Proof: Relationship between cross product and sin of angle

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:03.630
The goal of this video is to
start with our definition of

00:00:03.630 --> 00:00:06.690
the cross product and the result
that we started off

00:00:06.690 --> 00:00:09.370
with or that we got to in a
different video-- I think it

00:00:09.370 --> 00:00:12.610
was three videos ago-- where
we found out that the dot

00:00:12.610 --> 00:00:17.140
product of two nonzero vectors,
a dot b, is equal to

00:00:17.140 --> 00:00:19.070
the product of their lengths.

00:00:19.070 --> 00:00:23.540
So the product of the length
of a with the length of b

00:00:23.540 --> 00:00:25.780
times the cosine of the
angle between them.

00:00:25.780 --> 00:00:27.310
We're going to start with
these two things.

00:00:27.310 --> 00:00:30.680
This definition of a cross
product in R3, the only place

00:00:30.680 --> 00:00:33.290
it really is defined, and
then this result.

00:00:33.290 --> 00:00:42.300
And we want to get to the result
that the length of the

00:00:42.300 --> 00:00:44.870
cross product of two vectors.

00:00:44.870 --> 00:00:46.660
And so obviously, when
you take a cross

00:00:46.660 --> 00:00:47.760
product you get a vector.

00:00:47.760 --> 00:00:49.910
But if you take its length you
get a number again, you just

00:00:49.910 --> 00:00:53.580
get a scalar value, is equal to
the product of each of the

00:00:53.580 --> 00:00:55.180
vectors' lengths.

00:00:55.180 --> 00:00:57.610
It's the product of the length
of a times the product of the

00:00:57.610 --> 00:01:01.050
length of b times the sin of
the angle between them.

00:01:01.050 --> 00:01:03.590
Which is a pretty neat outcome
because it kind of shows that

00:01:03.590 --> 00:01:05.200
they're two sides of
the same coin.

00:01:05.200 --> 00:01:08.340
Dot product has cosine,
cross product has sin.

00:01:08.340 --> 00:01:11.320
I'm sure you've seen
this before.

00:01:11.320 --> 00:01:13.060
Well you definitely have seen
it if you've watched my

00:01:13.060 --> 00:01:16.360
physics playlist. And I even do
a whole video where I talk

00:01:16.360 --> 00:01:19.360
about the intuition behind
what this really means.

00:01:19.360 --> 00:01:20.790
And I encourage you to re-watch
that, and I'll

00:01:20.790 --> 00:01:23.180
probably do that again in the
linear algebra context.

00:01:23.180 --> 00:01:27.270
But the point of this video
is to prove this to you.

00:01:27.270 --> 00:01:30.380
Is to prove that with this and
this, I can get to this.

00:01:30.380 --> 00:01:33.100
Now if you're just believe me,
and you just say, oh I've seen

00:01:33.100 --> 00:01:33.630
that before.

00:01:33.630 --> 00:01:36.270
And I just think it definitely
is the case, then you don't

00:01:36.270 --> 00:01:38.380
have to watch the rest of this
video because I'll tell you

00:01:38.380 --> 00:01:40.270
right now, it's going
to get dirty.

00:01:40.270 --> 00:01:42.340
It's going to be a hairy,
hairy proof.

00:01:42.340 --> 00:01:48.530
But if you're willing to watch
and bear with me, let's start

00:01:48.530 --> 00:01:50.380
proving this result.

00:01:50.380 --> 00:01:54.910
So the place I'm going to start
is with the idea of

00:01:54.910 --> 00:02:01.570
taking the length of
a cross b squared.

00:02:01.570 --> 00:02:03.080
That's a cross b right there.

00:02:03.080 --> 00:02:06.460
So I'm essentially taking the
length of this vector squared.

00:02:06.460 --> 00:02:09.860
And we saw in many videos and
I've used this idea multiple

00:02:09.860 --> 00:02:13.550
times, that if I just have some
arbitrary vector, let me

00:02:13.550 --> 00:02:15.600
just say some arbitrary
vector.

00:02:15.600 --> 00:02:18.580
And I take its length squared,
that's just equal to that

00:02:18.580 --> 00:02:21.590
vector dotted with itself or
the square of each of its

00:02:21.590 --> 00:02:26.420
terms summed up all the
way to xn squared.

00:02:26.420 --> 00:02:28.410
So what will this be equal to?

00:02:28.410 --> 00:02:29.780
Well this is just equal
to that vector.

00:02:29.780 --> 00:02:31.640
And we only have three
components, so it's equal to

00:02:31.640 --> 00:02:35.960
the sum of the squares of each
of these components.

00:02:35.960 --> 00:02:37.010
Let me write this down.

00:02:37.010 --> 00:02:41.410
It's equal to this
term squared.

00:02:41.410 --> 00:02:43.020
So let me write that down.

00:02:43.020 --> 00:02:50.750
a2 b3 minus a3 b2 squared.

00:02:50.750 --> 00:02:52.990
Plus this term squared.

00:02:52.990 --> 00:03:02.490
So plus a3 b1 minus
a1 b3 squared.

00:03:02.490 --> 00:03:06.240
And then finally, plus
that term squared.

00:03:06.240 --> 00:03:15.560
So plus a1 b2 minus
a2 b1 squared.

00:03:15.560 --> 00:03:16.670
And what is this equal to?

00:03:16.670 --> 00:03:18.590
Well let's just expand it out.

00:03:18.590 --> 00:03:19.930
Let's expand that out.

00:03:19.930 --> 00:03:25.070
So this term right here, we're
just going to have to do our

00:03:25.070 --> 00:03:27.550
expansion of the square
of a binomial.

00:03:27.550 --> 00:03:28.960
And we've done this
multiple times.

00:03:28.960 --> 00:03:35.120
So this is going to be equal
to a2 squared b3 squared.

00:03:35.120 --> 00:03:37.180
And then we're going to have
these two multiplied by each

00:03:37.180 --> 00:03:38.230
other twice.

00:03:38.230 --> 00:03:40.030
So minus 2.

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I'm just multiplying this out.

00:03:41.150 --> 00:03:46.740
Minus 2 times a2 a3 b2 b3.

00:03:46.740 --> 00:03:49.460
I'm just rearranging them
to get the order right.

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Plus a3 squared b2 squared.

00:03:54.620 --> 00:03:55.870
That term squared.

00:03:58.200 --> 00:03:59.500
Then I have to add this term.

00:03:59.500 --> 00:04:08.490
So plus a3 squared b1 squared
minus 2 times both of these

00:04:08.490 --> 00:04:09.370
terms multiplied.

00:04:09.370 --> 00:04:16.019
Minus 2 times a1 a3 b1 b3.

00:04:16.019 --> 00:04:17.620
Plus that term squared.

00:04:17.620 --> 00:04:21.470
a1 squared b3 squared.

00:04:21.470 --> 00:04:24.190
And then finally, this
term squared.

00:04:24.190 --> 00:04:37.240
So plus a1 squared b2 squared
minus 2 times a1 a2 b1 b2.

00:04:37.240 --> 00:04:42.080
Plus a2 squared b1 squared.

00:04:42.080 --> 00:04:42.840
So there you go.

00:04:42.840 --> 00:04:45.450
And let's see if we can write
this in a form-- well, I'm

00:04:45.450 --> 00:04:47.090
going to write this in
a form that I know

00:04:47.090 --> 00:04:48.170
will be useful later.

00:04:48.170 --> 00:04:51.880
So what I'm going to do is I'm
going to factor out the a2,

00:04:51.880 --> 00:04:56.090
a1, a3 squared terms. So I could
write this as-- let me

00:04:56.090 --> 00:04:57.360
pick a new neutral color.

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So this is equal to, if I just
write a1 squared, where's my

00:05:01.940 --> 00:05:03.610
a1 squared terms?

00:05:03.610 --> 00:05:07.070
I got that one right there and
I have that one right there.

00:05:07.070 --> 00:05:14.720
So a1 squared times b2 squared
plus b3 squared.

00:05:14.720 --> 00:05:17.300
This would be 3 squared.

00:05:17.300 --> 00:05:18.180
Good enough.

00:05:18.180 --> 00:05:20.370
Now where are my a2
squared terms?

00:05:20.370 --> 00:05:25.800
a plus a2 squared times-- I have
that one and that one.

00:05:25.800 --> 00:05:28.470
So times b1 squared.

00:05:28.470 --> 00:05:29.050
That's that.

00:05:29.050 --> 00:05:30.300
Plus b3 squared.

00:05:33.480 --> 00:05:37.220
And then finally, let me
pick another new color.

00:05:37.220 --> 00:05:38.870
I can go back to yellow.

00:05:38.870 --> 00:05:42.690
Plus a3 squared times--
well that's that

00:05:42.690 --> 00:05:43.970
term and that term.

00:05:43.970 --> 00:05:45.510
So b1 and b2.

00:05:45.510 --> 00:05:49.830
So b1 squared plus b2 squared.

00:05:49.830 --> 00:05:52.760
And obviously, I can't forget
about all of that mess that I

00:05:52.760 --> 00:05:55.450
have in the middle, all of
this stuff right here.

00:05:58.470 --> 00:06:03.680
So plus, or maybe I should write
minus 2 times all of

00:06:03.680 --> 00:06:04.250
this stuff.

00:06:04.250 --> 00:06:11.580
Let me just write it real fast.
So it's a2 a3 b2 b3 plus

00:06:11.580 --> 00:06:17.340
a1 a3 b1 b3.

00:06:17.340 --> 00:06:24.940
Plus a1 a2 b1 b2.

00:06:24.940 --> 00:06:25.440
There you go.

00:06:25.440 --> 00:06:30.510
Now let's put this aside
for a little bit.

00:06:30.510 --> 00:06:34.540
Let me put this on the side
for a little bit.

00:06:34.540 --> 00:06:37.150
We'll let that equation rest
for a little while.

00:06:37.150 --> 00:06:40.990
And remember, this is just an
expansion of the length of a

00:06:40.990 --> 00:06:42.900
cross b squared.

00:06:42.900 --> 00:06:44.040
That's all this is.

00:06:44.040 --> 00:06:45.330
So just remember that.

00:06:45.330 --> 00:06:48.660
And now, let's do another
equally hairy and cumbersome

00:06:48.660 --> 00:06:50.350
computation.

00:06:50.350 --> 00:06:53.990
Let's take this result
up here.

00:06:53.990 --> 00:06:58.150
We know that the magnitude or
the length of a times the

00:06:58.150 --> 00:07:01.880
length of b times the angle
between them is

00:07:01.880 --> 00:07:04.680
equal to a dot b.

00:07:04.680 --> 00:07:07.860
Which is the same thing as
if we actually do the dot

00:07:07.860 --> 00:07:17.670
product, a1 times b1 plus a2
times b2 plus a3 times b3.

00:07:17.670 --> 00:07:22.830
Now, just to kind of make sure
that I get to do the hairiest

00:07:22.830 --> 00:07:27.800
problem possible, let's take
the square of both sides.

00:07:27.800 --> 00:07:30.600
If we square this side,
you get a squared b

00:07:30.600 --> 00:07:32.630
squared cosine squared.

00:07:32.630 --> 00:07:35.260
Then you got a dot b
squared or you get

00:07:35.260 --> 00:07:37.710
this whole thing squared.

00:07:37.710 --> 00:07:39.700
So what's this whole
thing squared?

00:07:39.700 --> 00:07:41.980
For me, it's easier to just
write out the thing again.

00:07:41.980 --> 00:07:47.020
Instead of writing a square,
just multiply that times a1 b1

00:07:47.020 --> 00:07:52.050
plus a2 b2 plus a3 b3.

00:07:52.050 --> 00:07:56.250
And let's do some polynomial
multiplication.

00:07:56.250 --> 00:07:59.270
So first, let's multiply this
guy times each of those guys.

00:07:59.270 --> 00:08:03.380
So you have a1 b1 times--
well there a1 b1.

00:08:03.380 --> 00:08:05.320
I'm going to do it right here.

00:08:05.320 --> 00:08:11.620
You get a1 squared b1 squared
plus a1-- plus this

00:08:11.620 --> 00:08:13.290
guy times this guy.

00:08:13.290 --> 00:08:22.650
Plus a1 a2 times b1 b2.

00:08:22.650 --> 00:08:24.530
Plus this guy times that guy.

00:08:24.530 --> 00:08:31.870
Plus a1 a3 times b1 b3.

00:08:31.870 --> 00:08:33.120
Fair enough.

00:08:33.120 --> 00:08:35.250
Now let's do the second term.

00:08:35.250 --> 00:08:37.960
We have to multiply this guy
times each of those guys.

00:08:37.960 --> 00:08:40.960
So a2 b2 times a1 b1.

00:08:40.960 --> 00:08:42.870
Well that's this
one right here.

00:08:42.870 --> 00:08:50.560
a2 b2 times a1 b1.

00:08:50.560 --> 00:08:52.700
I wrote it right here because
this is really the same term

00:08:52.700 --> 00:08:54.790
and eventually, we want
to simplify that.

00:08:54.790 --> 00:08:56.430
So that's that times that guy.

00:08:56.430 --> 00:08:59.990
Then we have this guy times
that over there.

00:08:59.990 --> 00:09:01.340
So let me write it over here.

00:09:01.340 --> 00:09:07.860
So that's a2 squared
b2 squared.

00:09:07.860 --> 00:09:09.740
Put a plus right there.

00:09:09.740 --> 00:09:13.910
And then finally, this middle
guy times this third guy.

00:09:13.910 --> 00:09:15.400
So let me write it over here.

00:09:15.400 --> 00:09:27.410
Plus-- so a2 a3 b2 b3.

00:09:27.410 --> 00:09:29.063
Now, we only have one left.

00:09:33.020 --> 00:09:34.940
Maybe I'll do it in
this blue color.

00:09:34.940 --> 00:09:37.620
I have to multiply this guy
times each of those guys.

00:09:37.620 --> 00:09:40.840
So a3 b3 times a1 b1.

00:09:40.840 --> 00:09:43.030
That's the same thing as
this term right here.

00:09:43.030 --> 00:09:45.240
Because you have a3.

00:09:45.240 --> 00:09:46.170
Let me write it right here.

00:09:46.170 --> 00:09:51.730
You have a3 b3 times a1 b1.

00:09:54.480 --> 00:09:57.150
Then you have this guy times
that guy, which is this

00:09:57.150 --> 00:10:04.710
because it's a3 b3
times a2 b2.

00:10:04.710 --> 00:10:06.390
Let me put a little
plus sign there.

00:10:06.390 --> 00:10:08.940
And then finally, you have
this guy times himself.

00:10:08.940 --> 00:10:14.170
So you have a3 squared
b3 squared.

00:10:14.170 --> 00:10:16.630
And so if you add up all
of this business

00:10:16.630 --> 00:10:17.880
here what do you get?

00:10:21.420 --> 00:10:24.050
I'll switch to another color.

00:10:24.050 --> 00:10:30.870
You have a1 squared
b1 squared.

00:10:30.870 --> 00:10:33.180
Plus, and I'm doing these colors
in a certain way on

00:10:33.180 --> 00:10:37.570
purpose, plus a2 squared
b2 squared.

00:10:40.670 --> 00:10:45.800
Plus a3 squared b3 squared.

00:10:45.800 --> 00:10:49.950
Plus, and let me do it in this,
I'll do it in white.

00:10:49.950 --> 00:10:51.250
Plus, what do you have here?

00:10:51.250 --> 00:10:53.640
You have this term times 2.

00:10:53.640 --> 00:10:56.880
You have this term times 2.

00:10:56.880 --> 00:10:58.700
And then you have this
term times 2.

00:10:58.700 --> 00:11:03.970
So plus 2 times a1-- let
me write that down.

00:11:03.970 --> 00:11:11.490
Plus 2 times a1 a2 b1 b2--
that's that term.

00:11:11.490 --> 00:11:12.920
Plus this one right here.

00:11:12.920 --> 00:11:18.040
Plus a1 a3 b1 b3.

00:11:18.040 --> 00:11:19.790
Finally, plus this one.

00:11:19.790 --> 00:11:24.860
a2 a3 plus b2 b3.

00:11:24.860 --> 00:11:28.000
And you might have noticed
something interesting already.

00:11:28.000 --> 00:11:32.640
If you compare this term right
here, if you compare that guy

00:11:32.640 --> 00:11:36.600
right there to this guy right
there, they're the same thing.

00:11:36.600 --> 00:11:40.260
You have an a1 a2 b1
b2, a1 a2 b1 b2.

00:11:40.260 --> 00:11:43.340
This term and that term
are the same.

00:11:43.340 --> 00:11:46.200
Let's look at the other terms.
Let me pick a nice color.

00:11:46.200 --> 00:11:49.300
a1 a3 b1 b3, a1 a3 b1 b3.

00:11:49.300 --> 00:11:52.140
That term and that
term is the same.

00:11:52.140 --> 00:11:57.130
And then finally, if you
compare a2 a3 b2 b3.

00:11:57.130 --> 00:11:57.920
This shouldn't be a plus.

00:11:57.920 --> 00:11:58.760
This is just this one.

00:11:58.760 --> 00:12:02.240
So a2 a3-- they're
all multiplied.

00:12:02.240 --> 00:12:05.530
a2 a3 b2 b3, a2 a3 b2 b3.

00:12:05.530 --> 00:12:08.020
This term and this
term is the same.

00:12:08.020 --> 00:12:10.850
And in this expression, when we
expanded it out, we have 2

00:12:10.850 --> 00:12:12.840
times this, positive
2 times this.

00:12:12.840 --> 00:12:15.780
And this term right here when
we expanded it out, we have

00:12:15.780 --> 00:12:19.190
minus 2 times this.

00:12:19.190 --> 00:12:22.220
So let's see if we can simplify
things a little bit.

00:12:22.220 --> 00:12:27.490
So what happens if we add
this guy to this guy?

00:12:27.490 --> 00:12:29.780
Let's do it.

00:12:29.780 --> 00:12:31.740
So it's a little exciting.

00:12:31.740 --> 00:12:37.150
So we get a cross b, the
length of that squared.

00:12:37.150 --> 00:12:40.310
And we're going to add to that
this expression right here, so

00:12:40.310 --> 00:12:47.130
plus the length of a squared
times the length of b squared

00:12:47.130 --> 00:12:50.180
times the cosine of the angle
between them squared.

00:12:50.180 --> 00:12:51.490
What's that going to equal?

00:12:51.490 --> 00:12:55.610
It's going to be equal to this
thing plus this thing.

00:12:55.610 --> 00:12:56.810
Let's do a simplification.

00:12:56.810 --> 00:12:58.680
What's this thing
plus this thing?

00:12:58.680 --> 00:13:01.560
Well we already said this is
the minus 2 times this.

00:13:01.560 --> 00:13:03.650
This is the plus 2 times this.

00:13:03.650 --> 00:13:07.000
So this guy-- let me
be very clear.

00:13:07.000 --> 00:13:10.200
This right here is going
to cancel out.

00:13:10.200 --> 00:13:12.360
When we add the two terms
it's going to cancel

00:13:12.360 --> 00:13:13.170
out with this guy.

00:13:13.170 --> 00:13:14.740
These guys are going
to cancel out.

00:13:14.740 --> 00:13:17.750
Thank God.

00:13:17.750 --> 00:13:18.550
Cancel out.

00:13:18.550 --> 00:13:19.730
Makes our life a little
bit easier.

00:13:19.730 --> 00:13:20.620
And what are we left with?

00:13:20.620 --> 00:13:25.810
We're left with this right here
plus that right there.

00:13:25.810 --> 00:13:28.160
Then we see we have an a1
squared term, so we just add

00:13:28.160 --> 00:13:29.690
the coefficients on
the a1 squared.

00:13:29.690 --> 00:13:31.300
We add the coefficients
of the a2 squared.

00:13:31.300 --> 00:13:32.900
We add the coefficients
of the a3 squared,

00:13:32.900 --> 00:13:34.280
and what do we get?

00:13:34.280 --> 00:13:41.560
We get a1 squared times this
coefficient plus this

00:13:41.560 --> 00:13:42.340
coefficient.

00:13:42.340 --> 00:13:48.040
So you get b1 squared plus b2
squared plus b3 squared.

00:13:48.040 --> 00:13:49.700
Things are starting to
look a little bit

00:13:49.700 --> 00:13:51.470
orderly all of a sudden.

00:13:51.470 --> 00:13:57.930
And then you have plus a2
squared times all of their

00:13:57.930 --> 00:13:58.900
coefficients added up.

00:13:58.900 --> 00:14:05.300
So b1 squared plus b2 squared
plus b3 squared.

00:14:05.300 --> 00:14:11.470
And then finally, in yellow,
you have plus a3-- sorry, I

00:14:11.470 --> 00:14:14.240
was trying to do
that in yellow.

00:14:14.240 --> 00:14:18.180
You have a3 squared
and you have that.

00:14:18.180 --> 00:14:21.410
You have b1 squared, b2 squared
and b3 squared.

00:14:21.410 --> 00:14:26.540
So b1 squared plus b2 squared
plus b3 squared.

00:14:26.540 --> 00:14:28.240
And as you see, we're
multiplying all of these

00:14:28.240 --> 00:14:31.543
things by this b1 squared plus
b2 squared plus b3 squared, so

00:14:31.543 --> 00:14:34.520
we can actually factor that out
and we get something very

00:14:34.520 --> 00:14:35.140
interesting.

00:14:35.140 --> 00:14:40.260
So this is equal to, if we
factor this thing out of all

00:14:40.260 --> 00:14:45.340
the terms, we get b1 squared
plus b2 squared plus b3

00:14:45.340 --> 00:14:51.010
squared times my a squared
terms. Times a1 squared plus

00:14:51.010 --> 00:14:55.250
a2 squared plus a-- I'm getting
excited, the home

00:14:55.250 --> 00:14:58.580
stretch is here-- a3 squared.

00:14:58.580 --> 00:15:01.360
So these two things are
equal to each other.

00:15:01.360 --> 00:15:02.270
But what's this thing?

00:15:02.270 --> 00:15:04.320
What's another way I
could write this?

00:15:04.320 --> 00:15:09.960
This is the same thing as b
dot b or the length of my

00:15:09.960 --> 00:15:11.720
vector b squared.

00:15:11.720 --> 00:15:12.980
And what's that?

00:15:12.980 --> 00:15:15.460
That's the length of my
vector a squared.

00:15:15.460 --> 00:15:18.850
This is my length of my
vector a squared.

00:15:18.850 --> 00:15:21.740
That's just a dot a.

00:15:21.740 --> 00:15:24.050
Let me rewrite everything.

00:15:24.050 --> 00:15:29.230
So we have the length of a--
that's a darker green.

00:15:29.230 --> 00:15:34.150
a cross b squared
plus this thing.

00:15:34.150 --> 00:15:40.320
Plus the length of-- plus,
let me actually just

00:15:40.320 --> 00:15:41.400
copy and paste it.

00:15:41.400 --> 00:15:42.720
It gets monotonous.

00:15:42.720 --> 00:15:44.370
Plus that thing right there.

00:15:47.620 --> 00:15:49.535
Why isn't it?

00:15:49.535 --> 00:15:52.010
If I control, copy and paste.

00:15:52.010 --> 00:15:53.370
So it's not working.

00:15:53.370 --> 00:15:57.520
All right, so plus that thing.

00:15:57.520 --> 00:16:01.200
The length of a squared times
the length of b squared times

00:16:01.200 --> 00:16:03.560
the cosine of the angle
squared between

00:16:03.560 --> 00:16:04.950
them is equal to that.

00:16:04.950 --> 00:16:10.120
Now what if we subtract
this from both sides?

00:16:10.120 --> 00:16:11.140
What do we get?

00:16:11.140 --> 00:16:17.720
We get the length of a cross
b squared is equal

00:16:17.720 --> 00:16:19.720
to this minus this.

00:16:19.720 --> 00:16:22.450
And we can factor-- so
let me write that.

00:16:22.450 --> 00:16:24.240
Actually, let me just subtract
this on this line.

00:16:24.240 --> 00:16:27.680
So if I subtract it from both
sides, I could get that out

00:16:27.680 --> 00:16:32.220
there and I'll put the minus the
length of a squared times

00:16:32.220 --> 00:16:36.430
the length of b squared times
the cosine squared of the

00:16:36.430 --> 00:16:37.680
angle between them.

00:16:37.680 --> 00:16:41.200
And we can factor this a squared
b squared, the lengths

00:16:41.200 --> 00:16:42.680
of the two vectors out, right?

00:16:42.680 --> 00:16:43.630
I'm just switching the order.

00:16:43.630 --> 00:16:49.170
So this is equal to the lengths
of a squared times the

00:16:49.170 --> 00:16:56.150
length of b squared times-- and
this is exciting-- time,

00:16:56.150 --> 00:17:00.100
when you factor this out of
this, you just get a 1, minus

00:17:00.100 --> 00:17:03.120
cosine squared of theta.

00:17:03.120 --> 00:17:06.380
And what is 1 minus cosine
squared of theta?

00:17:06.380 --> 00:17:09.290
Well, sin squared of theta plus
cosine-- this is the most

00:17:09.290 --> 00:17:10.460
basic trig identity.

00:17:10.460 --> 00:17:15.290
Sin squared of theta plus cosine
squared of theta is

00:17:15.290 --> 00:17:16.440
equal to 1.

00:17:16.440 --> 00:17:19.240
So if you subtract cosine
squared from both sides you

00:17:19.240 --> 00:17:23.140
get sin squared of theta is
equal to 1 minus cosine

00:17:23.140 --> 00:17:24.010
squared of theta.

00:17:24.010 --> 00:17:29.310
So this is sin squared
of theta.

00:17:29.310 --> 00:17:30.650
And then what happens if
you take the square

00:17:30.650 --> 00:17:32.510
root of both sides?

00:17:32.510 --> 00:17:34.250
This is really exciting.

00:17:34.250 --> 00:17:40.530
You get the length of vector a
crossed with vector b is equal

00:17:40.530 --> 00:17:46.450
to the length of vector a times
the length of vector b

00:17:46.450 --> 00:17:50.120
times the sin of the
angle between them.

00:17:50.120 --> 00:17:51.790
I just took the square root
of both sides of this.

00:17:51.790 --> 00:17:53.460
And we finally get our result.

00:17:53.460 --> 00:17:55.340
I never thought I
would get here.

00:17:55.340 --> 00:17:56.820
And so hopefully you're
satisfied.

00:17:56.820 --> 00:17:58.200
You never have to take
this as kind of a

00:17:58.200 --> 00:17:59.690
leap of faith anymore.

00:17:59.690 --> 00:18:01.470
And hopefully you're satisfied
with this.

00:18:01.470 --> 00:18:04.020
And I'm going to stop recording
this video before I

00:18:04.020 --> 00:18:06.410
make a careless mistake or the
power goes out; that would

00:18:06.410 --> 00:18:08.000
ruin everything.


Cross product introduction

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:05.050
We've learned a good bit
about the dot product.

00:00:05.050 --> 00:00:07.110
But when I first introduced it,
I mentioned that this was

00:00:07.110 --> 00:00:10.730
only one type of vector
multiplication, and the other

00:00:10.730 --> 00:00:14.150
type is the cross product, which
you're probably familiar

00:00:14.150 --> 00:00:16.840
with from your vector calculus
course or from

00:00:16.840 --> 00:00:18.090
your physics course.

00:00:21.120 --> 00:00:24.030
But the cross product is
actually much more limited

00:00:24.030 --> 00:00:24.940
than the dot product.

00:00:24.940 --> 00:00:27.280
It's useful, but it's
much more limited.

00:00:27.280 --> 00:00:32.970
The dot product is defined
in any dimension.

00:00:32.970 --> 00:00:39.810
So this is defined for any two
vectors that are in Rn.

00:00:39.810 --> 00:00:42.160
You could take the dot product
of vectors that have two

00:00:42.160 --> 00:00:42.650
components.

00:00:42.650 --> 00:00:45.020
You could take the dot product
of vectors that have a million

00:00:45.020 --> 00:00:45.840
components.

00:00:45.840 --> 00:00:55.430
The cross product is
only defined in R3.

00:00:55.430 --> 00:00:58.960
And the other, I guess, major
difference is the dot produc,

00:00:58.960 --> 00:01:01.350
and we're going to see this in a
second when I define the dot

00:01:01.350 --> 00:01:03.370
product for you, I haven't
defined it yet.

00:01:03.370 --> 00:01:06.140
The dot product results
in a scalar.

00:01:06.140 --> 00:01:07.850
You take the dot product
of two vectors,

00:01:07.850 --> 00:01:09.240
you just get a number.

00:01:09.240 --> 00:01:10.960
But in the cross product you're
going to see that we're

00:01:10.960 --> 00:01:13.130
going to get another vector.

00:01:13.130 --> 00:01:15.185
And the vector we're going to
get is actually going to be a

00:01:15.185 --> 00:01:17.570
vector that's orthogonal to
the two vectors that we're

00:01:17.570 --> 00:01:19.230
taking the cross product of.

00:01:19.230 --> 00:01:23.580
So now that I have you excited
with anticipation, let me

00:01:23.580 --> 00:01:24.370
define it for you.

00:01:24.370 --> 00:01:27.300
And you probably already have
seen this once or twice in

00:01:27.300 --> 00:01:28.960
your mathematical careers.

00:01:28.960 --> 00:01:32.200
Let's say I have the vector a.

00:01:32.200 --> 00:01:33.680
It has to be in R3, so
it only has three

00:01:33.680 --> 00:01:37.510
components: a1, a2 and a3.

00:01:37.510 --> 00:01:41.050
And I'm going to cross that with
the vector b and it has

00:01:41.050 --> 00:01:46.910
three components:
b1, b2 and b3.

00:01:46.910 --> 00:01:54.650
a cross b is defined
as a third vector.

00:01:54.650 --> 00:01:57.360
And now this is going to seem a
little bit bizarre and hard

00:01:57.360 --> 00:02:00.500
to essentially memorize because
this is a definition.

00:02:00.500 --> 00:02:03.120
But I'll show you how I think
about it when I have my

00:02:03.120 --> 00:02:05.220
vectors written in
this column form.

00:02:05.220 --> 00:02:07.960
If you watch the physics
playlist, I have a bunch of

00:02:07.960 --> 00:02:11.400
videos on the cross product
and I show you how I think

00:02:11.400 --> 00:02:15.070
about the cross product when I
have it in the i, j, k form.

00:02:15.070 --> 00:02:17.620
But when I have it like this,
the way you think about this

00:02:17.620 --> 00:02:20.880
first term up here, this is
going to be another three

00:02:20.880 --> 00:02:23.550
vector or another vector in R3,
so it's going to have 1,

00:02:23.550 --> 00:02:25.320
2, 3 terms.

00:02:25.320 --> 00:02:28.570
For the first term, what you do
is you ignore these top two

00:02:28.570 --> 00:02:32.330
terms of this vector and then
you look at the bottom two and

00:02:32.330 --> 00:02:37.670
you say, a2 times b3
minus a3 times b2.

00:02:40.800 --> 00:02:42.790
And I've made a few videos on
determinants, although I

00:02:42.790 --> 00:02:45.500
haven't formally done them in
kind of this linear algebra

00:02:45.500 --> 00:02:46.490
playlist yet.

00:02:46.490 --> 00:02:49.260
But if you remember kind of
co-factor-- finding out the

00:02:49.260 --> 00:02:52.410
co-factor terms for when
you're determining the

00:02:52.410 --> 00:02:54.780
determinant or if you're just
taking the determinant for a

00:02:54.780 --> 00:02:57.260
2x2 matrix, this might
seem very familiar.

00:02:57.260 --> 00:03:00.740
So this first term right here is
essentially the determinant

00:03:00.740 --> 00:03:05.770
of-- if you get rid of this
first row out of both of these

00:03:05.770 --> 00:03:11.370
guys right here, you take a2
times b3 minus a3 times b2.

00:03:11.370 --> 00:03:19.890
So it's a2 times b3
minus a3 times b2.

00:03:19.890 --> 00:03:22.110
That was hopefully pretty
straightforward.

00:03:22.110 --> 00:03:25.600
Now, not to make your life any
more complicated, when you do

00:03:25.600 --> 00:03:28.100
the second, when you do the
middle row, when you do this

00:03:28.100 --> 00:03:30.150
one right here, so you
cross that out.

00:03:30.150 --> 00:03:36.150
And you might want to do a1
times b3 minus a3 times b1.

00:03:36.150 --> 00:03:37.380
And that would be natural
because that's

00:03:37.380 --> 00:03:38.380
what we did up there.

00:03:38.380 --> 00:03:41.800
But the middle row you
do the opposite.

00:03:41.800 --> 00:03:46.310
You do a3 times b1 minus
a1 times b3.

00:03:46.310 --> 00:03:48.890
Or you can kind of view it as
the negative of what you would

00:03:48.890 --> 00:03:50.180
have done naturally.

00:03:50.180 --> 00:03:54.310
So you would have done
a1 b3 minus a3 b1.

00:03:54.310 --> 00:04:02.490
Now we're going to do
a3 b1 minus a1 b3.

00:04:06.140 --> 00:04:07.640
And that was only for
that middle row.

00:04:07.640 --> 00:04:11.120
And then, for the bottom
row, we cross that out

00:04:11.120 --> 00:04:12.300
again or ignore it.

00:04:12.300 --> 00:04:15.490
And we do a1 times b2, just like
we do with the first row.

00:04:19.000 --> 00:04:20.980
Times a2 b1.

00:04:20.980 --> 00:04:24.930
Or minus a2 b1.

00:04:24.930 --> 00:04:27.920
This seems all hard to-- and
it is hard to remember.

00:04:27.920 --> 00:04:30.280
That's why I kind of have to get
that system in place like

00:04:30.280 --> 00:04:31.130
I just talked to you about.

00:04:31.130 --> 00:04:33.650
But this might seem pretty
bizarre and hairy.

00:04:33.650 --> 00:04:35.870
So let me do a couple of
examples with you, just so you

00:04:35.870 --> 00:04:40.390
get the hang of our definition
of the dot product in R3.

00:04:40.390 --> 00:04:44.200
So let's say that I have the
vector-- let's say I'm

00:04:44.200 --> 00:04:46.950
crossing the vector.

00:04:46.950 --> 00:04:50.730
I have the vector 1,
minus 7, and 1.

00:04:50.730 --> 00:04:58.180
And I'm going to cross that
with the vector 5, 2, 4.

00:04:58.180 --> 00:05:00.475
So this is going to be equal
to a third vector.

00:05:03.810 --> 00:05:06.920
Let me get some space to
do my mathematics.

00:05:06.920 --> 00:05:10.170
So for the first element in
this vector, the first

00:05:10.170 --> 00:05:12.950
component, we just ignore the
first components of these

00:05:12.950 --> 00:05:20.610
vectors and we say minus 7
times 4 minus 1 times 2.

00:05:23.180 --> 00:05:24.680
And these are just regular
multiplication.

00:05:24.680 --> 00:05:25.830
I'm not taking the
dot product.

00:05:25.830 --> 00:05:27.570
These are just regular
numbers.

00:05:27.570 --> 00:05:31.050
Then for the middle term, we
ignore the middle terms here

00:05:31.050 --> 00:05:32.660
and then we do the opposite.

00:05:32.660 --> 00:05:39.270
We do 1 times 5 minus
1 times 4.

00:05:42.120 --> 00:05:45.410
Remember, you might have been
tempted to do 1 times 4 minus

00:05:45.410 --> 00:05:48.380
1 times 5 because that's how we
essentially did it in the

00:05:48.380 --> 00:05:48.900
first term.

00:05:48.900 --> 00:05:51.240
But the middle term
is the opposite.

00:05:51.240 --> 00:05:54.290
And then finally, the third
term you ignore the third

00:05:54.290 --> 00:05:56.030
terms here and then you do it
just like the first term.

00:05:56.030 --> 00:05:57.260
You start in the top left.

00:05:57.260 --> 00:06:04.340
1 times 2 minus 7.

00:06:04.340 --> 00:06:06.240
Put that in parentheses.

00:06:06.240 --> 00:06:09.500
Minus minus 7 times 5.

00:06:09.500 --> 00:06:15.070
And so that is equal
to-- let me see.

00:06:15.070 --> 00:06:16.500
What do we get?

00:06:16.500 --> 00:06:18.580
Minus 7 times 4.

00:06:18.580 --> 00:06:19.890
I don't want to make a careless
mistake here.

00:06:19.890 --> 00:06:22.960
That's minus 28.

00:06:22.960 --> 00:06:25.880
Minus 2.

00:06:25.880 --> 00:06:29.490
So this is minus 30 for
that first term.

00:06:29.490 --> 00:06:31.540
This one is 5 minus 4.

00:06:36.500 --> 00:06:38.530
5 minus 4 is just 1.

00:06:38.530 --> 00:06:43.430
And then 2 minus minus 35.

00:06:43.430 --> 00:06:45.680
So 2 minus minus 35.

00:06:45.680 --> 00:06:47.690
That's 2 plus 35.

00:06:47.690 --> 00:06:49.370
That's 37.

00:06:49.370 --> 00:06:50.220
So there you go.

00:06:50.220 --> 00:06:52.280
Hopefully you understand at
least the mechanics of the

00:06:52.280 --> 00:06:53.400
cross product.

00:06:53.400 --> 00:06:54.950
So the next thing you're
saying well, OK.

00:06:54.950 --> 00:06:58.000
I can find the cross product
of two things.

00:06:58.000 --> 00:06:59.790
But what is this good for?

00:06:59.790 --> 00:07:01.400
What does this do for me?

00:07:01.400 --> 00:07:04.580
And the answer is, is that this
third vector right here,

00:07:04.580 --> 00:07:07.170
and depending on whether I stay
in the abstract case or

00:07:07.170 --> 00:07:10.600
whether this case with numbers,
this is orthogonal to

00:07:10.600 --> 00:07:13.680
the two vectors that we took
the cross product of.

00:07:13.680 --> 00:07:21.920
So this vector right here is
orthogonal to a and b.

00:07:25.810 --> 00:07:27.330
Which is pretty neat.

00:07:27.330 --> 00:07:29.210
If you just go think about the
last video when we were

00:07:29.210 --> 00:07:33.250
talking about normal vectors
to a plane, we can define a

00:07:33.250 --> 00:07:36.080
vector by-- we can define
a plane by two vectors.

00:07:36.080 --> 00:07:40.130
If we define a plane-- let's say
that I have vector a right

00:07:40.130 --> 00:07:45.020
there, and then I
have vector b.

00:07:45.020 --> 00:07:49.680
Let me do vector b like this.

00:07:49.680 --> 00:07:52.650
Those define a plane in R3.

00:07:52.650 --> 00:07:54.510
Let me define your plane.

00:07:54.510 --> 00:07:57.530
So all the linear combinations
of those two guys, that's a

00:07:57.530 --> 00:08:00.010
plane in R3.

00:08:00.010 --> 00:08:02.370
You can kind of view it
as they might form

00:08:02.370 --> 00:08:03.940
a subspace in R3.

00:08:03.940 --> 00:08:05.630
That forms a plane.

00:08:05.630 --> 00:08:09.070
If you take a cross b, you
get a third vector that's

00:08:09.070 --> 00:08:12.840
orthogonal to those two.

00:08:12.840 --> 00:08:16.070
And so a cross b will
pop out like this.

00:08:16.070 --> 00:08:19.310
It'll be orthogonal to both of
them and look like that.

00:08:19.310 --> 00:08:22.940
And so this vector right
there is a cross b.

00:08:22.940 --> 00:08:25.500
And you might say, Sal, how did
you know-- I mean, there's

00:08:25.500 --> 00:08:27.070
multiple vectors that
are orthogonal.

00:08:27.070 --> 00:08:29.960
Obviously, the length of the
vector, and I didn't specify

00:08:29.960 --> 00:08:32.900
that there, but it could pop
straight up like that or why

00:08:32.900 --> 00:08:34.630
didn't it-- you know, you just
as easily could have popped

00:08:34.630 --> 00:08:35.610
straight down like that.

00:08:35.610 --> 00:08:38.130
That also would be orthogonal
to a and b.

00:08:38.130 --> 00:08:42.190
And the way that a cross b is
defined, you can essentially

00:08:42.190 --> 00:08:44.930
figure out the direction
visually by using what's

00:08:44.930 --> 00:08:46.390
called the right hand rule.

00:08:46.390 --> 00:08:51.880
And the way I think about it
is you take your right hand

00:08:51.880 --> 00:08:55.290
and let me see if I can draw
a suitable right hand.

00:08:55.290 --> 00:08:58.320
Point your index finger
in the direction of a.

00:08:58.320 --> 00:09:01.910
So if your index finger is in
the direction of a and then I

00:09:01.910 --> 00:09:04.340
point my middle finger in
the direction of b.

00:09:04.340 --> 00:09:06.150
So my middle finger, in this
case, is going to go

00:09:06.150 --> 00:09:07.370
something like that.

00:09:07.370 --> 00:09:09.250
My middle finger is going to
do something like that.

00:09:09.250 --> 00:09:11.560
And then my other fingers
do nothing.

00:09:11.560 --> 00:09:18.780
Then my thumb will go in the
direction of a cross b.

00:09:18.780 --> 00:09:19.710
You could see that there.

00:09:19.710 --> 00:09:24.750
My thumb is in the direction
of a cross b.

00:09:24.750 --> 00:09:28.890
And assuming that you are
anatomically similar to me,

00:09:28.890 --> 00:09:30.860
then you still get
the same result.

00:09:30.860 --> 00:09:32.270
Let me draw it all.

00:09:32.270 --> 00:09:34.810
So this is vector a.

00:09:34.810 --> 00:09:39.320
Vector b goes in
that direction.

00:09:39.320 --> 00:09:41.010
Hopefully you don't have a
thumb hanging down here.

00:09:41.010 --> 00:09:44.480
You know that a cross b in this
example will point up and

00:09:44.480 --> 00:09:48.160
it's orthogonal to both.

00:09:48.160 --> 00:09:50.740
To kind of satisfy you a little
bit, that the vector's

00:09:50.740 --> 00:09:53.570
definitely orthogonal or that
this thing is definitely

00:09:53.570 --> 00:09:55.720
orthogonal to both of these,
let's just play with it and

00:09:55.720 --> 00:09:57.960
see that that definitely
is the case.

00:09:57.960 --> 00:10:00.500
And what is orthogonal?

00:10:00.500 --> 00:10:04.490
What is in our context, the
definition of orthogonal?

00:10:04.490 --> 00:10:05.740
Orthogonal vectors.

00:10:09.240 --> 00:10:15.040
If a and b are orthogonal, that
means that a dot b is

00:10:15.040 --> 00:10:15.770
equal to 0.

00:10:15.770 --> 00:10:17.490
Remember, the difference
between orthogonal and

00:10:17.490 --> 00:10:20.090
perpendicular is that
orthogonal also

00:10:20.090 --> 00:10:21.340
applies to 0 vectors.

00:10:21.340 --> 00:10:23.550
So these could also
be 0 vectors.

00:10:23.550 --> 00:10:26.910
Notice that I didn't say that
any of these guys up here had

00:10:26.910 --> 00:10:28.140
to be nonzero.

00:10:28.140 --> 00:10:30.450
Well, in a little bit, we'll
talk about the angle between

00:10:30.450 --> 00:10:32.280
vectors and then you have
to assume nonzero.

00:10:32.280 --> 00:10:34.290
But if you're just taking a
cross product, nothing to stop

00:10:34.290 --> 00:10:36.580
you from taking-- no reason
why any of these

00:10:36.580 --> 00:10:38.180
numbers can't be 0.

00:10:38.180 --> 00:10:43.530
But let me show you that
a cross b is definitely

00:10:43.530 --> 00:10:45.710
orthogonal to both a and b.

00:10:45.710 --> 00:10:49.220
I think that might be somewhat
satisfying to you.

00:10:49.220 --> 00:10:52.780
So let me copy a cross b here.

00:10:52.780 --> 00:10:56.710
I don't feel like
rewriting it.

00:10:56.710 --> 00:10:58.250
OK.

00:10:58.250 --> 00:11:00.500
Let me paste it.

00:11:00.500 --> 00:11:03.310
OK, bundle up little other
stuff with it.

00:11:06.940 --> 00:11:12.800
Let me take the dot product of
that with just my vector a,

00:11:12.800 --> 00:11:17.730
which was just a1, a2, a3.

00:11:17.730 --> 00:11:19.530
So what does the dot
product look like?

00:11:19.530 --> 00:11:24.260
That term times that, so it's
a1-- let me get some space.

00:11:24.260 --> 00:11:31.710
It's a1 times a2 b3 minus
a1 times that.

00:11:31.710 --> 00:11:37.350
Minus a1 times a3 b2.

00:11:37.350 --> 00:11:40.110
And then you have plus
this times this.

00:11:40.110 --> 00:11:42.670
So plus a2 times a3.

00:11:42.670 --> 00:11:47.530
So plus a2 times a3 times b1.

00:11:47.530 --> 00:11:55.720
And then minus a2 a1 b3.

00:11:55.720 --> 00:11:59.180
And then finally, plus-- I'll
just continue it down here.

00:11:59.180 --> 00:12:10.050
Plus a3 times a1 b2 minus
a3 times a2 b1.

00:12:10.050 --> 00:12:12.680
All I did I just took the
cross-- the dot product of

00:12:12.680 --> 00:12:13.590
these two things.

00:12:13.590 --> 00:12:14.750
I just took each of this.

00:12:14.750 --> 00:12:18.010
This guy times that was equal
to those two terms. That guy

00:12:18.010 --> 00:12:20.370
times that was equal to the next
two terms, equal to those

00:12:20.370 --> 00:12:22.840
two terms. And then this guy
times that was equal to those

00:12:22.840 --> 00:12:26.740
two terms. And if these guys
are really orthogonal, then

00:12:26.740 --> 00:12:28.280
this should be equal to 0.

00:12:28.280 --> 00:12:30.020
So let's see if that's
the case.

00:12:30.020 --> 00:12:35.030
So I have an a1 a2 b3, a
positive here, and then I'm

00:12:35.030 --> 00:12:36.640
subtracting the same
thing here.

00:12:36.640 --> 00:12:39.940
This is the same thing as a1 a2
b3, but it's just a minus.

00:12:39.940 --> 00:12:43.380
So that will cancel
out with that.

00:12:43.380 --> 00:12:44.460
Let's see, what else
do we have?

00:12:44.460 --> 00:12:47.670
We have a minus a1 a3 b2.

00:12:47.670 --> 00:12:51.740
We have a plus a1 a3 b2 there,
so these two are going to

00:12:51.740 --> 00:12:52.950
cancel out.

00:12:52.950 --> 00:12:54.970
And I think you see where
this is going.

00:12:54.970 --> 00:12:59.960
You have a positive a2 a3 b1 and
then you have a negative

00:12:59.960 --> 00:13:02.050
a2 a3 b1 there.

00:13:02.050 --> 00:13:05.790
So these will also cancel out.

00:13:05.790 --> 00:13:08.370
Now, I just showed you that
it's orthogonal to a.

00:13:08.370 --> 00:13:11.110
Let me show you that it's
orthogonal to b.

00:13:11.110 --> 00:13:17.680
Let me get another version of
my-- the cross product of the

00:13:17.680 --> 00:13:18.940
two vectors.

00:13:18.940 --> 00:13:21.640
Probably scroll down
a little bit.

00:13:21.640 --> 00:13:23.730
And let me go back.

00:13:23.730 --> 00:13:28.570
And let me multiply that
times the vector b.

00:13:28.570 --> 00:13:32.440
b1, b2 and b3.

00:13:32.440 --> 00:13:33.700
I'll do it here just so
I have some space.

00:13:33.700 --> 00:13:43.430
So b1 times this whole thing
right here is b1 a2 b3 minus

00:13:43.430 --> 00:13:44.530
b1 times this.

00:13:44.530 --> 00:13:49.360
Minus b1 a3 b2.

00:13:49.360 --> 00:13:50.350
Let me switch colors.

00:13:50.350 --> 00:13:54.420
And then b2 times this thing
here is going to be b2-- so

00:13:54.420 --> 00:13:56.690
it's going to be plus.

00:13:56.690 --> 00:13:58.630
This is all really one
expression, I'm just writing

00:13:58.630 --> 00:13:59.430
it on multiple lines.

00:13:59.430 --> 00:14:00.630
This isn't a vector.

00:14:00.630 --> 00:14:03.030
Remember, when you take the dot
product of two things, you

00:14:03.030 --> 00:14:03.990
get a scalar quantity.

00:14:03.990 --> 00:14:06.020
So plus b2 times this thing.

00:14:06.020 --> 00:14:14.450
So b2 a3 b1 minus b2 a1 b3.

00:14:14.450 --> 00:14:17.730
And then finally,
b3 times this.

00:14:17.730 --> 00:14:30.620
So plus b3 a1 b2
minus b3 a2 b1.

00:14:30.620 --> 00:14:32.970
So if these guys are definitely
orthogonal, then

00:14:32.970 --> 00:14:34.460
this thing needs to equal 0.

00:14:34.460 --> 00:14:35.620
And let's see if that's
the case.

00:14:35.620 --> 00:14:39.400
We have a b1, a2, b3.

00:14:39.400 --> 00:14:42.750
So b1 and a b3.

00:14:42.750 --> 00:14:45.430
b1 a2 b3, that's a positive
one and this

00:14:45.430 --> 00:14:46.190
is a negative one.

00:14:46.190 --> 00:14:50.860
You have a b3, an a2 and a b1
so that and that cancel out.

00:14:50.860 --> 00:14:54.310
Here you have a minus
b1 a3 b2.

00:14:54.310 --> 00:14:56.550
So you have a b1 and a b2.

00:14:56.550 --> 00:14:59.110
It's a minus b1 a3 b2.

00:14:59.110 --> 00:15:01.530
This is a plus the same thing.

00:15:01.530 --> 00:15:03.400
Just switching the order
of the multiplication.

00:15:03.400 --> 00:15:04.620
But these two are
the same term.

00:15:04.620 --> 00:15:07.560
They're just opposites of each
other, so they cancel out.

00:15:07.560 --> 00:15:10.740
And then finally, you have
a b2, an a1 and a b3.

00:15:10.740 --> 00:15:11.410
It's a negative.

00:15:11.410 --> 00:15:14.000
And then you have a positive
version of the same thing.

00:15:14.000 --> 00:15:15.990
So these two guys cancel out.

00:15:15.990 --> 00:15:19.860
So you see that this
is also equal to 0.

00:15:19.860 --> 00:15:23.360
So hopefully you're satisfied
that this vector right here is

00:15:23.360 --> 00:15:25.800
definitely orthogonal
to both a and b.

00:15:25.800 --> 00:15:27.610
And that's because that's
how it was designed.

00:15:27.610 --> 00:15:28.980
This is a definition.

00:15:28.980 --> 00:15:31.510
You could do a little bit of
algebra and you could have

00:15:31.510 --> 00:15:34.680
without me explaining this
definition to you, you could

00:15:34.680 --> 00:15:37.440
have actually come up with this
definition on your own.

00:15:37.440 --> 00:15:40.000
But obviously this was kind
of designed to have other

00:15:40.000 --> 00:15:41.920
interesting properties to it.

00:15:41.920 --> 00:15:45.010
And I'll cover those in the next
few videos, so hopefully

00:15:45.010 --> 00:15:46.730
you found that helpful.


Defining a plane in R3 with a point and normal vector

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.810
Let's take a little bit of a
hiatus from our more rigorous

00:00:03.810 --> 00:00:06.570
math where we're building the
mathematics of vector algebra

00:00:06.570 --> 00:00:09.310
and just think a little bit
about something that you'll

00:00:09.310 --> 00:00:11.080
probably encounter if you ever
have to have to write a

00:00:11.080 --> 00:00:13.230
three-dimensional computer
program or have to do any

00:00:13.230 --> 00:00:15.300
mathematics dealing with
three dimensions.

00:00:15.300 --> 00:00:26.880
And that's the idea of just the
equation of a plane in R3.

00:00:26.880 --> 00:00:28.120
And you know what a plane is.

00:00:28.120 --> 00:00:29.880
I mean we live in a
three-dimensional world and we

00:00:29.880 --> 00:00:31.200
see planes all around us.

00:00:31.200 --> 00:00:34.580
The surface of your monitor, of
your computer monitor is a

00:00:34.580 --> 00:00:36.910
plane, regardless of what
angle you hold it at.

00:00:36.910 --> 00:00:38.750
And I can draw here in
three dimensions.

00:00:42.400 --> 00:00:44.840
Let me do a little bit
better job than that.

00:00:44.840 --> 00:00:49.220
So let's say that that
is the x-axis.

00:00:49.220 --> 00:00:51.850
This is my y-axis.

00:00:51.850 --> 00:00:55.110
And then that's my z-axis.

00:00:55.110 --> 00:00:57.820
And we know what a
plane looks like.

00:00:57.820 --> 00:00:59.840
it looks something like that.

00:00:59.840 --> 00:01:01.950
I'm just drawing it at an
arbitrary angle and it goes

00:01:01.950 --> 00:01:04.980
off in every direction.

00:01:04.980 --> 00:01:06.870
Now, the equation of a
plane-- and you've

00:01:06.870 --> 00:01:09.430
probably seen this before.

00:01:09.430 --> 00:01:11.390
It's a linear function
of x, y and z.

00:01:11.390 --> 00:01:18.450
So it's ax plus by plus
cz is equal to d.

00:01:18.450 --> 00:01:20.720
If this is the graph on that
plane, then that means that

00:01:20.720 --> 00:01:28.090
every point on this plane, every
x, y and z on this plane

00:01:28.090 --> 00:01:30.085
satisfies this equation.

00:01:30.085 --> 00:01:34.150
Now another way, just as valid
a way to specify a point or

00:01:34.150 --> 00:01:38.480
specify a plane is to give you
an actual point on the plane.

00:01:38.480 --> 00:01:42.670
So we'll say, look, that's
the point on the plane.

00:01:42.670 --> 00:01:48.800
Let's say that this is the
point x0, y0, and z0.

00:01:48.800 --> 00:01:51.420
It could be one of the instances
of this point right

00:01:51.420 --> 00:01:52.800
here, but I'm just saying
this is another point

00:01:52.800 --> 00:01:54.260
that's on the plane.

00:01:54.260 --> 00:01:57.940
That obviously by itself isn't
going to specify the plane.

00:01:57.940 --> 00:02:00.030
You could pivot the plane
around that point in an

00:02:00.030 --> 00:02:01.470
infinite number of ways.

00:02:01.470 --> 00:02:05.090
But if you specify that point
and you specify a vector

00:02:05.090 --> 00:02:09.240
that's perpendicular to the
plane-- and I can draw that

00:02:09.240 --> 00:02:11.460
starting from here, but I can
shift a vector wherever.

00:02:11.460 --> 00:02:12.950
But let me just draw
it right there.

00:02:12.950 --> 00:02:18.210
So if I also specify a normal
vector to the plane.

00:02:18.210 --> 00:02:20.400
And I just used a word that I
haven't defined for you yet.

00:02:20.400 --> 00:02:22.280
But when I say a normal
vector, so

00:02:22.280 --> 00:02:25.280
n is a normal vector.

00:02:28.100 --> 00:02:30.990
Or which just means that it's
perpendicular to the plane.

00:02:30.990 --> 00:02:33.450
It's perpendicular to everything
on the plane.

00:02:33.450 --> 00:02:36.680
It's perpendicular to every
vector on the plane.

00:02:36.680 --> 00:02:39.270
Perpendicular to, I guess the
best way to say it, I'll just

00:02:39.270 --> 00:02:40.850
say it in very imprecise
terms--

00:02:40.850 --> 00:02:45.570
everything on the plane.

00:02:45.570 --> 00:02:49.730
So if you have some vectors
lying on the plane, if I have

00:02:49.730 --> 00:02:52.050
some vector here, let's say
that's lying on the plane.

00:02:52.050 --> 00:02:54.960
If you imagine the plane as
a piece of cardboard, that

00:02:54.960 --> 00:02:58.450
yellow arrow I just drew-- I
would've actually drawn that

00:02:58.450 --> 00:02:59.240
on the cardboard.

00:02:59.240 --> 00:03:00.670
It's sitting on the plane.

00:03:00.670 --> 00:03:05.280
If this yellow vector, let me
call it vector a, then if this

00:03:05.280 --> 00:03:07.970
is just some arbitrary vector
sitting on the plane and this

00:03:07.970 --> 00:03:11.810
is the normal vector through
the plane, we know from our

00:03:11.810 --> 00:03:15.810
definition of vector angles that
this is perpendicular to

00:03:15.810 --> 00:03:23.090
this if and only if, n dot a--
only if the dot product of

00:03:23.090 --> 00:03:25.350
these two things
are equal to 0.

00:03:25.350 --> 00:03:28.050
And that's true for any vector
that we pick that actually

00:03:28.050 --> 00:03:29.680
lies on the plane.

00:03:29.680 --> 00:03:35.470
So let's see if we can use this
definition of a plane, if

00:03:35.470 --> 00:03:40.870
we can use the-- I'll call it
the normal or let me call it

00:03:40.870 --> 00:03:44.700
the n plus-- let me
do it this way.

00:03:44.700 --> 00:03:54.390
The n plus some x0,
y0, z0 definition.

00:03:54.390 --> 00:03:59.950
And if I can go from that to
just the standard linear

00:03:59.950 --> 00:04:07.010
equation definition, ax plus
by plus cz is equal to 0.

00:04:07.010 --> 00:04:09.420
Let's see if there's some way
based on what we already know

00:04:09.420 --> 00:04:11.730
and using this information
that we can do that.

00:04:11.730 --> 00:04:14.350
So the way to think about it,
this point, this little blue

00:04:14.350 --> 00:04:16.570
point that lies on the plane,
I can specify it

00:04:16.570 --> 00:04:18.390
by a position vector.

00:04:18.390 --> 00:04:22.950
So let me set some position
vector x0 to be equal to-- so

00:04:22.950 --> 00:04:27.470
I'm going to define x0
to be equal to the

00:04:27.470 --> 00:04:30.855
scalar x0, y0, z0.

00:04:30.855 --> 00:04:32.630
And I want to be very clear.

00:04:32.630 --> 00:04:36.240
This specifies the coordinate
that lies on the plane.

00:04:36.240 --> 00:04:39.750
This vector does not lie
entirely on the plane.

00:04:39.750 --> 00:04:41.620
The way I drew it right here,
it's starting at the origin.

00:04:41.620 --> 00:04:42.870
It's a position vector.

00:04:42.870 --> 00:04:45.840
And the way I drew it it's
behind the plane.

00:04:45.840 --> 00:04:48.180
The tip of its arrow
sits on the plane.

00:04:48.180 --> 00:04:50.330
But this vector itself,
it's not necessarily

00:04:50.330 --> 00:04:51.640
drawn on the plane.

00:04:51.640 --> 00:04:54.210
This plane might not even go
through the origin, while this

00:04:54.210 --> 00:04:56.270
vector does touch
on the origin.

00:04:56.270 --> 00:04:59.300
It just specifies some
point on the plane.

00:04:59.300 --> 00:05:01.440
Similarly, let me define
another vector.

00:05:01.440 --> 00:05:04.210
I said this was some other
arbitrary point on the plane

00:05:04.210 --> 00:05:07.800
xyz, and so you know,
this is true for any

00:05:07.800 --> 00:05:08.380
point on the plane.

00:05:08.380 --> 00:05:11.490
Let me define another vector
x, and I'm going to define

00:05:11.490 --> 00:05:14.710
that as x, y, and z.

00:05:14.710 --> 00:05:17.980
So once again, like x0, the
vector x-- let me draw it

00:05:17.980 --> 00:05:19.790
right there.

00:05:19.790 --> 00:05:23.690
This vector x does not
lie on the plane.

00:05:23.690 --> 00:05:24.890
It goes from the origin.

00:05:24.890 --> 00:05:27.260
It's a position vector
that specifies a

00:05:27.260 --> 00:05:29.320
point on the plane.

00:05:29.320 --> 00:05:31.240
So it goes from the origin
and it goes out.

00:05:31.240 --> 00:05:34.200
And you can almost view them
as if the plane was like a

00:05:34.200 --> 00:05:37.060
coffee table that this would
kind of be these vectors.

00:05:37.060 --> 00:05:39.020
Let me see if I can draw it.

00:05:39.020 --> 00:05:43.470
If this was the flat surface of
the plane that the vector

00:05:43.470 --> 00:05:46.380
x0 is going from the origin
to specify some

00:05:46.380 --> 00:05:48.180
point on the plane.

00:05:48.180 --> 00:05:51.570
And then the vector x is also
going from the origin to

00:05:51.570 --> 00:05:54.550
specify-- let me do it
in a different color.

00:05:54.550 --> 00:05:59.410
The vector x is also going from
the origin to specify

00:05:59.410 --> 00:06:01.970
some other point on the
plane right there.

00:06:01.970 --> 00:06:04.870
I just took the plane and made
it flat, so you see it right

00:06:04.870 --> 00:06:06.790
along its side.

00:06:06.790 --> 00:06:09.240
If you could imagine sitting
right at the surface of the

00:06:09.240 --> 00:06:12.170
plane and then you could see
that these guys clearly do not

00:06:12.170 --> 00:06:13.680
lie in the plane.

00:06:13.680 --> 00:06:16.460
But using these guys, I can
construct a vector that does

00:06:16.460 --> 00:06:19.080
lie on the plane.

00:06:19.080 --> 00:06:23.253
What does the vector x
minus x0 look like?

00:06:26.080 --> 00:06:31.420
Well, I just drew a little
triangle here. x minus x0--

00:06:31.420 --> 00:06:33.300
I'll do it in this
green color.

00:06:33.300 --> 00:06:37.130
It'll look exactly like this. x
minus x0 will be this green

00:06:37.130 --> 00:06:39.270
line right here.

00:06:39.270 --> 00:06:42.670
This is x minus x0.

00:06:42.670 --> 00:06:46.460
We could view x0 plus this
vector plus x minus x0 is

00:06:46.460 --> 00:06:47.590
going to be equal to x.

00:06:47.590 --> 00:06:49.345
If I were to do it on this
graph, it's going

00:06:49.345 --> 00:06:50.130
to look like this.

00:06:50.130 --> 00:06:53.540
It's going to be like this.

00:06:53.540 --> 00:06:55.550
So let me draw it better
than that.

00:06:55.550 --> 00:06:58.250
It's going to go from that
point from x0, the point

00:06:58.250 --> 00:07:01.500
specified by x0 to the
point specified by x.

00:07:01.500 --> 00:07:03.660
And it's going to lie
along the plane.

00:07:03.660 --> 00:07:07.220
So this right here
is x minus x0.

00:07:07.220 --> 00:07:10.110
I know this drawing is getting
very dirty, but you can see

00:07:10.110 --> 00:07:13.060
that this is definitely
lying on the plane.

00:07:13.060 --> 00:07:20.555
So this vector right here must
be perpendicular to n.

00:07:23.592 --> 00:07:29.980
Perpendicular to our
normal vector.

00:07:29.980 --> 00:07:36.150
Now, if my normal vector-- let's
say my normal vector n.

00:07:36.150 --> 00:07:38.940
So this vector is perpendicular
to this guy

00:07:38.940 --> 00:07:39.440
right here.

00:07:39.440 --> 00:07:43.770
It's perpendicular to the
vector n1, n2, n3.

00:07:43.770 --> 00:07:48.260
Now using this information, how
can we get to this type of

00:07:48.260 --> 00:07:52.730
an expression, just this linear
equation of x, y and z?

00:07:52.730 --> 00:07:58.680
Well, we know that n-- let me
switch to a neutral color.

00:07:58.680 --> 00:08:01.885
We know that n-- actually, I
didn't want to do this caret

00:08:01.885 --> 00:08:03.950
here. n is not a unit vector.

00:08:03.950 --> 00:08:08.170
But let's say n, so it's
perpendicular to this.

00:08:08.170 --> 00:08:10.620
So the dot product of it,
we saw that right there.

00:08:10.620 --> 00:08:11.600
We saw it in the
previous video.

00:08:11.600 --> 00:08:15.640
The dot product of n with this
vector right here-- actually,

00:08:15.640 --> 00:08:16.650
let me draw it.

00:08:16.650 --> 00:08:18.740
You know, I already drew the
plane sideways, so I can

00:08:18.740 --> 00:08:20.300
actually draw my n vector.

00:08:20.300 --> 00:08:22.010
My n vector's going to look
something like this.

00:08:22.010 --> 00:08:24.130
It's going to be popping
straight out of the plane.

00:08:24.130 --> 00:08:26.320
And I could shift it over, but
it's always going to be in

00:08:26.320 --> 00:08:27.030
that same direction.

00:08:27.030 --> 00:08:30.300
It's going to be perpendicular
to this vector right here.

00:08:30.300 --> 00:08:33.659
So n is perpendicular
to x minus x0.

00:08:38.450 --> 00:08:42.260
Which means that their dot
product is equal to 0.

00:08:42.260 --> 00:08:45.280
Well what does x minus
x0 look like?

00:08:45.280 --> 00:08:48.240
So this is going to look
like this expression.

00:08:48.240 --> 00:08:53.090
If I write out the vectors
themselves it's the vector n1,

00:08:53.090 --> 00:09:00.570
n2, n3 being dotted with-- well,
if I take x minus x0,

00:09:00.570 --> 00:09:04.980
that's just the scalar x
minus the scalar x0.

00:09:04.980 --> 00:09:06.740
The first term subtracted.

00:09:06.740 --> 00:09:10.950
And then the scalar y
minus the scalar y0.

00:09:10.950 --> 00:09:14.920
And then the scalar z
minus the scalar z0.

00:09:14.920 --> 00:09:17.540
We know that this whole thing
has to be equal to 0 because

00:09:17.540 --> 00:09:18.880
they're perpendicular.

00:09:18.880 --> 00:09:25.756
And then if we take the dot
product here, we get n1 times

00:09:25.756 --> 00:09:34.130
x minus x0 plus n2 times y minus
y0 plus n3 times z minus

00:09:34.130 --> 00:09:37.510
z0 is equal to 0.

00:09:37.510 --> 00:09:40.420
And you might not completely
recognize it, but this is--

00:09:40.420 --> 00:09:42.960
you'll have to do a little
algebra to clean it up-- but

00:09:42.960 --> 00:09:52.160
this is the form ax plus by
plus cz is equal to d.

00:09:52.160 --> 00:09:53.570
And actually, I think I
made a mistake here.

00:09:53.570 --> 00:09:58.110
This should be not 0,
this is equal to d.

00:09:58.110 --> 00:10:03.160
This is the general form
for a plane in R3.

00:10:03.160 --> 00:10:05.070
A plane is just a linear
surface in R3.

00:10:05.070 --> 00:10:06.770
I shouldn't have written
a 0 there.

00:10:06.770 --> 00:10:09.570
So this does take this form.

00:10:09.570 --> 00:10:11.360
And if you don't believe
me, we can do it

00:10:11.360 --> 00:10:13.020
with an actual example.

00:10:13.020 --> 00:10:16.080
So let's say we have--
I gave you my

00:10:16.080 --> 00:10:17.590
normal-- I specify a plane.

00:10:17.590 --> 00:10:20.420
If I give you a normal vector
and I tell you that normal

00:10:20.420 --> 00:10:25.110
vector is the point 1, 3, minus
2 and I say that it

00:10:25.110 --> 00:10:30.840
intersects the point, or a point
that lies on the plane--

00:10:30.840 --> 00:10:33.060
The normal vector and the point
don't necessarily have

00:10:33.060 --> 00:10:34.140
to intersect.

00:10:34.140 --> 00:10:36.920
But let's say for a point that
lies on the plane, I have the

00:10:36.920 --> 00:10:39.380
point 1, 2 and 3.

00:10:39.380 --> 00:10:42.920
And I say give me the equation
for this plane.

00:10:42.920 --> 00:10:46.430
Well, I would say well, if I
take any other point on that

00:10:46.430 --> 00:10:52.115
plane-- so if I take any other
point on that plane, xyz and

00:10:52.115 --> 00:10:56.110
it's specified by this vector,
the vector that's defined by

00:10:56.110 --> 00:10:58.190
the difference between
these two is going

00:10:58.190 --> 00:10:59.840
to lie on the plane.

00:10:59.840 --> 00:11:03.270
This point and this point
lie on the plane, so the

00:11:03.270 --> 00:11:05.870
difference between these two
vectors, the whole vector will

00:11:05.870 --> 00:11:08.190
lie on the plane.

00:11:08.190 --> 00:11:09.180
Let me take the difference.

00:11:09.180 --> 00:11:22.530
So x minus x0 is equal to x
minus 1, y minus 2, and

00:11:22.530 --> 00:11:25.310
then z minus 3.

00:11:25.310 --> 00:11:29.320
I'm saying this will
lie on the plane.

00:11:29.320 --> 00:11:32.800
This is on the plane.

00:11:32.800 --> 00:11:34.230
And it's going to
be perpendicular

00:11:34.230 --> 00:11:35.480
to our normal vector.

00:11:37.760 --> 00:11:44.590
So if I dot my normal vector, 1,
3, minus 2, with this thing

00:11:44.590 --> 00:11:51.000
right here, with x minus 1,
y minus 2, z minus 3,

00:11:51.000 --> 00:11:52.660
I should get 0.

00:11:52.660 --> 00:11:54.910
Because this has to be
perpendicular to anything that

00:11:54.910 --> 00:11:56.710
lies on the plane.

00:11:56.710 --> 00:11:57.320
So what do we get?

00:11:57.320 --> 00:12:00.990
We get 1 times x minus
1 is x minus 1.

00:12:00.990 --> 00:12:04.710
Plus 3 times y minus 2.

00:12:04.710 --> 00:12:06.010
Just taking the dot product.

00:12:06.010 --> 00:12:11.050
Minus 2 times z minus
3 is equal to 0.

00:12:11.050 --> 00:12:14.380
Let's if we can do a little bit
of algebra here to clean

00:12:14.380 --> 00:12:15.720
this up a little bit.

00:12:15.720 --> 00:12:24.920
I get x minus 1 plus 3y
minus 6 minus 2z plus

00:12:24.920 --> 00:12:28.870
6 is equal to 0.

00:12:28.870 --> 00:12:29.400
And let's see.

00:12:29.400 --> 00:12:32.200
Minus 6 and a plus
6 cancel out.

00:12:32.200 --> 00:12:34.380
And then I can take
this minus 1.

00:12:34.380 --> 00:12:41.540
I could add 1 to both sides and
I get x plus 3y minus 2z--

00:12:41.540 --> 00:12:44.300
add that 1 to both sidess--
is equal to 1.

00:12:44.300 --> 00:12:45.750
And there you have it.

00:12:45.750 --> 00:12:49.330
Just by using the simple fact
that this is a point on the

00:12:49.330 --> 00:12:51.720
plane and this is a normal
vector, I was able to use the

00:12:51.720 --> 00:12:54.930
idea that this has to be normal
or its dot product with

00:12:54.930 --> 00:12:58.130
any point, with any vector that
lies on the plane, I was

00:12:58.130 --> 00:13:00.570
able to get this right here.

00:13:00.570 --> 00:13:02.000
I didn't have to go
through this whole

00:13:02.000 --> 00:13:02.700
business right here.

00:13:02.700 --> 00:13:05.450
You could have just used this
formula right here.

00:13:05.450 --> 00:13:11.370
You could have just said n1
is 1 times x minus x1.

00:13:11.370 --> 00:13:13.180
Or x0 I could call it.

00:13:13.180 --> 00:13:23.260
So x minus this 1 plus n2, 3
times y minus 2 plus minus 2

00:13:23.260 --> 00:13:28.070
times z minus 3 is equal to 0.

00:13:28.070 --> 00:13:29.700
And then if you just did
a little bit of math.

00:13:29.700 --> 00:13:32.130
a little bit of algebra, you
would've gotten there.

00:13:32.130 --> 00:13:34.150
So hopefully find this
reasonable useful.

00:13:34.150 --> 00:13:37.230
This is actually quite useful
if you ever have to do

00:13:37.230 --> 00:13:40.660
anything that involves any
type of three-dimensional

00:13:40.660 --> 00:13:41.520
mathematics.

00:13:41.520 --> 00:13:44.970
And if you ever become a game
programmer this would be--

00:13:44.970 --> 00:13:46.540
there's thousands of
other applications.

00:13:46.540 --> 00:13:49.680
But this is kind of a useful
byproduct of some of the

00:13:49.680 --> 00:13:52.740
formal mathematics that
we've been doing.


Defining the angle between vectors

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:05.350
A couple of videos ago we
introduced the idea of the

00:00:05.350 --> 00:00:07.240
length of a vector.

00:00:07.240 --> 00:00:09.210
That equals the length.

00:00:09.210 --> 00:00:12.560
And this was a neat idea because
we're used to the

00:00:12.560 --> 00:00:15.720
length of things in two- or
three-dimensional space, but

00:00:15.720 --> 00:00:18.730
it becomes very abstract when
we get to n dimensions.

00:00:18.730 --> 00:00:21.250
If this has a hundred
components, at least for me,

00:00:21.250 --> 00:00:24.210
it's hard to visualize a hundred
dimension vector.

00:00:24.210 --> 00:00:26.410
But we've actually defined
it's notion of length.

00:00:26.410 --> 00:00:30.900
And we saw that this is actually
a scalar value.

00:00:30.900 --> 00:00:32.270
It's just a number.

00:00:32.270 --> 00:00:36.530
In this video, I want to attempt
to define the notion

00:00:36.530 --> 00:00:40.125
of an angle between vectors.

00:00:45.400 --> 00:00:48.880
As you can see, we're building
up this mathematics of vectors

00:00:48.880 --> 00:00:51.495
from the ground up, and we can't
just say, oh, I know

00:00:51.495 --> 00:00:53.590
what an angle is because
everything we know about

00:00:53.590 --> 00:00:56.410
angles and even lengths, it
just applies to what we

00:00:56.410 --> 00:00:58.650
associate with two- or
three-dimensional space.

00:00:58.650 --> 00:01:02.230
But the whole study of linear
algebra is abstracting these

00:01:02.230 --> 00:01:05.010
ideas into multi-dimensional
space.

00:01:05.010 --> 00:01:07.240
And I haven't even defined what
dimension is yet, but I

00:01:07.240 --> 00:01:11.030
think you understand that idea
to some degree already.

00:01:11.030 --> 00:01:14.020
When people talk about one or
two or three dimensions.

00:01:14.020 --> 00:01:18.020
So let's say that I have some
vector-- let's say I have two

00:01:18.020 --> 00:01:20.530
vectors, vectors a and b.

00:01:23.940 --> 00:01:26.460
They're nonzero and they're
members of Rn.

00:01:30.800 --> 00:01:36.790
And I don't have a notion of
the angle between them yet,

00:01:36.790 --> 00:01:38.430
but let me just draw them out.

00:01:38.430 --> 00:01:40.390
Let me just draw them as if
I could draw them in two

00:01:40.390 --> 00:01:40.870
dimensions.

00:01:40.870 --> 00:01:44.960
So that would be vector
a right there.

00:01:44.960 --> 00:01:48.400
Maybe that's vector
b right there.

00:01:48.400 --> 00:01:50.980
And then this vector right
there would be the

00:01:50.980 --> 00:01:53.810
vector a minus b.

00:01:53.810 --> 00:01:57.050
And you can verify that just the
way we've learned to add

00:01:57.050 --> 00:01:58.200
and subtract vectors.

00:01:58.200 --> 00:02:00.310
Or you know, this is
heads to tails.

00:02:00.310 --> 00:02:03.840
So b plus a minus b is of
course, going to be vector a.

00:02:03.840 --> 00:02:06.490
And that all just
works out there.

00:02:06.490 --> 00:02:09.050
To help us define this notion
of angle, let me construct

00:02:09.050 --> 00:02:12.345
another triangle that's going
to look a lot like this one.

00:02:12.345 --> 00:02:17.470
But remember, I'm just doing
this for our simple minds to

00:02:17.470 --> 00:02:19.140
imagine it in two dimensions.

00:02:19.140 --> 00:02:23.030
But these aren't necessarily
two-dimensional beasts.

00:02:23.030 --> 00:02:25.000
These each could have a
hundred components.

00:02:25.000 --> 00:02:26.555
But let me make another
triangle.

00:02:30.150 --> 00:02:33.440
Well, it should look similar.

00:02:33.440 --> 00:02:35.000
Say it looks like that.

00:02:35.000 --> 00:02:38.040
And I'm going to define the
sides of the triangles to be

00:02:38.040 --> 00:02:40.090
the lengths of each
of these vectors.

00:02:40.090 --> 00:02:42.250
Remember, the lengths of each of
these vectors, I don't care

00:02:42.250 --> 00:02:43.460
how many components there
are, they're just

00:02:43.460 --> 00:02:44.680
going to be your numbers.

00:02:44.680 --> 00:02:46.980
So the length of this side right
here is just going to be

00:02:46.980 --> 00:02:48.530
the length of a.

00:02:48.530 --> 00:02:50.580
The length of this side right
here is just going to be the

00:02:50.580 --> 00:02:54.695
length of vector a
minus vector b.

00:02:57.610 --> 00:02:59.780
And the length of this side
right here is going to be the

00:02:59.780 --> 00:03:02.010
length of vector b.

00:03:02.010 --> 00:03:03.990
Now the first thing we want to
make sure is that we can

00:03:03.990 --> 00:03:07.500
always construct a triangle
like that.

00:03:07.500 --> 00:03:09.940
And so under what circumstances
could we not

00:03:09.940 --> 00:03:11.280
construct a triangle
like this?

00:03:11.280 --> 00:03:13.550
Well we wouldn't be able
construct a triangle like this

00:03:13.550 --> 00:03:15.820
if this side.

00:03:15.820 --> 00:03:19.410
if b, if the magnitude-- so
let me write this down.

00:03:19.410 --> 00:03:20.710
It's kind of a subtle
point, but I want to

00:03:20.710 --> 00:03:22.520
make this very clear.

00:03:22.520 --> 00:03:24.810
In order to define an angle, I
want to be comfortable that I

00:03:24.810 --> 00:03:27.300
can always make this
construction.

00:03:27.300 --> 00:03:30.680
And I need to make sure that--
let me write reasons why I

00:03:30.680 --> 00:03:39.490
couldn't make this
construction.

00:03:39.490 --> 00:03:45.310
Well what if the magnitude of
b was greater than, or the

00:03:45.310 --> 00:03:49.450
length of vector b was greater
than the length of vector a

00:03:49.450 --> 00:03:54.760
plus the length of
vector a minus b?

00:03:54.760 --> 00:03:57.260
In two dimensions, I could never
draw a triangle like

00:03:57.260 --> 00:03:59.950
that then because you would
have this length plus this

00:03:59.950 --> 00:04:02.680
length would be shorter than
this thing right here.

00:04:02.680 --> 00:04:04.010
So you could never
construct it.

00:04:04.010 --> 00:04:05.200
And I could do with
all the sides.

00:04:05.200 --> 00:04:08.470
What if this length was larger
than one of these two sides?

00:04:08.470 --> 00:04:10.080
Or what if that length
was larger than one

00:04:10.080 --> 00:04:10.760
of those two sides?

00:04:10.760 --> 00:04:14.300
I could just never draw
a two-dimensional

00:04:14.300 --> 00:04:15.560
triangle that way.

00:04:15.560 --> 00:04:17.800
So what I'm going to do is I'm
going to use the triangle--

00:04:17.800 --> 00:04:21.544
the vector triangle inequality
to prove that each of these

00:04:21.544 --> 00:04:27.560
sides is less than or equal to
the sum of the other sides.

00:04:27.560 --> 00:04:28.540
I could do the same thing.

00:04:28.540 --> 00:04:29.620
Let me make the point clear.

00:04:29.620 --> 00:04:33.500
I could show that if a, for
whatever reason, was greater

00:04:33.500 --> 00:04:39.190
than the other sides plus b,
then I wouldn't be able to

00:04:39.190 --> 00:04:39.990
create a triangle.

00:04:39.990 --> 00:04:44.020
And the last one of course is
if a minus b, for whatever

00:04:44.020 --> 00:04:46.200
reason, was greater than the
other two sides, I just

00:04:46.200 --> 00:04:49.310
wouldn't be able to draw
a triangle in a plus b.

00:04:49.310 --> 00:04:53.590
So I need to show that for any
vectors, any real vectors--

00:04:53.590 --> 00:04:56.770
nonzero, real vectors that are
members of Rn-- that none of

00:04:56.770 --> 00:04:58.180
these can ever happen.

00:04:58.180 --> 00:04:59.970
I need to prove that none
of those can happen.

00:04:59.970 --> 00:05:02.580
So what does the triangle
inequality tell us?

00:05:02.580 --> 00:05:06.680
The triangle inequality tells
us that if I have the sum of

00:05:06.680 --> 00:05:12.250
two vectors, if I take the
length of the sum of two

00:05:12.250 --> 00:05:18.400
vectors, that that is always
going to be less than-- and

00:05:18.400 --> 00:05:19.360
these are nonzero vectors.

00:05:19.360 --> 00:05:22.710
This is always going to be less
than or equal to the sum

00:05:22.710 --> 00:05:24.510
of each of their individual
lengths.

00:05:28.220 --> 00:05:31.020
So let's see if we can
apply that to this

00:05:31.020 --> 00:05:32.930
triangle right here.

00:05:32.930 --> 00:05:41.250
So what is the magnitude,
the length of a?

00:05:41.250 --> 00:05:42.780
Well I can rewrite vector a.

00:05:42.780 --> 00:05:44.110
What is vector a equal to?

00:05:44.110 --> 00:05:53.380
Vector a is equal to vector
b plus vector a minus b.

00:05:53.380 --> 00:05:55.450
I mean I'm just rewriting
the vector here.

00:05:55.450 --> 00:05:59.970
I'm just rewriting a here as a
sum of the other two vectors.

00:05:59.970 --> 00:06:00.910
Nothing fancy there.

00:06:00.910 --> 00:06:02.790
I haven't used the triangle
inequality or anything.

00:06:02.790 --> 00:06:06.060
I've just used my definition
of vector addition.

00:06:06.060 --> 00:06:09.630
But here now, if I put little
parentheses here, now I can

00:06:09.630 --> 00:06:10.820
apply the triangle inequality.

00:06:10.820 --> 00:06:11.760
And I say, well,
you know what?

00:06:11.760 --> 00:06:14.440
This is going to be, by the
triangle inequality, which

00:06:14.440 --> 00:06:18.180
we've proved, it's going to be
less than or equal to the

00:06:18.180 --> 00:06:20.740
lengths of each of
these vectors.

00:06:20.740 --> 00:06:27.410
Vector b plus the length
of vector a minus b.

00:06:27.410 --> 00:06:31.400
So we know that the length of a
is less than the sum of that

00:06:31.400 --> 00:06:32.560
one and that one.

00:06:32.560 --> 00:06:35.980
So we don't have to worry about
this being our problem.

00:06:35.980 --> 00:06:38.270
We know that that is not true.

00:06:38.270 --> 00:06:42.610
Now let's look at b.

00:06:42.610 --> 00:06:46.040
So is there any way that I can
rewrite b as a sum of two

00:06:46.040 --> 00:06:46.790
other vectors?

00:06:46.790 --> 00:06:47.650
Well sure.

00:06:47.650 --> 00:06:55.800
I can write it as a sum of a
plus, let me put it this way.

00:06:55.800 --> 00:06:59.610
If that vector right there is a
minus b, the same vector in

00:06:59.610 --> 00:07:03.720
the reverse direction is going
to be the vector b minus a.

00:07:03.720 --> 00:07:10.770
So a plus the vector
b minus a.

00:07:10.770 --> 00:07:11.890
That's the same thing as b.

00:07:11.890 --> 00:07:13.290
And you can see it right here.

00:07:13.290 --> 00:07:14.580
The a's would cancel
out and you're just

00:07:14.580 --> 00:07:15.990
left with the b there.

00:07:15.990 --> 00:07:19.710
Now by the triangle inequality,
we know that this

00:07:19.710 --> 00:07:25.960
is less than or equal to the
length of vector a plus the

00:07:25.960 --> 00:07:29.350
length of vector b minus a.

00:07:29.350 --> 00:07:31.930
Now you're saying hey, Sal,
you're dealing with b minus a.

00:07:31.930 --> 00:07:34.200
This is the length
of a minus b.

00:07:34.200 --> 00:07:36.630
And I can leave this for you
to prove it based on our

00:07:36.630 --> 00:07:45.400
definition of vector lengths,
but the length of b minus a is

00:07:45.400 --> 00:07:50.830
equal to minus 1 times
a minus b.

00:07:50.830 --> 00:07:54.050
And I'll leave it to you
to say that look,

00:07:54.050 --> 00:07:57.250
these lengths are equal.

00:07:57.250 --> 00:08:00.780
Because essentially-- I could
leave that, but I think you

00:08:00.780 --> 00:08:05.030
can take that based on just the
visual depiction of them

00:08:05.030 --> 00:08:07.510
that they're the exact same
vectors, just in different

00:08:07.510 --> 00:08:07.930
directions.

00:08:07.930 --> 00:08:09.850
And I have to be careful with
length because it's not just

00:08:09.850 --> 00:08:10.780
in two dimensions.

00:08:10.780 --> 00:08:14.290
But I think you get the idea and
I'll leave that for you to

00:08:14.290 --> 00:08:16.450
prove that these lengths
are the same thing.

00:08:16.450 --> 00:08:18.850
So we know that b is less
than the length

00:08:18.850 --> 00:08:20.270
of those two things.

00:08:20.270 --> 00:08:25.000
So we don't have to worry about
that one right there.

00:08:25.000 --> 00:08:28.595
Finally, a minus b.

00:08:28.595 --> 00:08:35.919
The magnitude or the length
of vector a minus b.

00:08:35.919 --> 00:08:41.940
Well I can write that as the
length of-- or I can write

00:08:41.940 --> 00:08:46.140
that as vector a plus
vector minus b.

00:08:49.920 --> 00:08:52.610
If we just put a minus b right
there and go in the other

00:08:52.610 --> 00:08:55.560
directions, we could say minus
b, which would be in that

00:08:55.560 --> 00:09:01.550
direction plus a would give
us our vector a minus b.

00:09:01.550 --> 00:09:02.680
Actually, I don't even
have to go there.

00:09:02.680 --> 00:09:03.650
That's obvious from this.

00:09:03.650 --> 00:09:06.040
I just kind of put the negative
in the parentheses.

00:09:06.040 --> 00:09:08.260
Well the triangle inequality,
and this might seem a little

00:09:08.260 --> 00:09:11.050
mundane to you, but it really
shows us that we can always

00:09:11.050 --> 00:09:13.970
define a regular planar triangle
based on these

00:09:13.970 --> 00:09:15.430
vectors in this way.

00:09:15.430 --> 00:09:19.070
It tells us this is less than or
equal to the length of our

00:09:19.070 --> 00:09:24.150
vector a plus the length
of minus b.

00:09:24.150 --> 00:09:26.680
And I just said and you could
prove it to yourself, that

00:09:26.680 --> 00:09:29.660
this is the same thing
as the length of b.

00:09:29.660 --> 00:09:32.130
So we just saw that
this is definitely

00:09:32.130 --> 00:09:32.870
less than those two.

00:09:32.870 --> 00:09:34.480
This is definitely less
than those two.

00:09:34.480 --> 00:09:37.290
And that is definitely
less than those two.

00:09:37.290 --> 00:09:40.130
None of the reasons that would
keep us from constructing a

00:09:40.130 --> 00:09:41.080
triangle are valid.

00:09:41.080 --> 00:09:44.710
So we can always construct a
triangle in this way from any

00:09:44.710 --> 00:09:48.340
arbitrary nonzero
vectors in Rn.

00:09:48.340 --> 00:09:50.260
We can always construct this.

00:09:50.260 --> 00:09:57.390
Now, to define an angle, let
me redraw it down here.

00:09:57.390 --> 00:10:00.820
Let me redraw the vectors, maybe
a little bit bigger.

00:10:00.820 --> 00:10:03.920
That's vector a.

00:10:03.920 --> 00:10:05.390
This is vector b.

00:10:05.390 --> 00:10:07.680
And then let me just
draw it this way.

00:10:07.680 --> 00:10:09.300
This is the vector
right there.

00:10:09.300 --> 00:10:12.170
That is the vector a minus b.

00:10:12.170 --> 00:10:14.360
And we said we're going to
define a corresponding

00:10:14.360 --> 00:10:18.910
regular, run of the mill,
vanilla triangle whose lengths

00:10:18.910 --> 00:10:22.600
are defined by the lengths
of the vectors,

00:10:22.600 --> 00:10:23.710
by the vector lengths.

00:10:23.710 --> 00:10:26.900
So this is the length
of b, that side.

00:10:26.900 --> 00:10:31.340
This is the length
of a minus b.

00:10:31.340 --> 00:10:33.890
And then this is the
length of a.

00:10:33.890 --> 00:10:37.370
Now that I know that I can
always construct a triangle

00:10:37.370 --> 00:10:41.080
like this, I can attempt to
define-- or actually, I will

00:10:41.080 --> 00:10:45.480
define my definition of an angle
between two vectors.

00:10:45.480 --> 00:10:47.860
So we know what an angle
means in this context.

00:10:47.860 --> 00:10:52.040
This is just a regular, run of
the mill, geometric triangle.

00:10:52.040 --> 00:10:55.430
Now, my definition of an angle
between two vectors I'm going

00:10:55.430 --> 00:10:59.120
to say-- so this is what
I'm trying to define.

00:10:59.120 --> 00:11:01.900
This is what I'm going
to define.

00:11:01.900 --> 00:11:03.860
These can have arbitrary number
of components, so it's

00:11:03.860 --> 00:11:04.720
hard to visualize.

00:11:04.720 --> 00:11:09.500
But I'm going to define this
angle as the corresponding

00:11:09.500 --> 00:11:13.900
angle in a regular, run of the
mill triangle where the sides

00:11:13.900 --> 00:11:16.460
of the run of the mill triangle
are the two vectors

00:11:16.460 --> 00:11:19.420
and then the opposite side is
the subtraction, is the length

00:11:19.420 --> 00:11:21.720
of the difference between
the two vectors.

00:11:21.720 --> 00:11:24.560
This is just the definition.

00:11:24.560 --> 00:11:28.935
I'm defining this, the angle
between two vectors in Rn that

00:11:28.935 --> 00:11:32.210
could have an arbitrary number
of components, I'm defining

00:11:32.210 --> 00:11:36.770
this angle to be the same as
this angle, the angle between

00:11:36.770 --> 00:11:40.860
the two sides, the two lengths
of those vectors in just a

00:11:40.860 --> 00:11:43.380
regular, run of the
mill triangle.

00:11:43.380 --> 00:11:45.560
Now, what can I do with this?

00:11:45.560 --> 00:11:48.300
Well, can we find a relationship
between all of

00:11:48.300 --> 00:11:50.230
these things right here?

00:11:50.230 --> 00:11:50.780
Well sure.

00:11:50.780 --> 00:11:53.910
If you remember from your
trigonometry class, and if you

00:11:53.910 --> 00:11:56.810
don't, I've proved it
in the playlist. You

00:11:56.810 --> 00:11:58.060
have the law of cosines.

00:12:02.680 --> 00:12:04.860
And I'll do it with an arbitrary
triangle right here

00:12:04.860 --> 00:12:06.320
just because I don't want
to confuse you.

00:12:06.320 --> 00:12:12.040
So if this is side a, b, and c
and this is theta, the law of

00:12:12.040 --> 00:12:17.140
cosines tells us that c squared
is equal to a squared

00:12:17.140 --> 00:12:23.080
plus b squared minus 2ab
cosine of theta.

00:12:23.080 --> 00:12:26.220
I always think of it as kind
of a broader Pythagorean

00:12:26.220 --> 00:12:29.000
theorem because this
thing does not have

00:12:29.000 --> 00:12:30.460
to be a right angle.

00:12:30.460 --> 00:12:31.640
It accounts for all angles.

00:12:31.640 --> 00:12:34.100
If this becomes a right angle,
then this term disappears and

00:12:34.100 --> 00:12:36.130
you're just left with the
Pythagorean theorem.

00:12:36.130 --> 00:12:37.050
But we've proven this.

00:12:37.050 --> 00:12:41.200
This applies to just regular,
run of the mill triangles.

00:12:41.200 --> 00:12:44.870
And lucky for us, we have a
regular, run of the mill

00:12:44.870 --> 00:12:46.080
triangle here.

00:12:46.080 --> 00:12:48.580
So let's apply the law
of cosines to this

00:12:48.580 --> 00:12:51.650
triangle right here.

00:12:51.650 --> 00:12:53.950
And the way I drew it,
they correspond.

00:12:53.950 --> 00:12:56.900
The length of this
side squared.

00:12:56.900 --> 00:13:03.420
So that means the length
of a minus b squared.

00:13:03.420 --> 00:13:06.290
Length of vector a minus vector
b, that's just the

00:13:06.290 --> 00:13:07.160
length of that side.

00:13:07.160 --> 00:13:10.490
So I'm just squaring
that side.

00:13:10.490 --> 00:13:16.790
It equals the length of vector
b squared plus the length of

00:13:16.790 --> 00:13:23.150
vector a squared minus 2 times
the length of-- I'll just

00:13:23.150 --> 00:13:27.150
write two times length of vector
a times the length of

00:13:27.150 --> 00:13:31.360
vector b times the cosine of
this angle right here.

00:13:31.360 --> 00:13:32.940
Times the cosine
of that angle.

00:13:32.940 --> 00:13:35.650
And I'm defining this angle
between these two vectors to

00:13:35.650 --> 00:13:37.630
be the same as this
angle right there.

00:13:37.630 --> 00:13:40.790
So if we know this angle, by
definition, we know that angle

00:13:40.790 --> 00:13:41.970
right there.

00:13:41.970 --> 00:13:45.210
Well, we know that the square
of our lengths of a vector

00:13:45.210 --> 00:13:47.525
when we use our factor
definition of length, that

00:13:47.525 --> 00:13:50.960
this is just the same thing as
a vector dotted with itself.

00:13:50.960 --> 00:13:58.080
So that's a minus
b dot a minus b.

00:13:58.080 --> 00:14:00.530
It's all going to be equal to
this whole stuff on the

00:14:00.530 --> 00:14:01.370
right-hand side.

00:14:01.370 --> 00:14:02.780
But let me simplify
the left-hand

00:14:02.780 --> 00:14:04.230
side of this equation.

00:14:04.230 --> 00:14:10.350
a minus b dot a minus b, this is
the same thing as a dot a--

00:14:10.350 --> 00:14:15.180
those two terms--
minus a dot b.

00:14:18.065 --> 00:14:20.650
And then I have minus b dot a.

00:14:24.680 --> 00:14:26.390
Those two terms right there.

00:14:26.390 --> 00:14:29.010
And then you have the
minus b dot minus b.

00:14:29.010 --> 00:14:32.530
That's the same thing
as a plus b dot b.

00:14:32.530 --> 00:14:34.260
Remember, this is just a
simplification of the

00:14:34.260 --> 00:14:35.640
left-hand side.

00:14:35.640 --> 00:14:36.850
And I can rewrite this.

00:14:36.850 --> 00:14:42.010
a dot a, we know that's just
the length of a squared.

00:14:42.010 --> 00:14:44.210
a dot b and b dot a are
the same thing, so

00:14:44.210 --> 00:14:45.460
we have two of these.

00:14:45.460 --> 00:14:51.400
So this right here, this term
right there will simplify to

00:14:51.400 --> 00:14:57.370
minus 2 times a dot b.

00:14:57.370 --> 00:15:00.090
And then finally, b dot b.

00:15:00.090 --> 00:15:03.120
We know that that's just the
length of b squared.

00:15:03.120 --> 00:15:05.700
I just simplified or maybe I
just expanded-- that's a

00:15:05.700 --> 00:15:06.650
better word.

00:15:06.650 --> 00:15:09.680
When you go from one term here
to three terms, you can't say

00:15:09.680 --> 00:15:10.680
you simplified it.

00:15:10.680 --> 00:15:13.400
But I expanded just the
left-hand side and so this has

00:15:13.400 --> 00:15:16.520
to be equal to the right-hand
side by the law of cosines.

00:15:16.520 --> 00:15:21.570
So that is equal to-- I almost
feel like instead of rewriting

00:15:21.570 --> 00:15:23.130
it, let me just copy
and paste it.

00:15:28.280 --> 00:15:29.490
What did I just do?

00:15:29.490 --> 00:15:33.240
Copy, edit.

00:15:33.240 --> 00:15:35.620
Copy and paste.

00:15:35.620 --> 00:15:36.810
There you go.

00:15:36.810 --> 00:15:37.870
I don't know it that
was worth it.

00:15:37.870 --> 00:15:40.350
But maybe I saved a little
bit of time.

00:15:40.350 --> 00:15:44.810
So that is equal to
that right there.

00:15:44.810 --> 00:15:46.170
And then we can simplify.

00:15:46.170 --> 00:15:48.610
We have a length of a
squared here, length

00:15:48.610 --> 00:15:49.590
of a squared there.

00:15:49.590 --> 00:15:51.040
Subtract it from both sides.

00:15:51.040 --> 00:15:54.270
The length of b squared here,
length of b squared there.

00:15:54.270 --> 00:15:56.120
Subtract it from both sides.

00:15:56.120 --> 00:16:00.000
And then, what can we do?

00:16:00.000 --> 00:16:02.770
We can divide both sides by
minus 2 because everything

00:16:02.770 --> 00:16:04.100
else has disappeared.

00:16:04.100 --> 00:16:08.960
And so that term and that term
will both become 1's.

00:16:08.960 --> 00:16:16.470
And all we're left with is the
vector a dot the vector b.

00:16:16.470 --> 00:16:18.490
And this is interesting because
all of a sudden we're

00:16:18.490 --> 00:16:19.860
getting a relationship
between the dot

00:16:19.860 --> 00:16:21.200
products of two vectors.

00:16:21.200 --> 00:16:24.460
We've kind of gone away from
their definition by lengths.

00:16:24.460 --> 00:16:29.740
But the dot product of two
vectors is equal to the

00:16:29.740 --> 00:16:36.550
product of their lengths,
their vector lengths.

00:16:36.550 --> 00:16:39.110
And they can have an arbitrary
number of components.

00:16:39.110 --> 00:16:43.450
Times the cosine of the
angle between them.

00:16:43.450 --> 00:16:46.300
Remember, this theta, I said
this is the same as when you

00:16:46.300 --> 00:16:48.490
draw this kind of analogous,
regular triangle.

00:16:48.490 --> 00:16:50.530
But I'm defining the angle
between them to

00:16:50.530 --> 00:16:52.230
be the same as that.

00:16:52.230 --> 00:16:56.610
So I can say that this is
the angle between them.

00:16:56.610 --> 00:17:00.290
And obviously, the idea of
between two vectors, it's hard

00:17:00.290 --> 00:17:02.110
to visualize if you go beyond
three dimensions.

00:17:02.110 --> 00:17:04.289
But now we have it at least,
mathematically defined.

00:17:07.099 --> 00:17:11.310
So if you give me two vectors we
can now, using this formula

00:17:11.310 --> 00:17:14.920
that we've proved using this
definition up here, we can now

00:17:14.920 --> 00:17:19.410
calculate the angle between
any two vectors using this

00:17:19.410 --> 00:17:20.510
right here.

00:17:20.510 --> 00:17:28.319
And just to make it clear, what
happens if a is a-- and

00:17:28.319 --> 00:17:30.270
maybe it's not clear from that
definition, so I'll make it

00:17:30.270 --> 00:17:36.020
clearer here that by definition,
if a is equal to

00:17:36.020 --> 00:17:40.660
some scalar multiple of b where
c is greater than 0,

00:17:40.660 --> 00:17:46.190
we'll define theta
to be equal to 0.

00:17:46.190 --> 00:17:50.000
And if c is less than 0, so a is
collinear, but goes in the

00:17:50.000 --> 00:17:53.330
exact opposite direction, we'll
define theta to be equal

00:17:53.330 --> 00:17:55.440
to 180 degrees.

00:17:55.440 --> 00:17:58.080
And that's consistent with what
we understand about just

00:17:58.080 --> 00:18:00.620
two-dimensional vectors.

00:18:00.620 --> 00:18:03.220
If they're collinear and
kind of the scalar

00:18:03.220 --> 00:18:03.790
multiples the same.

00:18:03.790 --> 00:18:07.460
That means a looks something
like that and b looks

00:18:07.460 --> 00:18:08.940
something like that.

00:18:08.940 --> 00:18:10.810
So we say oh, that's
a 0 angle.

00:18:10.810 --> 00:18:15.010
And if they go the other way,
if a looks something like--

00:18:15.010 --> 00:18:16.870
this is the case where a is
just going in the other

00:18:16.870 --> 00:18:18.130
direction from b.

00:18:18.130 --> 00:18:21.540
a goes like that and b goes like
that, we define the angle

00:18:21.540 --> 00:18:24.310
between them to be
180 degrees.

00:18:24.310 --> 00:18:26.690
But everything else is pretty
well defined by

00:18:26.690 --> 00:18:27.750
the triangle example.

00:18:27.750 --> 00:18:29.880
I had to make the special case
of these because it's not

00:18:29.880 --> 00:18:32.620
clear you really get a triangle
in these cases

00:18:32.620 --> 00:18:34.870
because the triangle
kind of disappears.

00:18:34.870 --> 00:18:38.320
It flattens out if a and b are
on top of each other or if

00:18:38.320 --> 00:18:39.940
they're going in the exact
opposite direction.

00:18:39.940 --> 00:18:45.150
So that's why I wanted to make
a little bit of a side note

00:18:45.150 --> 00:18:47.080
right there.

00:18:47.080 --> 00:18:53.130
Now, using this definition of
the angle between the vectors,

00:18:53.130 --> 00:18:57.440
we can now define the idea
of perpendicular vectors.

00:18:57.440 --> 00:18:59.660
So we can now say perpendicular
vectors-- this

00:18:59.660 --> 00:19:06.100
is another definition-- and
this won't be earth

00:19:06.100 --> 00:19:09.220
shattering, but it kind of is
because we've generalized this

00:19:09.220 --> 00:19:11.570
to vectors that have an
arbitrary number of

00:19:11.570 --> 00:19:12.000
components.

00:19:12.000 --> 00:19:16.590
We're defining perpendicular
to mean the theta between--

00:19:16.590 --> 00:19:22.890
two vectors a and b are
perpendicular if the angle

00:19:22.890 --> 00:19:29.520
between them is 90 degrees.

00:19:29.520 --> 00:19:30.550
And we can define that.

00:19:30.550 --> 00:19:34.880
We can take two vectors,
dot them.

00:19:34.880 --> 00:19:36.110
Take their dot product.

00:19:36.110 --> 00:19:38.660
Figure out their two lengths and
then you could figure out

00:19:38.660 --> 00:19:39.760
the angle between them.

00:19:39.760 --> 00:19:41.080
And if it's 90 degrees.

00:19:41.080 --> 00:19:45.460
you can say that they are
perpendicular angles.

00:19:45.460 --> 00:19:47.880
And I want to be very clear here
that this is actually not

00:19:47.880 --> 00:19:50.640
defined for the 0 vector
right here.

00:19:50.640 --> 00:19:57.240
So this situation right here,
not defined for the 0 vector.

00:19:57.240 --> 00:19:59.660
Because if you have the 0
vector, then this quantity

00:19:59.660 --> 00:20:02.760
right here is going to be 0 and
then this quantity right

00:20:02.760 --> 00:20:03.810
here is going to be 0.

00:20:03.810 --> 00:20:06.600
And there's no clear definition
for your angle.

00:20:06.600 --> 00:20:11.300
If this is 0 right here, you
did 0 is equal to 0 times

00:20:11.300 --> 00:20:12.940
cosine of theta.

00:20:12.940 --> 00:20:15.610
And so if you wanted to solve
for theta you'd get cosine of

00:20:15.610 --> 00:20:19.810
theta is equal to 0/0,
which is undefined.

00:20:24.540 --> 00:20:29.200
But what we can do is create a
slightly more general word

00:20:29.200 --> 00:20:31.300
than the word perpendicular.

00:20:31.300 --> 00:20:35.530
So you have to have a defined
angle to even talk about

00:20:35.530 --> 00:20:36.400
perpendicular.

00:20:36.400 --> 00:20:39.010
If the angle between two vectors
is 90 degrees, we're

00:20:39.010 --> 00:20:43.170
saying by definition, those two
vectors are perpendicular.

00:20:43.170 --> 00:20:46.360
But what if we made the
statement and we can-- if you

00:20:46.360 --> 00:20:48.460
look at them, if the angle
between two vectors is 90

00:20:48.460 --> 00:20:51.190
degrees, what does that mean?

00:20:51.190 --> 00:20:53.180
So let's say that theta
is 90 degrees.

00:20:53.180 --> 00:20:56.370
Let me draw a line here.

00:20:56.370 --> 00:20:58.060
Let's say that theta
is 90 degrees.

00:20:58.060 --> 00:21:00.420
Theta is equal to 90 degrees.

00:21:00.420 --> 00:21:02.260
What does this formula
tell us?

00:21:02.260 --> 00:21:07.960
It tells us that a dot b is
equal to the length of a times

00:21:07.960 --> 00:21:12.450
the length of b times cosine
of 90 degrees.

00:21:12.450 --> 00:21:14.180
What's cosine of 90 degrees?

00:21:14.180 --> 00:21:15.220
It's 0.

00:21:15.220 --> 00:21:18.610
You can review your unit circle
if that doesn't make a

00:21:18.610 --> 00:21:18.850
lot of sense.

00:21:18.850 --> 00:21:21.070
But that is equal to 0, so this
whole term is going to be

00:21:21.070 --> 00:21:22.370
equal to 0.

00:21:22.370 --> 00:21:27.400
So if theta is equal to
90 degrees, then a dot

00:21:27.400 --> 00:21:31.220
b is equal to 0.

00:21:31.220 --> 00:21:36.320
And so this is another
interesting takeaway.

00:21:36.320 --> 00:21:45.690
If a and b are perpendicular,
then their dot product is

00:21:45.690 --> 00:21:49.020
going to be equal to 0.

00:21:49.020 --> 00:21:52.330
Now if their dot product is
equal to 0, can we necessarily

00:21:52.330 --> 00:21:54.470
say that they are
perpendicular?

00:21:54.470 --> 00:21:59.120
Well what if a or b
is the 0 vector?

00:21:59.120 --> 00:22:02.820
The 0 vector-- let me call
it z for 0 vector.

00:22:02.820 --> 00:22:04.070
Or I could just draw.

00:22:06.350 --> 00:22:10.590
The 0 vector dot anything is
always going to be equal to 0.

00:22:10.590 --> 00:22:12.970
So does that mean that the 0
vector is perpendicular to

00:22:12.970 --> 00:22:13.670
everything?

00:22:13.670 --> 00:22:14.570
Well no.

00:22:14.570 --> 00:22:19.320
Because the 0 vector I said, we
have to have the notion of

00:22:19.320 --> 00:22:21.550
an angle between things in
order to use the word

00:22:21.550 --> 00:22:22.170
perpendicular.

00:22:22.170 --> 00:22:25.340
So we can't use the 0 vector.

00:22:25.340 --> 00:22:28.280
We can't say just because two
vectors dot products are equal

00:22:28.280 --> 00:22:30.200
to 0 that they are
perpendicular.

00:22:30.200 --> 00:22:32.460
And that's because the 0 vector
would mess that up

00:22:32.460 --> 00:22:34.500
because the 0 vector
is not defined.

00:22:34.500 --> 00:22:38.810
But if we say, and we have been
saying, that a and b are

00:22:38.810 --> 00:22:47.660
nonzero, if they are nonzero
vectors, then we can say that

00:22:47.660 --> 00:22:50.350
if a and b are nonzero and their
dot product is equal to

00:22:50.350 --> 00:22:52.370
0, then a and b are
perpendicular.

00:22:52.370 --> 00:22:54.190
So now it goes both ways.

00:22:54.190 --> 00:22:56.960
But what if we just have this
condition right here?

00:22:56.960 --> 00:23:00.440
What if we just have the
condition that a dot b is

00:23:00.440 --> 00:23:01.200
equal to 0?

00:23:01.200 --> 00:23:05.040
It seems like that's kind of
just a simple, pure condition.

00:23:05.040 --> 00:23:06.740
And we can write a
word for that.

00:23:06.740 --> 00:23:08.790
And these words are often used
synonymously, but hopefully

00:23:08.790 --> 00:23:10.480
you understand the
distinction now.

00:23:10.480 --> 00:23:17.110
We can say that if two vectors
dot product is equal to 0, we

00:23:17.110 --> 00:23:18.930
will call them orthogonal.

00:23:25.580 --> 00:23:29.250
As I always say, spelling
isn't my best subject.

00:23:29.250 --> 00:23:30.880
But this is kind
of a neat idea.

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This tells us that-- well, that
all perpendicular vectors

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are orthogonal.

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And it also tells us that the
0 vector is orthogonal to

00:23:50.780 --> 00:23:52.460
everything else.

00:23:52.460 --> 00:23:55.570
To everything, even to itself.

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The 0 dot 0 vector
you still get 0.

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So by definition,
it's orthogonal.

00:23:59.820 --> 00:24:03.080
So for the first time probably
in your mathematical career,

00:24:03.080 --> 00:24:06.110
you're seeing that the words--
you know, every time you first

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got exposed to the words
perpendicular and orthogonal

00:24:08.220 --> 00:24:10.950
in geometry or maybe in physics
or wherever else, they

00:24:10.950 --> 00:24:13.200
were always kind of
the same words.

00:24:13.200 --> 00:24:16.230
But now I'm introducing a nice,
little distinction here

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and you can kind of be a little

00:24:17.880 --> 00:24:19.610
smart aleck with teachers.

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Oh, you know, it's perpendicular
only is the

00:24:22.040 --> 00:24:24.810
vectors aren't-- if neither
of them are 0 vector.

00:24:24.810 --> 00:24:27.290
Otherwise, if their dot product
is 0, you can only say

00:24:27.290 --> 00:24:28.250
that they're orthogonal.

00:24:28.250 --> 00:24:29.730
But if they're nonzero you
could say that they're

00:24:29.730 --> 00:24:31.105
orthogonal and perpendicular.

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But anyway, I thought that I
would introduce this little

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distinction for you in case you
have someone that likes to

00:24:35.455 --> 00:24:37.050
trip you up with words.

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But it also I think highlights
that we are building a

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mathematics from the ground up
and we have to be careful

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about the words we use.

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And we have to be very precise
about our definitions.

00:24:46.040 --> 00:24:47.770
Because if we're not precise
about our definitions and we

00:24:47.770 --> 00:24:50.200
build up a bunch of mathematics
on top of this and

00:24:50.200 --> 00:24:53.260
do a bunch of proofs, one day we
might scratch our heads and

00:24:53.260 --> 00:24:55.040
read some type of
weird ambiguity.

00:24:55.040 --> 00:24:57.660
And it might have all came out
of the fact that we weren't

00:24:57.660 --> 00:25:01.110
precise enough in defining what
some of these terms mean.

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Well anyway, hopefully you
found this useful.

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We can now take the angle or we
can now determine the angle

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between vectors with an
arbitrary number of

00:25:09.340 --> 00:25:10.590
components.