0000,0000,0000,,then what's 20 minus minus 6?
Dialogue: 0,0:08:43.72,0:08:46.60,Default,,0000,0000,0000,,Well that's 20 plus\Nplus 6, so 26.
Dialogue: 0,0:08:46.60,0:08:47.59,Default,,0000,0000,0000,,And then we have a\Nminus out here.
Dialogue: 0,0:08:47.59,0:08:51.64,Default,,0000,0000,0000,,So minus 26j.
Dialogue: 0,0:08:51.64,0:08:54.34,Default,,0000,0000,0000,,And that was 35 minus\N12, that's 23.
Dialogue: 0,0:08:54.34,0:08:57.19,Default,,0000,0000,0000,,Plus 23k.
Dialogue: 0,0:08:57.19,0:08:58.69,Default,,0000,0000,0000,,So that's the cross product.
Dialogue: 0,0:08:58.69,0:09:01.15,Default,,0000,0000,0000,,And if you were to graph this in\Nthree dimensions, you will
Dialogue: 0,0:09:01.15,0:09:03.71,Default,,0000,0000,0000,,see-- and this is what's\Ninteresting-- you will see
Dialogue: 0,0:09:03.71,0:09:09.41,Default,,0000,0000,0000,,that vector, if my math is\Ncorrect, minus 35i, minus 26j,
Dialogue: 0,0:09:09.41,0:09:15.75,Default,,0000,0000,0000,,plus 23k, is perpendicular\Nto both of these vectors.
Dialogue: 0,0:09:15.75,0:09:19.44,Default,,0000,0000,0000,,I think I'll leave you there for\Nnow, and I will see you in
Dialogue: 0,0:09:19.44,0:09:20.05,Default,,0000,0000,0000,,the next video.
Dialogue: 0,0:09:20.05,0:09:22.14,Default,,0000,0000,0000,,And hopefully, I can track down\Na vector graphic program.
Dialogue: 0,0:09:22.14,0:09:25.88,Default,,0000,0000,0000,,Because I think it'll be fun to\Nboth calculate the dot and
Dialogue: 0,0:09:25.88,0:09:29.13,Default,,0000,0000,0000,,the cross products using the\Nmethods I just showed you and
Dialogue: 0,0:09:29.13,0:09:29.84,Default,,0000,0000,0000,,then to graph them.
Dialogue: 0,0:09:29.84,0:09:31.32,Default,,0000,0000,0000,,And to show that it\Nreally does work.
Dialogue: 0,0:09:31.32,0:09:36.93,Default,,0000,0000,0000,,That this vector really is\Nperpendicular to both of these
Dialogue: 0,0:09:36.93,0:09:40.82,Default,,0000,0000,0000,,and pointing in the direction as\Nyou would predict using the
Dialogue: 0,0:09:40.82,0:09:42.52,Default,,0000,0000,0000,,right hand rule.
Dialogue: 0,0:09:42.52,0:09:43.99,Default,,0000,0000,0000,,I'll see you in the next video.