0:00:00.840,0:00:03.270
Let's say that I have[br]two nonzero vectors.

0:00:05.137,0:00:07.005
Let's say the first vector is[br]x, the second vector is y.

0:00:09.920,0:00:10.757
They are both a part of our set.

0:00:11.595,0:00:13.270
They're both in the set Rn[br]and they're nonzero.

0:00:17.450,0:00:22.133
It turns out that the absolute[br]value of their-- let me do it

0:00:22.133,0:00:25.320
in a different color.

0:00:25.320,0:00:26.800
This color's nice.

0:00:26.800,0:00:31.080
The absolute value of their[br]dot product of the two

0:00:31.080,0:00:35.280
vectors-- and remember, this is[br]just a scalar quantity-- is

0:00:35.280,0:00:40.840
less than or equal to the[br]product of their lengths.

0:00:40.840,0:00:43.140
And we've defined the dot[br]product and we've defined

0:00:43.140,0:00:44.230
lengths already.

0:00:44.230,0:00:47.390
It's less than or equal to the[br]product of their lengths and

0:00:47.390,0:00:50.730
just to push it even further,[br]the only time that this is

0:00:50.730,0:00:57.930
equal, so the dot product of the[br]two vectors is only going

0:00:57.930,0:01:01.575
to be equal to the lengths of[br]this-- the equal and the less

0:01:01.575,0:01:05.470
than or equal apply only in the[br]situation-- let me write

0:01:05.470,0:01:11.460
that down-- where one of these[br]vectors is a scalar multiple

0:01:11.460,0:01:11.920
of the other.

0:01:11.920,0:01:13.580
Or they're collinear.

0:01:13.580,0:01:16.430
You know, one's just kind of the[br]longer or shorter version

0:01:16.430,0:01:17.530
of the other one.

0:01:17.530,0:01:22.250
So only in the situation where[br]let's just say x is equal to

0:01:22.250,0:01:24.875
some scalar multiple of y.

0:01:27.970,0:01:30.760
These inequalities or I guess[br]the equality of this

0:01:30.760,0:01:33.215
inequality, this is called the[br]Cauchy-Schwarz Inequality.

0:01:33.215,0:01:43.200
Cauchy-Shwarz Inequality

0:01:43.200,0:01:45.580
So let's prove it because you[br]can't take something like this

0:01:45.580,0:01:46.680
just at face value.

0:01:46.680,0:01:49.040
You shouldn't just[br]accept that.

0:01:49.040,0:01:53.280
So let me just construct a[br]somewhat artificial function.

0:01:53.280,0:01:58.180
Let me construct some function[br]of-- that's a function of some

0:01:58.180,0:02:00.410
variables, some scalar t.

0:02:00.410,0:02:04.710
Let me define p of t to be equal[br]to the length of the

0:02:04.710,0:02:12.420
vector t times the vector-- some[br]scalar t times the vector

0:02:12.420,0:02:15.880
y minus the vector x.

0:02:15.880,0:02:17.300
It's the length of[br]this vector.

0:02:17.300,0:02:19.320
This is going to be[br]a vector now.

0:02:19.320,0:02:20.920
That squared.

0:02:20.920,0:02:23.130
Now before I move forward[br]I want to make one

0:02:23.130,0:02:23.790
little point here.

0:02:23.790,0:02:29.740
If I take the length of any[br]vector, I'll do it here.

0:02:29.740,0:02:32.890
Let's say I take the length[br]of some vector v.

0:02:32.890,0:02:36.820
I want you to accept that this[br]is going to be a positive

0:02:36.820,0:02:39.150
number, or it's at least greater[br]than or equal to 0.

0:02:39.150,0:02:42.940
Because this is just going to be[br]each of its terms squared.

0:02:42.940,0:02:45.340
v2 squared all the way[br]to vn squared.

0:02:45.340,0:02:46.640
All of these are real numbers.

0:02:46.640,0:02:49.550
When you square a real number,[br]you get something greater than

0:02:49.550,0:02:50.770
or equal to 0.

0:02:50.770,0:02:52.290
When you sum them up, you're[br]going to have something

0:02:52.290,0:02:53.670
greater than or equal to 0.

0:02:53.670,0:02:55.840
And you take the square root[br]of it, the principal square

0:02:55.840,0:02:57.370
root, the positive square root,[br]you're going to have

0:02:57.370,0:02:59.270
something greater than[br]or equal to 0.

0:02:59.270,0:03:02.930
So the length of any real vector[br]is going to be greater

0:03:02.930,0:03:04.180
than or equal to 0.

0:03:04.180,0:03:06.690
So this is the length[br]of a real vector.

0:03:06.690,0:03:11.230
So this is going to be greater[br]than or equal to 0.

0:03:11.230,0:03:14.400
Now, in the previous video, I[br]think it was two videos ago, I

0:03:14.400,0:03:18.860
also showed that the magnitude[br]or the length of a vector

0:03:18.860,0:03:22.950
squared can also be rewritten[br]as the dot product of that

0:03:22.950,0:03:24.570
vector with itself.

0:03:24.570,0:03:26.830
So let's rewrite this[br]vector that way.

0:03:29.920,0:03:32.750
The length of this vector[br]squared is equal to the dot

0:03:32.750,0:03:34.230
product of that vector[br]with itself.

0:03:34.230,0:03:44.880
So it's ty minus x[br]dot ty minus x.

0:03:44.880,0:03:49.120
In the last video, I showed[br]you that you can treat a

0:03:49.120,0:03:52.050
multiplication or you can treat[br]the dot product very

0:03:52.050,0:03:54.330
similar to regular[br]multiplication when it comes

0:03:54.330,0:03:57.420
to the associative, distributive[br]and commutative

0:03:57.420,0:03:58.330
properties.

0:03:58.330,0:04:00.200
So when you multiplied these,[br]you know, you could kind of

0:04:00.200,0:04:02.340
view this as multiplying[br]these two binomials.

0:04:02.340,0:04:05.440
You can do it the same way as[br]you would just multiply two

0:04:05.440,0:04:07.360
regular algebraic binomials.

0:04:07.360,0:04:11.030
You're essentially just using[br]the distributive property.

0:04:11.030,0:04:13.760
But remember, this isn't just[br]regular multiplication.

0:04:13.760,0:04:15.480
This is the dot product[br]we're doing.

0:04:15.480,0:04:18.149
This is vector multiplication[br]or one version of vector

0:04:18.149,0:04:19.220
multiplication.

0:04:19.220,0:04:24.730
So if we distribute it out, this[br]will become ty dot ty.

0:04:24.730,0:04:25.850
So let me write that out.

0:04:25.850,0:04:30.850
That'll be ty dot ty.

0:04:30.850,0:04:36.430
And then we'll get a minus--[br]let me do it this way.

0:04:36.430,0:04:42.580
Then we get the minus[br]x times this ty.

0:04:42.580,0:04:44.640
Instead of saying times,[br]I should be very

0:04:44.640,0:04:45.720
careful to say dot.

0:04:45.720,0:04:52.400
So minus x dot ty.

0:04:52.400,0:04:58.810
And then you have this ty[br]times this minus x.

0:04:58.810,0:05:04.860
So then you have[br]minus ty dot x.

0:05:04.860,0:05:08.900
And then finally, you have the[br]x's dot with each other.

0:05:08.900,0:05:12.680
And you can view them as[br]minus 1x dot minus 1x.

0:05:12.680,0:05:16.080
You could say plus minus 1x.

0:05:16.080,0:05:21.500
I could just view this as plus[br]minus 1 or plus minus 1.

0:05:21.500,0:05:26.260
So this is minus 1x[br]dot minus 1x.

0:05:26.260,0:05:26.860
So let's see.

0:05:26.860,0:05:29.940
So this is what my whole[br]expression simplified to or

0:05:29.940,0:05:30.710
expanded to.

0:05:30.710,0:05:32.890
I can't really call this[br]a simplification.

0:05:32.890,0:05:35.410
But we can use the fact that[br]this is commutative and

0:05:35.410,0:05:38.230
associative to rewrite this[br]expression right here.

0:05:38.230,0:05:45.430
This is equal to y dot[br]y times t squared.

0:05:45.430,0:05:46.680
t is just a scalar.

0:05:49.270,0:05:50.750
Minus-- and actually,[br]this is 2.

0:05:50.750,0:05:52.810
These two things[br]are equivalent.

0:05:52.810,0:05:55.370
They're just rearrangements of[br]the same thing and we saw that

0:05:55.370,0:05:57.290
the dot product is[br]associative.

0:05:57.290,0:06:06.260
So this is just equal to 2[br]times x dot y times t.

0:06:06.260,0:06:09.230
And I should do that in maybe[br]a different color.

0:06:09.230,0:06:13.080
So these two terms result in[br]that term right there.

0:06:13.080,0:06:16.570
And then if you just rearrange[br]these you have a minus 1

0:06:16.570,0:06:17.400
times a minus 1.

0:06:17.400,0:06:20.070
They cancel out, so those will[br]become plus and you're just

0:06:20.070,0:06:25.140
left with plus x dot x.

0:06:25.140,0:06:27.740
And I should do that in a[br]different color as well.

0:06:27.740,0:06:29.690
I'll do that in an[br]orange color.

0:06:29.690,0:06:32.820
So those terms end up[br]with that term.

0:06:32.820,0:06:35.620
Then of course, that term[br]results in that term.

0:06:35.620,0:06:37.880
And remember, all I did[br]is I rewrote this

0:06:37.880,0:06:38.490
thing and said, look.

0:06:38.490,0:06:41.990
This has got to be greater[br]than or equal to 0.

0:06:41.990,0:06:44.620
So I could rewrite that here.

0:06:44.620,0:06:46.070
This thing is still just[br]the same thing.

0:06:46.070,0:06:47.450
I've just rewritten it.

0:06:47.450,0:06:52.620
So this is all going to be[br]greater than or equal to 0.

0:06:52.620,0:06:54.990
Now let's make a little bit of[br]a substitution just to clean

0:06:54.990,0:06:56.590
up our expression[br]a little bit.

0:06:56.590,0:06:59.280
And we'll later back substitute[br]into this.

0:06:59.280,0:07:02.480
Let's define this as a.

0:07:02.480,0:07:07.860
Let's define this piece[br]right here as b.

0:07:07.860,0:07:10.380
So the whole thing[br]minus 2x dot y.

0:07:10.380,0:07:11.780
I'll leave the t there.

0:07:11.780,0:07:17.020
And let's define this or[br]let me just define this

0:07:17.020,0:07:17.825
right here as c.

0:07:17.825,0:07:20.130
X dot x as c.

0:07:20.130,0:07:22.060
So then, what does our[br]expression become?

0:07:22.060,0:07:29.910
It becomes a times t squared[br]minus-- I want to be careful

0:07:29.910,0:07:35.480
with the colors-- b[br]times t plus c.

0:07:39.180,0:07:41.050
And of course, we know that it's[br]going to be greater than

0:07:41.050,0:07:41.780
or equal to 0.

0:07:41.780,0:07:43.660
It's the same thing as[br]this up here, greater

0:07:43.660,0:07:44.270
than or equal to 0.

0:07:44.270,0:07:47.125
I could write p of t here.

0:07:47.125,0:07:50.890
Now this is greater than or[br]equal to 0 for any t that I

0:07:50.890,0:07:51.530
put in here.

0:07:51.530,0:07:53.995
For any real t that[br]I put in there.

0:08:00.640,0:08:05.190
Let me evaluate our function[br]at b over 2a.

0:08:05.190,0:08:07.570
And I can definitely do this[br]because what was a?

0:08:07.570,0:08:10.700
I just have to make sure I'm not[br]dividing by 0 any place.

0:08:10.700,0:08:13.850
So a was this vector[br]dotted with itself.

0:08:13.850,0:08:16.290
And we said this was[br]a nonzero vector.

0:08:16.290,0:08:18.680
So this is the square[br]of its length.

0:08:18.680,0:08:21.610
It's a nonzero vector, so some[br]of these terms up here would

0:08:21.610,0:08:23.790
end up becoming positively[br]when you take its length.

0:08:23.790,0:08:25.710
So this thing right[br]here is nonzero.

0:08:25.710,0:08:27.310
This is a nonzero vector.

0:08:27.310,0:08:30.880
Then 2 times the dot product[br]with itself is also going to

0:08:30.880,0:08:31.450
be nonzero.

0:08:31.450,0:08:32.309
So we can do this.

0:08:32.309,0:08:34.990
We don't worry about dividing[br]by 0, whatever else.

0:08:34.990,0:08:37.049
But what will this[br]be equal to?

0:08:37.049,0:08:39.200
This'll be equal to-- and I'll[br]just stick to the green.

0:08:39.200,0:08:42.110
It takes too long to keep[br]switching between colors.

0:08:42.110,0:08:45.230
This is equal to a times this[br]expression squared.

0:08:45.230,0:08:49.340
So it's b squared[br]over 4a squared.

0:08:49.340,0:08:52.090
I just squared 2a to[br]get the 4a squared.

0:08:52.090,0:08:55.270
Minus b times this.

0:08:55.270,0:08:58.760
So b times-- this is just[br]regular multiplication.

0:08:58.760,0:09:01.650
b times b over 2a.

0:09:01.650,0:09:03.780
Just write regular[br]multiplication there.

0:09:03.780,0:09:05.130
Plus c.

0:09:05.130,0:09:07.870
And we know all of that is[br]greater than or equal to 0.

0:09:07.870,0:09:12.080
Now if we simplify this a little[br]bit, what do we get?

0:09:12.080,0:09:15.290
Well this a cancels out with[br]this exponent there and you

0:09:15.290,0:09:18.480
end up with a b squared[br]right there.

0:09:18.480,0:09:26.150
So we get b squared over 4a[br]minus b squared over 2a.

0:09:26.150,0:09:27.650
That's that term over there.

0:09:27.650,0:09:31.730
Plus c is greater than[br]or equal to 0.

0:09:31.730,0:09:33.440
Let me rewrite this.

0:09:33.440,0:09:36.810
If I multiply the numerator and[br]denominator of this by 2,

0:09:36.810,0:09:38.010
what do I get?

0:09:38.010,0:09:41.110
I get 2b squared over 4a.

0:09:41.110,0:09:43.050
And the whole reason I did[br]that is to get a common

0:09:43.050,0:09:44.660
denominator here.

0:09:44.660,0:09:45.670
So what do you get?

0:09:45.670,0:09:50.050
You get b squared over 4a minus[br]2b squared over 4a.

0:09:50.050,0:09:52.600
So what do these two[br]terms simplify to?

0:09:52.600,0:09:54.930
Well the numerator is b squared[br]minus 2b squared.

0:09:54.930,0:10:01.200
So that just becomes minus b[br]squared over 4a plus c is

0:10:01.200,0:10:02.710
greater than or equal to 0.

0:10:02.710,0:10:06.570
These two terms add up to[br]this one right here.

0:10:06.570,0:10:11.490
Now if we add this to both sides[br]of the equation, we get

0:10:11.490,0:10:16.260
c is greater than or equal[br]to b squared over 4a.

0:10:16.260,0:10:17.720
It was a negative on[br]the left-hand side.

0:10:17.720,0:10:19.570
If I add it to both sides it's[br]going to be a positive on the

0:10:19.570,0:10:21.760
right-hand side.

0:10:21.760,0:10:24.470
We're approaching something that[br]looks like an inequality,

0:10:24.470,0:10:28.380
so let's back substitute our[br]original substitutions to see

0:10:28.380,0:10:30.060
what we have now.

0:10:30.060,0:10:32.660
So where was my original[br]substitutions that I made?

0:10:32.660,0:10:35.790
It was right here.

0:10:35.790,0:10:37.970
And actually, just to simplify[br]more, let me multiply both

0:10:37.970,0:10:39.220
sides by 4a.

0:10:41.440,0:10:43.020
I said a, not only[br]is it nonzero,

0:10:43.020,0:10:44.130
it's going to be positive.

0:10:44.130,0:10:46.070
This is the square[br]of its length.

0:10:46.070,0:10:49.670
And I already showed you that[br]the length of any real

0:10:49.670,0:10:51.170
vector's going to be positive.

0:10:51.170,0:10:53.460
And the reason why I'm taking[br]great pains to show that a is

0:10:53.460,0:10:55.510
positive is because if I[br]multiply both sides of it I

0:10:55.510,0:10:57.690
don't want to change the[br]inequality sign.

0:10:57.690,0:10:59.910
So let me multiply both sides[br]of this by a before I

0:10:59.910,0:11:00.380
substitute.

0:11:00.380,0:11:07.750
So we get 4ac is greater than[br]or equal to b squared.

0:11:07.750,0:11:08.160
There you go.

0:11:08.160,0:11:09.890
And remember, I took[br]great pains.

0:11:09.890,0:11:13.400
I just said a is definitely a[br]positive number because it is

0:11:13.400,0:11:16.830
essentially the square of the[br]length. y dot y is the square

0:11:16.830,0:11:19.450
of the length of y, and that's[br]a positive value.

0:11:19.450,0:11:20.320
It has to be positive.

0:11:20.320,0:11:21.970
We're dealing with[br]real vectors.

0:11:21.970,0:11:24.470
Now let's back substitute[br]this.

0:11:24.470,0:11:30.030
So 4 times a, 4 times y dot y.

0:11:30.030,0:11:33.420
y dot y is also-- I might as[br]well just write it there.

0:11:33.420,0:11:39.470
y dot y is the same thing as[br]the magnitude of y squared.

0:11:39.470,0:11:40.490
That's y dot y.

0:11:40.490,0:11:42.760
This is a.

0:11:42.760,0:11:45.690
y dot y, I showed you that[br]in the previous video.

0:11:45.690,0:11:46.800
Times c.

0:11:46.800,0:11:49.760
c is x dot x.

0:11:49.760,0:11:53.640
Well x dot x is the[br]same thing as the

0:11:53.640,0:11:56.200
length of vector x squared.

0:11:56.200,0:11:57.420
So this was c.

0:11:57.420,0:12:01.030
So 4 times a times c is going[br]to be greater than

0:12:01.030,0:12:03.920
or equal to b squared.

0:12:03.920,0:12:06.760
Now what was b? b was[br]this thing here.

0:12:06.760,0:12:14.610
So b squared would be 2[br]times x dot y squared.

0:12:14.610,0:12:17.880
So we've gotten to this[br]result so far.

0:12:17.880,0:12:19.620
And so what can we[br]do with this?

0:12:19.620,0:12:21.200
Oh sorry, and this whole[br]thing is squared.

0:12:21.200,0:12:23.240
This whole thing right[br]here is b.

0:12:23.240,0:12:25.220
So let's see if we can[br]simplify this.

0:12:25.220,0:12:27.620
So we get-- let me switch[br]to a different color.

0:12:27.620,0:12:34.870
4 times the length of y squared[br]times the length of x

0:12:34.870,0:12:37.660
squared is greater than or equal[br]to-- if we squared this

0:12:37.660,0:12:46.270
quantity right here, we[br]get 4 times x dot y.

0:12:46.270,0:12:54.620
4 times x dot y times x dot y.

0:12:54.620,0:12:56.510
Actually, even better, let me[br]just write it like this.

0:12:56.510,0:13:01.090
Let me just write 4 times[br]x dot y squared.

0:13:01.090,0:13:02.950
Now we can divide[br]both sides by 4.

0:13:02.950,0:13:04.600
That won't change[br]our inequality.

0:13:04.600,0:13:06.250
So that just cancels[br]out there.

0:13:06.250,0:13:08.050
And now let's take the[br]square root of both

0:13:08.050,0:13:09.840
sides of this equation.

0:13:09.840,0:13:13.040
So the square roots of both[br]sides of this equation-- these

0:13:13.040,0:13:15.380
are positive values, so the[br]square root of this side is

0:13:15.380,0:13:16.840
the square root of each[br]of its terms. That's

0:13:16.840,0:13:18.325
just an exponent property.

0:13:18.325,0:13:21.150
So if you take the square root[br]of both sides you get the

0:13:21.150,0:13:28.010
length of y times the length of[br]x is greater than or equal

0:13:28.010,0:13:29.570
to the square root of this.

0:13:29.570,0:13:31.650
And we're going to take the[br]positive square root.

0:13:31.650,0:13:33.480
We're going to take the positive[br]square root on both

0:13:33.480,0:13:34.400
sides of this equation.

0:13:34.400,0:13:36.600
That keeps us from having to[br]mess with anything on the

0:13:36.600,0:13:38.260
inequality or anything[br]like that.

0:13:38.260,0:13:42.640
So the positive square root is[br]going to be the absolute value

0:13:42.640,0:13:44.370
of x dot y.

0:13:44.370,0:13:46.060
And I want to be very careful[br]to say this is the absolute

0:13:46.060,0:13:51.060
value because it's possible that[br]this thing right here is

0:13:51.060,0:13:51.755
a negative value.

0:13:51.755,0:13:55.520
But when you square it, you want[br]to be careful that when

0:13:55.520,0:13:56.710
you take the square root[br]of it that you

0:13:56.710,0:13:57.960
stay a positive value.

0:13:57.960,0:14:02.050
Because otherwise when we take[br]the principal square root, we

0:14:02.050,0:14:04.070
might mess with the inquality.

0:14:04.070,0:14:07.130
We're taking the positive square[br]root, which will be--

0:14:07.130,0:14:08.965
so if you take the absolute[br]value, you're ensuring that

0:14:08.965,0:14:10.590
it's going to be positive.

0:14:10.590,0:14:12.000
But this is our result.

0:14:12.000,0:14:15.800
The absolute value of the dot[br]product of our vectors is less

0:14:15.800,0:14:19.510
than the product of the[br]two vectors lengths.

0:14:19.510,0:14:21.586
So we got our Cauchy-Schwarz[br]inequality.

0:14:27.600,0:14:36.630
Now the last thing I said is[br]look, what happens if x is

0:14:36.630,0:14:39.640
equal to some scalar[br]multiple of y?

0:14:39.640,0:14:41.385
Well in that case, what's[br]the absolute value?

0:14:41.385,0:14:45.930
The absolute value of x dot y?

0:14:45.930,0:14:48.520
Well that equals--[br]that equals what?

0:14:48.520,0:14:51.080
If we make the substitution that[br]equals the absolute value

0:14:51.080,0:14:52.850
of c times y.

0:14:52.850,0:14:59.160
That's just x dot y, which[br]is equal to just from the

0:14:59.160,0:15:00.730
associative property.

0:15:00.730,0:15:04.800
It's equal to the absolute value[br]of c times-- we want to

0:15:04.800,0:15:07.910
make sure our absolute value,[br]keep everything positive.

0:15:07.910,0:15:10.970
y dot y.

0:15:10.970,0:15:22.320
Well this is just equal to c[br]times the magnitude of y-- the

0:15:22.320,0:15:24.210
length of y squared.

0:15:24.210,0:15:31.260
Well that just is equal to the[br]magnitude of c times-- or the

0:15:31.260,0:15:34.986
absolute value of our scalar[br]c times our length of y.

0:15:40.000,0:15:44.290
Well this right here,[br]I can rewrite this.

0:15:44.290,0:15:46.610
I mean you can prove this to[br]yourself if you don't believe

0:15:46.610,0:15:49.790
it, but this-- we could put the[br]c inside of the magnitude

0:15:49.790,0:15:51.630
and that could be a good[br]exercise for you to prove.

0:15:51.630,0:15:52.300
But it's pretty straightforward.

0:15:52.300,0:15:54.180
You just do the definition[br]of length.

0:15:54.180,0:15:56.050
And you multiply it by c.

0:15:56.050,0:16:01.530
This is equal to the magnitude[br]of cy times-- let me say the

0:16:01.530,0:16:06.600
length of cy times[br]the length of y.

0:16:06.600,0:16:10.620
I've lost my vector notation[br]someplace over here.

0:16:10.620,0:16:11.580
There you go.

0:16:11.580,0:16:13.920
Now, this is x.

0:16:13.920,0:16:19.370
So this is equal to the length[br]of x times the length of y.

0:16:19.370,0:16:21.360
So I showed you kind of[br]the second part of the

0:16:21.360,0:16:24.650
Cauchy-Schwarz Inequality that[br]this is only equal to each

0:16:24.650,0:16:28.510
other if one of them is a scalar[br]multiple of the other.

0:16:28.510,0:16:29.540
If you're a little uncomfortable[br]with some of

0:16:29.540,0:16:31.600
these steps I took, it might[br]be a good exercise to

0:16:31.600,0:16:32.370
actually prove it.

0:16:32.370,0:16:35.720
For example, to prove that the[br]absolute value of c times the

0:16:35.720,0:16:39.270
length of the vector y is[br]the same thing as the

0:16:39.270,0:16:41.890
length of c times y.

0:16:41.890,0:16:43.790
Anyway, hopefully you found[br]this pretty useful.

0:16:43.790,0:16:47.180
The Cauchy-Schwarz Inequality[br]we'll use a lot when we prove

0:16:47.180,0:16:49.580
other results in[br]linear algebra.

0:16:49.580,0:16:51.250
And in a future video, I'll[br]give you a little more

0:16:51.250,0:16:53.960
intuition about why this makes a[br]lot of sense relative to the

0:16:53.960,0:16:55.500
dot product.