Complex determinant example

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.450
Let omega be the complex
number cosine of 2 pi over 3

00:00:04.450 --> 00:00:07.660
plus i sine of 2 pi
over 3, then the number

00:00:07.660 --> 00:00:12.800
of distinct complex numbers
z satisfying this determinant

00:00:12.800 --> 00:00:13.850
equaling 0.

00:00:13.850 --> 00:00:17.470
So we have this 3 by 3
determinant equaling 0.

00:00:17.470 --> 00:00:19.340
So let's just evaluate
this determinant

00:00:19.340 --> 00:00:21.270
and see if we can solve
for z, or figure out

00:00:21.270 --> 00:00:24.860
how many complex numbers z
that we get satisfying this--

00:00:24.860 --> 00:00:26.450
what's essentially an equation.

00:00:26.450 --> 00:00:28.140
So let's evaluate
the determinant.

00:00:28.140 --> 00:00:32.290
So if we start with
this term up here

00:00:32.290 --> 00:00:35.790
it's going to be this term times
the determinant of this sub 2

00:00:35.790 --> 00:00:37.220
by two matrix.

00:00:37.220 --> 00:00:44.830
So it's going to be
z plus 1, times z

00:00:44.830 --> 00:00:56.060
plus omega squared, times
z plus omega, minus 1.

00:00:56.060 --> 00:01:00.400
So this is the determinant
of the sub 2 by 2 matrix

00:01:00.400 --> 00:01:02.830
right over here-- the sub 2
by 2 determinant-- and that's

00:01:02.830 --> 00:01:04.300
the z plus 1.

00:01:04.300 --> 00:01:06.310
And then we have to swap signs.

00:01:06.310 --> 00:01:07.150
We go here.

00:01:07.150 --> 00:01:09.220
We want to put a
negative sign out front.

00:01:09.220 --> 00:01:10.650
You have that
checkerboard pattern

00:01:10.650 --> 00:01:11.970
when you evaluate determinants.

00:01:11.970 --> 00:01:16.370
So minus omega times-- you
block out that row, that column.

00:01:16.370 --> 00:01:21.540
So the sub-determinant is
omega times z plus omega.

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Let me just multiply
that out right now.

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So that's z omega plus omega
squared, minus omega squared,

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times 1.

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So that's pretty neat.

00:01:33.510 --> 00:01:37.490
We already got something
that simplifies a little bit.

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And then we have to worry
about this omega squared.

00:01:40.930 --> 00:01:47.180
So it's going to be plus omega
squared times omega times

00:01:47.180 --> 00:01:50.930
one-- omega times 1
is omega-- minus omega

00:01:50.930 --> 00:01:52.560
squared times z plus omega.

00:01:52.560 --> 00:02:00.235
So minus omega squared z
minus omega to the fourth.

00:02:04.140 --> 00:02:05.790
And remember this
whole expression

00:02:05.790 --> 00:02:08.490
needs to be equal to 0.

00:02:08.490 --> 00:02:11.890
This whole thing needs
to be equal to 0.

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Now let's see if we can simplify
this thing a little bit.

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So let me first multiply
these two guys in here.

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So z times z is z squared.

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Z times omega is z omega.

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Omega squared times
z is omega squared z.

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And then omega
squared times omega

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is omega to the third power--
plus omega to the third power.

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And then we have
this minus 1 here.

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And then all of that's going
to be multiplied by z plus 1.

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Let me just continue
with this blue part

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since we're already
focused here.

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So this is going to be
z times all of this.

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So it's going to be z to
the third power plus z

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squared omega, plus
omega squared z squared,

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plus omega to the
third z minus z.

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I just multiplied z
times all of this.

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And then plus 1
times all of this.

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So plus this thing again,
z squared plus z omega,

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plus omega squared z, plus
omega to the third, minus 1.

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And then let's simplify
this over here.

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So that in green,
these canceled out.

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So we're just left
with-- I'll do

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the same green-- negative
omega times z omega.

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So it's minus z omega squared.

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And then over here
in magenta we have

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plus omega to the third power
minus omega to the third z.

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Oh, let me be careful.

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That would be omega
to the fourth.

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Omega squared
times omega squared

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is omega to the fourth--
omega to the fourth z.

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And then omega squared
times omega to the fourth

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is omega to the sixth.

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And we have a negative sign.

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So negative omega to the sixth.

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And of course, this whole
thing needs to be equal to 0.

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Well, let's see if
we can simplify this.

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Let's just group the terms--
the different powers of z.

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So we have a z to
the third here.

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And that is the only z to
the third that we have.

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And then let's group
the z squared terms.

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So this is a z squared term.

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This is a z squared term.

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That's a z squared term.

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And then we don't have
any more z squared terms.

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And so the
coefficients-- this is

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going to be the same thing as
omega squared-- that's this one

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right over here-- plus omega,
plus 1, times z squared.

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So we've taken
care of everything

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that I've underlined
in pink over here.

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Now let's worry
about the z terms.

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I'll do it in this color.

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So this is a z term.

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We're just multiplying times z.

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This is a z term.

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This is a z term.

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That is also a z term.

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And then do we have
any other z terms?

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Well, this is a z
term right over here.

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And in fact, these
two cancel out.

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You have negative
z omega squared,

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positive z omega squared.

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So this and this cancel out.

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And so our only z terms
are these three over here.

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So we have plus
omega to the third.

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And then we have plus
omega to the third.

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Oh let me-- this
color is too close.

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Plus omega to the third.

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And then our next z is minus 1.

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But then we actually
have this omega here.

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So plus omega minus 1.

00:05:51.737 --> 00:05:53.570
I was getting confused
because this white is

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so close to this pink.

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So it's omega to the third
minus 1, plus omega, times z.

00:06:02.470 --> 00:06:04.514
And then oh, we
can't forget this.

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We have this term over here.

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Omega to the fourth times z.

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So we have minus 1
omega to the third z.

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Then we have minus 1 z.

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Then we have plus omega z.

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And then we have negative
omega to the fourth z.

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So I'll just put it out here.

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I'm not doing it in a nice
descending power order,

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but I don't want to have
to rewrite all of this.

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So then we took care
of all of the z terms.

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And then what we have
left is just what

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we can view-- the constant
terms, from a z point of view.

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So we have omega--
So plus omega cubed.

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Actually, we have
two omega cubed.

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We have an omega cubed here
and an omega cubed there.

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We've taken care of those.

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And then we have minus
omega to the sixth.

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And then we have minus 1.

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And all of this
needs to equal 0.

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Now, this might look
pretty daunting,

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especially when omega looks
like this fairly complex-- well,

00:07:05.550 --> 00:07:07.930
it is a complex expression.

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But maybe, or probably,
it popped into your mind

00:07:11.070 --> 00:07:14.270
that this can also be expressed
as an exponential using

00:07:14.270 --> 00:07:17.470
Euler's formula.

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We know that e to the i theta is
equal to cosine of theta plus i

00:07:24.610 --> 00:07:27.680
sine of theta.

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So omega can be
rewritten as, omega

00:07:30.580 --> 00:07:38.480
is the exact same thing
as e to the 2 pi over 3 i.

00:07:38.480 --> 00:07:39.615
That's what omega is.

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And what is e to
the 2 pi over 3 i?

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We can evaluate that here.

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It's cosine 2 pi over 3.

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And we can get our
unit circle out.

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2 pi over 3 corresponds
to 120 degrees.

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So let me get my
unit circle out just

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so we can figure out
what the values are.

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And the reason why I'm
doing this-- I'm actually

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going to evaluate it here
just to get a number here--

00:08:02.688 --> 00:08:04.990
but the reason I'm doing this
is it becomes much easier

00:08:04.990 --> 00:08:09.060
to take the powers of
e to the 2 pi over 3 i

00:08:09.060 --> 00:08:11.104
than to take the powers
of this thing over here.

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But if we take the
unit circle, if we're

00:08:12.770 --> 00:08:17.090
at 120 degrees on the
unit circle like this,

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this is 2 pi over 3.

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Then we have a situation.

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This angle over here is
going to be 60 degrees.

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And so our height is going to
be square root of 3 over 2.

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And our x-coordinate is
going to be negative 1/2.

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So this is going to be
equal to negative 1/2.

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Cosine of 2 pi over
3 is negative 1/2.

00:08:37.840 --> 00:08:40.919
And then our sine of 2 pi over
3 is square root of 3 over 2.

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But that's being
multiplied by an i.

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So plus square
root of 3 over 2 i.

00:08:47.320 --> 00:08:49.120
So that is omega.

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Now, let's think
about omega squared.

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Omega squared is
going to be this

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raised to the second power.

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So it's e to the-- we just
multiply it by 2. e to the 4 pi

00:09:00.580 --> 00:09:03.000
over 3 i.

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We just multiply this by 2.

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So it's equivalent
to 240 degrees.

00:09:06.390 --> 00:09:09.829
So it's equivalent to
this right over here.

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So let me write it.

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This is the same thing
as cosine of 4 pi over 3,

00:09:16.240 --> 00:09:21.110
plus i sine of four pi over 3.

00:09:21.110 --> 00:09:24.060
And here our x-coordinate
is going to be the same.

00:09:24.060 --> 00:09:25.360
It's going to be negative 1/2.

00:09:25.360 --> 00:09:27.350
So this is equal
to negative 1/2.

00:09:27.350 --> 00:09:34.620
And our y-coordinate is negative
square root of 3 over 2 i.

00:09:34.620 --> 00:09:37.280
Now let's think of omega
to the third power.

00:09:37.280 --> 00:09:45.050
Omega to the third power is
equal to e to the 2 pi i, which

00:09:45.050 --> 00:09:51.010
is equal to cosine of 2
pi plus i sine of 2 pi.

00:09:51.010 --> 00:09:52.400
Well, sine of 2 pi is just 0.

00:09:52.400 --> 00:09:58.120
Cosine of 2 pi is equal to 1.

00:09:58.120 --> 00:10:00.650
Then we go to omega
to the fourth power,

00:10:00.650 --> 00:10:03.540
because that also shows
up in these expressions.

00:10:03.540 --> 00:10:08.640
So we have omega to the
fourth power is equal to-- we

00:10:08.640 --> 00:10:17.040
just multiply it times 4--
it's e to the 8 pi over 3 i.

00:10:17.040 --> 00:10:21.280
8 pi over 3 is equivalent
to-- so this was 2 pi over 3.

00:10:21.280 --> 00:10:23.030
This is 4 pi over 3.

00:10:23.030 --> 00:10:24.560
This is essentially 6 pi over 3.

00:10:24.560 --> 00:10:26.910
And then 8 pi over 3
becomes-- is essentially

00:10:26.910 --> 00:10:32.010
the same angle as 2 pi
over 3-- so its value is

00:10:32.010 --> 00:10:32.950
going to be the same.

00:10:32.950 --> 00:10:38.110
Negative 1/2 plus square
root of 3 over 2 i.

00:10:38.110 --> 00:10:40.170
And then we don't have
an omega to the fifth,

00:10:40.170 --> 00:10:41.940
but we do have an omega
to the sixth here.

00:10:41.940 --> 00:10:43.290
So let's just figure that out.

00:10:43.290 --> 00:10:46.095
Omega to the sixth-- well,
that's just omega cubed

00:10:46.095 --> 00:10:46.595
squared.

00:10:46.595 --> 00:10:48.180
And omega cubed is 1.

00:10:48.180 --> 00:10:49.850
So it's just going
to be 1 squared,

00:10:49.850 --> 00:10:51.245
or it's also going to be 1.

00:10:51.245 --> 00:10:52.870
Now with that out of
the way, let's see

00:10:52.870 --> 00:10:54.710
what these things simplify to.

00:10:54.710 --> 00:10:57.070
Hopefully they
simplify to something.

00:10:57.070 --> 00:10:58.740
So this expression
right here-- omega

00:10:58.740 --> 00:11:00.489
squared-- we have--
I'll write it up here.

00:11:00.489 --> 00:11:04.810
We have negative 1/2 minus
square root of 3 over 2 i.

00:11:04.810 --> 00:11:06.230
That's omega squared.

00:11:06.230 --> 00:11:08.140
Then we have plus omega.

00:11:08.140 --> 00:11:09.770
Omega is right here.

00:11:09.770 --> 00:11:14.500
Negative 1/2 plus square
root of 3 over 2 i.

00:11:14.500 --> 00:11:16.450
And then we have a plus 1.

00:11:16.450 --> 00:11:18.270
So the negative square
root of 3 over 2 i

00:11:18.270 --> 00:11:20.820
and the positive square root
of 3 over 2 i cancel out.

00:11:20.820 --> 00:11:24.370
Negative 1/2 minus 1/2 is
negative one, plus one.

00:11:24.370 --> 00:11:25.510
Those cancel out.

00:11:25.510 --> 00:11:28.182
So this whole expression
comes out to be 0.

00:11:28.182 --> 00:11:28.890
So we can ignore.

00:11:28.890 --> 00:11:31.970
This is going to be
0 no matter what.

00:11:31.970 --> 00:11:33.440
Now let's look at
this over here.

00:11:33.440 --> 00:11:36.410
This is omega cubed.

00:11:36.410 --> 00:11:40.910
Omega cubed we know
is 1, plus omega,

00:11:40.910 --> 00:11:44.790
which is-- so I
should write minus 1/2

00:11:44.790 --> 00:11:49.160
plus square root of
3 over 2 i, minus 1,

00:11:49.160 --> 00:11:51.505
minus omega to the fourth.

00:11:51.505 --> 00:11:54.290
Well, omega to the fourth
is the same thing as omega.

00:11:54.290 --> 00:11:57.670
Omega to the fourth is
the same thing as omega.

00:11:57.670 --> 00:11:59.367
So you have omega
minus something

00:11:59.367 --> 00:12:00.700
that's the same thing as itself.

00:12:00.700 --> 00:12:02.100
Those are going to cancel out.

00:12:02.100 --> 00:12:03.079
Omega cubed is 1.

00:12:03.079 --> 00:12:04.870
So the 1 and the 1 are
going to cancel out.

00:12:04.870 --> 00:12:07.230
So this is also equal to 0.

00:12:07.230 --> 00:12:08.890
I think I see a pattern here.

00:12:08.890 --> 00:12:11.348
And then finally, let's look
at these characters over here.

00:12:11.348 --> 00:12:12.930
2 times omega
cubed-- this is just

00:12:12.930 --> 00:12:16.740
going to be a 2 here--
minus omega to the sixth.

00:12:16.740 --> 00:12:18.190
Well, that's just minus 1.

00:12:18.190 --> 00:12:19.860
And then you have a minus 1.

00:12:19.860 --> 00:12:22.090
So this 2, minus 1, minus 1.

00:12:22.090 --> 00:12:25.270
This is also equal to 0.

00:12:25.270 --> 00:12:29.410
So that whole determinant that
whole equation has simplified

00:12:29.410 --> 00:12:34.020
to z to the third
power is equal to 0.

00:12:34.020 --> 00:12:35.920
And the only number,
that when they take it

00:12:35.920 --> 00:12:38.580
to the third power-- real,
or complex, or anything--

00:12:38.580 --> 00:12:40.290
is going to be 0.

00:12:40.290 --> 00:12:43.610
z equals 0 is the only solution.

00:12:43.610 --> 00:12:44.910
But they're not asking us that.

00:12:44.910 --> 00:12:48.070
They're asking the number
of distinct complex numbers

00:12:48.070 --> 00:12:49.720
z satisfying this.

00:12:49.720 --> 00:12:52.340
So z equals 0 is the only
one that satisfies this.

00:12:52.340 --> 00:12:56.810
So the number of complex
numbers satisfying it is 1.


Periodic Definite Integral

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.450
For any real number
x, let brackets

00:00:03.450 --> 00:00:06.380
around x denote the
largest integer less than

00:00:06.380 --> 00:00:10.200
or equal to x, often known as
the greatest integer function.

00:00:10.200 --> 00:00:12.730
Let f be a real valued
function defined

00:00:12.730 --> 00:00:14.600
on the interval
negative 10 to 10,

00:00:14.600 --> 00:00:16.580
including the
boundaries by f of x

00:00:16.580 --> 00:00:21.610
is equal to x minus the greatest
integer of x, if the greatest

00:00:21.610 --> 00:00:25.340
integer of x is odd, and 1
plus the greatest integer

00:00:25.340 --> 00:00:29.500
of x minus x, if the greatest
integer of x is even.

00:00:29.500 --> 00:00:32.700
Then the value of pi
squared over 10 times

00:00:32.700 --> 00:00:34.530
a definite integral
from negative 10 to 10

00:00:34.530 --> 00:00:38.860
of f of x cosine of pi of
x dx is-- so before even

00:00:38.860 --> 00:00:41.340
try to attempt to
evaluate this integral,

00:00:41.340 --> 00:00:44.060
let's see if we can at least
visualize this function,

00:00:44.060 --> 00:00:46.110
f of x, right over here.

00:00:46.110 --> 00:00:48.160
So let's do our best
to visualize it.

00:00:48.160 --> 00:00:52.910
So let me draw my x-axis.

00:00:52.910 --> 00:00:55.980
And let me draw my y-axis.

00:00:55.980 --> 00:00:59.000
So let me draw my y-axis.

00:00:59.000 --> 00:01:02.500
And then let's think about what
this function will look like.

00:01:02.500 --> 00:01:06.840
So this is x is equal to
0, this is x is equal to 0,

00:01:06.840 --> 00:01:11.700
this is x is equal to 1, x is
equal to 2, x is equal to 3.

00:01:11.700 --> 00:01:15.170
We could go down to
negative 1, negative 2.

00:01:15.170 --> 00:01:17.370
We could just keep
going, if we like.

00:01:17.370 --> 00:01:19.160
Hopefully we'll see
some type of pattern,

00:01:19.160 --> 00:01:21.790
because it seems to
change from odd to even.

00:01:21.790 --> 00:01:30.864
So between 0 and 1, what is
the greatest integer of x?

00:01:30.864 --> 00:01:32.280
So let me just
write it over here.

00:01:32.280 --> 00:01:36.790
So between 0 and 1,
until you get to 1--

00:01:36.790 --> 00:01:41.740
so maybe I should do this--
from including 0 until 1,

00:01:41.740 --> 00:01:46.470
the greatest integer
of x is equal to 0.

00:01:46.470 --> 00:01:51.830
If I'm at 0.5, the greatest
integer below 0.5 is 0.

00:01:51.830 --> 00:02:00.120
As I go from 1 to 2,
this brackets around x

00:02:00.120 --> 00:02:00.717
is equal to 1.

00:02:00.717 --> 00:02:01.800
It's the greatest integer.

00:02:01.800 --> 00:02:03.910
If I'm at 1.5, the
greatest integer is 1.

00:02:03.910 --> 00:02:06.600
If I'm at 1.9, the
greatest integer is 1.

00:02:06.600 --> 00:02:14.410
And then if I go to above
from between 2 and 3,

00:02:14.410 --> 00:02:17.750
then the greatest
integer is going to be 2.

00:02:17.750 --> 00:02:20.504
If I'm at 2.5, greatest
integer is going to be 2.

00:02:20.504 --> 00:02:22.170
So with that, let's
try to at least draw

00:02:22.170 --> 00:02:24.270
this function over
these intervals.

00:02:24.270 --> 00:02:27.620
So between 0 and 1, the
greatest integer is 0.

00:02:27.620 --> 00:02:31.300
0 we can consider to be even.

00:02:31.300 --> 00:02:33.660
0 is even, especially
if we're alternating.

00:02:33.660 --> 00:02:36.230
1 is odd, 2 is even, 3 is odd.

00:02:36.230 --> 00:02:37.410
So 0 is even.

00:02:37.410 --> 00:02:40.440
So we would look at this
circumstance right over here,

00:02:40.440 --> 00:02:41.820
if x is even.

00:02:41.820 --> 00:02:46.210
And then over this time frame
or over this part of the x-axis,

00:02:46.210 --> 00:02:48.190
the greatest integer
of x is just 0.

00:02:48.190 --> 00:02:51.890
So the equation or the
line or the function

00:02:51.890 --> 00:02:55.440
is just going to be 1
minus x over this interval,

00:02:55.440 --> 00:02:57.180
because the greatest
integer is 0.

00:02:57.180 --> 00:02:59.670
So 1 minus x will
look like this.

00:02:59.670 --> 00:03:05.410
If this is 1, 1 minus x
just goes down like that.

00:03:05.410 --> 00:03:07.640
That's what it looks
like from 0 to 1.

00:03:07.640 --> 00:03:11.530
Now let's think about what
happens as we go from 1 to 2.

00:03:11.530 --> 00:03:15.450
As we go from 1 to 2, not
including 2 but including 1,

00:03:15.450 --> 00:03:17.700
all the way up to
2, not including it,

00:03:17.700 --> 00:03:20.960
the greatest integer is 1.

00:03:20.960 --> 00:03:23.640
The greatest integer is odd.

00:03:23.640 --> 00:03:25.130
So we use this case.

00:03:25.130 --> 00:03:28.060
And over here, we're going
to have x minus the greatest

00:03:28.060 --> 00:03:29.990
integer of x over this interval.

00:03:29.990 --> 00:03:32.110
The greatest integer of x is 1.

00:03:32.110 --> 00:03:36.010
So it's going to be
the graph of x minus 1.

00:03:36.010 --> 00:03:39.440
So x minus 1 at 1
is going to be 0.

00:03:39.440 --> 00:03:41.770
And at 2, it's
going to be 1 again.

00:03:41.770 --> 00:03:43.990
So it's going to be this.

00:03:43.990 --> 00:03:46.180
It's going to look
just like that.

00:03:46.180 --> 00:03:48.660
So this right here is x minus 1.

00:03:48.660 --> 00:03:51.806
And then this over here,
essentially was 1 minus x.

00:03:51.806 --> 00:03:52.930
And we could keep doing it.

00:03:52.930 --> 00:03:57.370
As we go from 2 to 3, the
greatest integer of x is 2.

00:03:57.370 --> 00:03:58.860
We would look at this case.

00:03:58.860 --> 00:04:01.620
So we're going to have 1 plus 2.

00:04:01.620 --> 00:04:06.100
So we're going to
have 3 here minus x.

00:04:06.100 --> 00:04:07.800
So when we start
over here when x

00:04:07.800 --> 00:04:11.150
is 2 or a little bit above that,
we're going to have 3 minus 2.

00:04:11.150 --> 00:04:12.170
We're going go to 1.

00:04:12.170 --> 00:04:13.580
It's going to be right at 1.

00:04:13.580 --> 00:04:17.470
And then as x is equal
to 3, 3 minus 3 is 0.

00:04:17.470 --> 00:04:19.709
It's going to oscillate
back down like this.

00:04:19.709 --> 00:04:21.630
I think we have an
appreciation for what

00:04:21.630 --> 00:04:23.830
this graph is
going to look like.

00:04:23.830 --> 00:04:26.249
It's going to keep going
up and down like this,

00:04:26.249 --> 00:04:28.790
with a slope of negative 1, then
positive 1, then negative 1,

00:04:28.790 --> 00:04:29.850
then positive 1.

00:04:29.850 --> 00:04:33.340
It's just going to keep doing
that over and over again.

00:04:33.340 --> 00:04:35.340
You can keep trying it
out with other intervals,

00:04:35.340 --> 00:04:37.613
but it's pretty clear
that this is the pattern.

00:04:37.613 --> 00:04:41.100
Now, what we want to do is
evaluate the integral from 10

00:04:41.100 --> 00:04:45.270
to negative 10 of this function
times cosine of pi of x.

00:04:45.270 --> 00:04:47.400
So let's think about
cosine of pi of x

00:04:47.400 --> 00:04:50.230
and think about whether
that also is periodic.

00:04:50.230 --> 00:04:51.950
And of course, it is.

00:04:51.950 --> 00:04:53.690
And then if we can
simplify this integral

00:04:53.690 --> 00:04:57.620
so we don't have to evaluate
it over this entire period

00:04:57.620 --> 00:05:01.270
over here, maybe we can simplify
it into a simpler integral.

00:05:01.270 --> 00:05:08.760
So cosine of pi x,
cosine of 0 is 1.

00:05:08.760 --> 00:05:11.070
Don't want you to
get that wrong.

00:05:11.070 --> 00:05:12.470
Cosine pi 0 is cosine of 0.

00:05:12.470 --> 00:05:13.640
So that's 1.

00:05:13.640 --> 00:05:16.890
Cosine of pi is negative 1.

00:05:16.890 --> 00:05:21.390
So when x is equal to 1,
this becomes cosine of pi.

00:05:21.390 --> 00:05:24.240
So then the value of the
function is negative 1.

00:05:24.240 --> 00:05:25.540
It'll be over here.

00:05:25.540 --> 00:05:30.750
And then cosine of 2 pi, 2
times pi, is then 1 again.

00:05:30.750 --> 00:05:32.590
So it'll look like this.

00:05:35.270 --> 00:05:36.430
This is at 1/2.

00:05:36.430 --> 00:05:38.513
When you put it over here,
it'll become pi over 2.

00:05:38.513 --> 00:05:40.730
Cosine of pi over 2 is 0.

00:05:40.730 --> 00:05:43.910
So it'll look like this.

00:05:43.910 --> 00:05:46.490
Let me draw it as
neatly as possible.

00:05:46.490 --> 00:05:49.640
So it will look like this.

00:05:49.640 --> 00:05:51.830
Cosine, and then
it'll keep doing that,

00:05:51.830 --> 00:05:53.860
and then it'll go like this.

00:05:53.860 --> 00:05:56.570
So it is also periodic.

00:05:56.570 --> 00:06:02.320
So if we wanted to figure out
the integral of the product

00:06:02.320 --> 00:06:04.520
of these two periodic
functions from all the way

00:06:04.520 --> 00:06:07.930
from negative 10 to 10,
can we simplify that?

00:06:07.930 --> 00:06:10.392
And it looks like
it would just be,

00:06:10.392 --> 00:06:12.100
because we have this
interval, let's look

00:06:12.100 --> 00:06:15.560
at this interval over here.

00:06:15.560 --> 00:06:18.370
Let's look at just from 0 to 1.

00:06:18.370 --> 00:06:21.830
So just from 0 to 1, we're
going to take this function

00:06:21.830 --> 00:06:24.430
and take the product
of this cosine

00:06:24.430 --> 00:06:27.430
times essentially
1 minus x, and then

00:06:27.430 --> 00:06:31.040
find the area under that
curve, whatever it might be.

00:06:31.040 --> 00:06:33.790
Then when we go
from 1 to 2, when

00:06:33.790 --> 00:06:37.390
we take the product
of this and x minus 1,

00:06:37.390 --> 00:06:42.830
it's actually going to be the
same area, because these two,

00:06:42.830 --> 00:06:46.380
going from 0 to 1 and
going from 1 to 2,

00:06:46.380 --> 00:06:48.710
it's completely symmetric.

00:06:48.710 --> 00:06:51.000
You can flip it over
this line of symmetry,

00:06:51.000 --> 00:06:55.020
and both functions are
completely symmetric.

00:06:55.020 --> 00:06:56.770
So you're going to
have the same area when

00:06:56.770 --> 00:06:58.630
you take their product.

00:06:58.630 --> 00:07:01.850
So what we see is,
over every interval,

00:07:01.850 --> 00:07:05.159
when you go from 2 to 3,
first of all, the integral

00:07:05.159 --> 00:07:06.700
from 2 to 3 is
clearly the same thing

00:07:06.700 --> 00:07:08.500
as the integral from 0 to 1.

00:07:08.500 --> 00:07:12.520
Both functions look
identical over that interval.

00:07:12.520 --> 00:07:15.630
But it will also be the
same as going from 1 to 2,

00:07:15.630 --> 00:07:17.440
because it's
completely symmetric.

00:07:17.440 --> 00:07:19.530
When you take the
products of the function,

00:07:19.530 --> 00:07:22.645
that function will be completely
symmetric around this axis.

00:07:22.645 --> 00:07:24.020
So the integral
from here to here

00:07:24.020 --> 00:07:27.070
will be the same as the
integral from there to there.

00:07:27.070 --> 00:07:31.300
So with that said, we can
rewrite this thing over here.

00:07:31.300 --> 00:07:35.510
So what we want to evaluate,
pi squared over 10 times

00:07:35.510 --> 00:07:43.260
the integral from negative 10
to 10 of f of x cosine of pi x,

00:07:43.260 --> 00:07:45.130
using the logic we
just talked about.

00:07:45.130 --> 00:07:47.020
This is going to be
the same thing as being

00:07:47.020 --> 00:07:57.010
equal to pi squared over 10
times the integral from 0 to 1,

00:07:57.010 --> 00:07:59.060
but 20 times that,
because we have

00:07:59.060 --> 00:08:01.060
20 integers between
negative 10 and 10.

00:08:01.060 --> 00:08:03.580
We have 20 intervals
of length 1.

00:08:03.580 --> 00:08:10.596
So times 20 times the
integral from 0 to 1

00:08:10.596 --> 00:08:16.860
of f of x cosine of pi x dx.

00:08:16.860 --> 00:08:18.612
I forgot to write
the dx over there.

00:08:18.612 --> 00:08:20.070
I want to make sure
you understand,

00:08:20.070 --> 00:08:22.290
because this is really the
hard part of the problem,

00:08:22.290 --> 00:08:25.560
just realizing that the integral
over this interval is just 1/20

00:08:25.560 --> 00:08:28.606
of the whole thing, because over
every interval, from 0 to 1,

00:08:28.606 --> 00:08:30.730
the integral is going to
evaluate to the same thing

00:08:30.730 --> 00:08:33.188
as going from 1 to 2, which
will be the same thing as going

00:08:33.188 --> 00:08:35.897
from 2 to 3, or going from
negative 2 to negative 1.

00:08:35.897 --> 00:08:37.480
So instead of doing
the whole interval

00:08:37.480 --> 00:08:38.896
from negative 10
to 10, we're just

00:08:38.896 --> 00:08:42.210
doing 20 times the
interval from 0 to 1.

00:08:42.210 --> 00:08:46.070
From negative 10 to 10, there's
a difference of 20 here.

00:08:46.070 --> 00:08:47.370
So we're multiplying by 20.

00:08:47.370 --> 00:08:48.850
And this simplifies
it a good bit.

00:08:48.850 --> 00:08:51.810
First of all, this part
over here simplifies to 20

00:08:51.810 --> 00:08:53.270
divided by 10 is 2.

00:08:53.270 --> 00:08:55.610
So it's 2 pi squared.

00:08:55.610 --> 00:08:57.400
So it becomes 2 pi
squared-- that's just

00:08:57.400 --> 00:09:01.250
this part over here-- times
the integral from 0 to 1.

00:09:01.250 --> 00:09:03.340
Now, from 0 to 1,
what is f of x?

00:09:03.340 --> 00:09:06.890
Well, we just figured out,
from 0 to 1, f of x is just 1

00:09:06.890 --> 00:09:08.720
minus x.

00:09:08.720 --> 00:09:14.190
f of x is just 1 minus x
from 0 to 1 times cosine

00:09:14.190 --> 00:09:18.040
of pi x, cosine of pi x dx.

00:09:18.040 --> 00:09:21.360
And now we just have to evaluate
this integral right over here.

00:09:21.360 --> 00:09:22.830
So let's do that.

00:09:22.830 --> 00:09:25.050
So 1 minus x times
cosine of pi x

00:09:25.050 --> 00:09:32.500
is the same thing as cosine of
pi x minus x cosine of pi x.

00:09:32.500 --> 00:09:38.842
Now, this right here,
well, let's just

00:09:38.842 --> 00:09:40.300
focus on taking
the antiderivative.

00:09:40.300 --> 00:09:41.280
This is pretty easy.

00:09:41.280 --> 00:09:42.946
But let's try to do
this one, because it

00:09:42.946 --> 00:09:44.920
seems a little bit
more complicated.

00:09:44.920 --> 00:09:52.670
So let's take the antiderivative
of x cosine of pi x dx.

00:09:52.670 --> 00:09:54.620
And what should jump
in your mind is,

00:09:54.620 --> 00:09:55.829
well, this isn't that simple.

00:09:55.829 --> 00:09:57.786
But if I were able to
take the derivative of x,

00:09:57.786 --> 00:09:58.620
that would simplify.

00:09:58.620 --> 00:10:00.745
It's very easy to take the
antiderivative of cosine

00:10:00.745 --> 00:10:03.380
of pi x without making
it more complicated.

00:10:03.380 --> 00:10:05.970
So maybe integration by parts.

00:10:05.970 --> 00:10:07.720
And remember,
integration by parts

00:10:07.720 --> 00:10:10.300
tells us that the integral--
I'll write it up here--

00:10:10.300 --> 00:10:20.097
the integral of udv is equal to
uv minus the integral of vdu.

00:10:20.097 --> 00:10:21.180
And we'll apply that here.

00:10:21.180 --> 00:10:24.530
But I've done many, many videos
where I prove this and show

00:10:24.530 --> 00:10:27.050
examples of exactly
what that means.

00:10:27.050 --> 00:10:29.262
But let's apply it
right over here.

00:10:29.262 --> 00:10:30.720
And in general,
we're going to take

00:10:30.720 --> 00:10:32.590
the derivative of
whatever the u thing is.

00:10:32.590 --> 00:10:34.060
So we want u to be
something that's

00:10:34.060 --> 00:10:36.000
simpler when I take
the derivative.

00:10:36.000 --> 00:10:38.434
And then we're going to take
the antiderivative of dv.

00:10:38.434 --> 00:10:40.850
So we want something that does
not become more complicated

00:10:40.850 --> 00:10:42.655
when I take the antiderivative.

00:10:42.655 --> 00:10:44.030
So the thing that
becomes simpler

00:10:44.030 --> 00:10:45.730
when I take this
derivative is x.

00:10:45.730 --> 00:10:49.380
So if I set u is
equal to x, then

00:10:49.380 --> 00:10:53.110
clearly du is equal to just dx.

00:10:53.110 --> 00:10:54.890
Or you say du dx is equal to 1.

00:10:54.890 --> 00:10:57.180
So du is equal to dx.

00:10:57.180 --> 00:11:01.500
And then dv is going
to be the rest of this.

00:11:01.500 --> 00:11:05.060
This whole thing over
here is going to be dv.

00:11:05.060 --> 00:11:12.660
dv is equal to cosine pi x dx.

00:11:12.660 --> 00:11:15.210
And so v would just
be the antiderivative

00:11:15.210 --> 00:11:21.990
of this with respect to x.
v is going to be equal to 1

00:11:21.990 --> 00:11:25.550
over pi sine of pi x.

00:11:25.550 --> 00:11:27.930
If I took the derivative here,
derivative of the inside,

00:11:27.930 --> 00:11:30.000
you get a pi, times 1
over pi, cancels out.

00:11:30.000 --> 00:11:33.460
Derivative of sine of pi
x becomes cosine of pi x.

00:11:33.460 --> 00:11:38.710
So that's our u,
that's our v. This

00:11:38.710 --> 00:11:44.050
is going to be equal to u times
v. So it's equal to x, this x

00:11:44.050 --> 00:11:45.030
times this.

00:11:45.030 --> 00:11:56.670
So x over pi sine of pi x
minus the integral of v,

00:11:56.670 --> 00:12:03.140
which is 1 over pi
sine of pi x du.

00:12:03.140 --> 00:12:05.890
du is just dx.

00:12:05.890 --> 00:12:07.400
And this is pretty
straightforward.

00:12:07.400 --> 00:12:12.150
The antiderivative of sine of
pi x is 1 over pi or negative 1

00:12:12.150 --> 00:12:13.746
over pi cosine of pi x.

00:12:13.746 --> 00:12:15.120
And you could take
the derivative

00:12:15.120 --> 00:12:16.690
if you don't believe me.

00:12:16.690 --> 00:12:17.882
You could do u substitution.

00:12:17.882 --> 00:12:20.090
But hopefully you can start
to do these in your head,

00:12:20.090 --> 00:12:22.040
especially if you're going to
take the IIT joint entrance

00:12:22.040 --> 00:12:22.500
exam.

00:12:22.500 --> 00:12:23.875
So this whole
expression is going

00:12:23.875 --> 00:12:25.360
to be, this part
over here is going

00:12:25.360 --> 00:12:33.150
to be x over pi sine of pi x.

00:12:33.150 --> 00:12:37.170
And then this over here
is going to be-- well,

00:12:37.170 --> 00:12:39.080
you have the antiderivative
of sine of pi

00:12:39.080 --> 00:12:43.150
x is negative 1 over
pi cosine of pi x.

00:12:43.150 --> 00:12:44.450
The negatives cancel out.

00:12:44.450 --> 00:12:48.230
So you have plus, and then
the 1 over pi times 1 over pi,

00:12:48.230 --> 00:12:53.287
1 over pi squared
cosine of pi x.

00:12:53.287 --> 00:12:54.870
That's the antiderivative
right there.

00:12:54.870 --> 00:12:55.661
And you can verify.

00:12:55.661 --> 00:12:58.410
Derivative of cosine
of pi x is going

00:12:58.410 --> 00:13:01.185
to be negative pi sine of x.

00:13:01.185 --> 00:13:02.580
One pi will cancel out here.

00:13:02.580 --> 00:13:05.100
You get a negative sign, and
then you have sine of pi x.

00:13:05.100 --> 00:13:07.600
So this is the
antiderivative of that.

00:13:07.600 --> 00:13:09.580
And so if we want
the antiderivative

00:13:09.580 --> 00:13:13.770
of this whole thing
right over here,

00:13:13.770 --> 00:13:15.630
this is what we
care about from 0

00:13:15.630 --> 00:13:19.820
to 1 dx, the
antiderivative of cosine pi

00:13:19.820 --> 00:13:21.280
x-- pretty straightforward.

00:13:21.280 --> 00:13:23.238
We've actually already
done it right over here.

00:13:23.238 --> 00:13:29.030
It is 1 over pi sine of pi x.

00:13:29.030 --> 00:13:30.250
That's this first term.

00:13:30.250 --> 00:13:32.460
And the antiderivative
of x cosine pi x

00:13:32.460 --> 00:13:33.650
is this thing over here.

00:13:33.650 --> 00:13:35.150
But we're subtracting it.

00:13:35.150 --> 00:13:37.320
So we'll put a negative
sign out front.

00:13:37.320 --> 00:13:48.940
So minus x over pi sine of pi x
minus 1 over pi squared cosine

00:13:48.940 --> 00:13:49.975
of pi x.

00:13:49.975 --> 00:13:51.725
And of course, we took
the antiderivative,

00:13:51.725 --> 00:13:53.120
but it's a definite integral.

00:13:53.120 --> 00:13:57.000
We need to evaluate
it from 0 to 1.

00:13:57.000 --> 00:13:59.900
And we don't want to forget
that 2 pi squared out front.

00:14:03.600 --> 00:14:05.400
So let's evaluate this.

00:14:05.400 --> 00:14:10.260
So the first thing, we're going
to have 1 over pi sine of 1 pi.

00:14:10.260 --> 00:14:13.630
Sine of 1 pi is 0.

00:14:13.630 --> 00:14:18.070
So you have 0 minus 1 over
pi times sine of 1 pi again.

00:14:18.070 --> 00:14:25.400
That's again 0, minus 1 over
pi squared cosine of pi,

00:14:25.400 --> 00:14:26.480
or cosine of 1 pi.

00:14:26.480 --> 00:14:28.252
Cosine of pi is negative 1.

00:14:28.252 --> 00:14:30.390
Negative 1 times negative
1 over pi squared

00:14:30.390 --> 00:14:33.270
is plus 1 over pi squared.

00:14:33.270 --> 00:14:35.100
So we've evaluated it at 1.

00:14:35.100 --> 00:14:39.200
And from that, we want to
subtract it evaluated at 0.

00:14:39.200 --> 00:14:43.710
Sine of 0 is 0 minus--
this is clearly 0,

00:14:43.710 --> 00:14:46.360
because you have a 0
out front-- minus 0.

00:14:46.360 --> 00:14:50.180
And then you have
minus cosine of 0.

00:14:50.180 --> 00:14:52.630
Cosine of 0 is 1.

00:14:52.630 --> 00:14:55.170
So then you have a
minus 1 over pi squared.

00:14:57.730 --> 00:15:01.400
And so this term, we could
just say this is a negative.

00:15:01.400 --> 00:15:04.030
These don't matter--
negative, positive, positive.

00:15:04.030 --> 00:15:07.140
And we're just left with
a 1 over pi squared plus 1

00:15:07.140 --> 00:15:14.700
over pi squared, which is
equal to 2 over pi squared.

00:15:14.700 --> 00:15:16.440
That's what this
part evaluates to.

00:15:16.440 --> 00:15:17.900
It's 2 over pi squared.

00:15:20.394 --> 00:15:21.810
We can't forget
that we were going

00:15:21.810 --> 00:15:24.105
to multiply this whole
thing times 2 pi squared.

00:15:27.460 --> 00:15:29.370
That's this thing
out front here.

00:15:29.370 --> 00:15:31.650
And so the pi squared
cancels out the pi squared.

00:15:31.650 --> 00:15:36.330
We're left with 2 times
2, which is equal to 4.

00:15:36.330 --> 00:15:37.772
And we're done.

00:15:37.772 --> 00:15:39.480
This thing that looked
pretty complicated

00:15:39.480 --> 00:15:42.392
just evaluates out to 4.


Distance between planes

WEBVTT
Kind: captions
Language: en

00:00:01.602 --> 00:00:09.985
If the distance between the plane Ax-2y+z = d and the plane containing the lines,

00:00:09.985 --> 00:00:12.585
and they give us two lines here in three-dimensions,

00:00:12.585 --> 00:00:18.084
if that distance is square-root of 6, then the absolute value of d is...

00:00:18.084 --> 00:00:25.472
So let's think about it for a little bit. They're talking about the distance between this plane and some

00:00:25.472 --> 00:00:28.625
plane that contains these two line.

00:00:28.625 --> 00:00:33.331
So in order to talk realistically about the distance between the planes, those planes will have to be

00:00:33.331 --> 00:00:37.928
parallel, because if they're not parallel - if they intersect with each other, the distance is clearly

00:00:37.928 --> 00:00:41.821
zero, and they're telling us here that the distance is square-root of 6.

00:00:41.821 --> 00:00:46.143
So we have a situation so that the planes can't intersect they must be parallel.

00:00:46.143 --> 00:00:53.431
So you have this plane up here, you have this plane up here, you could call this the equation here

00:00:53.431 --> 00:01:01.510
is ax - 2y + z = d, and then you're going to have another plane, that's going to be parallel to it,

00:01:01.510 --> 00:01:07.403
maybe it looks something like this. You have the other plane that is parallel, and it's going to contain

00:01:07.403 --> 00:01:14.843
both of these lines. So maybe it has this line - so this line is in green, maybe this line looks something

00:01:14.843 --> 00:01:23.045
like this, it's on that blue plane. And then this line, maybe in magenta, is also going to be on the

00:01:23.045 --> 00:01:26.912
blue plane. So how can we figure out the distances?

00:01:26.912 --> 00:01:33.996
Well a good starting point would be to try to figure out the equation for this blue plane here.

00:01:33.996 --> 00:01:40.659
And since these planes are parallel, this equation should look very much like this orange equation -

00:01:40.659 --> 00:01:44.597
at least on the left-hand-side, it might just have a different d-value, and that's because it has the

00:01:44.597 --> 00:01:52.338
exact same inclination. And then once we figure out the equation for this plane over here, then we could

00:01:52.338 --> 00:01:58.852
actually probably figure out what 'a' is, then we could find some point on the blue plane and then use

00:01:58.852 --> 00:02:05.550
our knowledge of finding the distance points and planes to figure out the actual distance from any point

00:02:05.550 --> 00:02:07.454
to this orange plane.

00:02:07.454 --> 00:02:12.354
So let's figure out the equation of this blue plane first, and a good place to start is just to try to

00:02:12.354 --> 00:02:19.624
figure out two vectors on this blue plane, then we can take the cross-product of those two vectors to

00:02:19.624 --> 00:02:24.707
find out a normal to this blue plane, and then use that information to actually figure out the

00:02:24.707 --> 00:02:26.982
equation for the blue plane.

00:02:26.982 --> 00:02:31.047
So let's figure out some points that sit on the blue plane.

00:02:31.047 --> 00:02:37.315
So on this green line right over here you have, see if I want all of these to be equal to zero, you'd

00:02:37.315 --> 00:02:46.492
have the point x is equal to 1, y is equal to 2, z is equal to 3, so you have the point (1,2,3) - that definitely sits

00:02:46.492 --> 00:02:52.199
on the blue plane. Let's come up with another point. let's see if I want all of these to be equal to

00:02:52.199 --> 00:02:59.931
one, I could make this, if I want this to evaluate to two, I'd have the point three, I would have the point,

00:02:59.931 --> 00:03:07.942
if I want this to evaluate to one, I would want five minus two over three, so 3, 5, and then I would want this

00:03:07.942 --> 00:03:16.557
to be seven. Seven minus three over four is also 1, so that's another point, actually both of these points sit on this

00:03:16.557 --> 00:03:24.508
line right over here. 1,2,3 and then 3,5,7. And then let's do the same thing for this - let's find two

00:03:24.508 --> 00:03:29.723
points on this plane, and actually we just need to find one point on this plane, because if you have

00:03:29.723 --> 00:03:35.677
three points that's enough to figure out two different vectors -vectors that aren't scalar multiples

00:03:35.677 --> 00:03:39.638
of each other, which would be enough to figure out the normal to this plane.

00:03:39.638 --> 00:03:44.215
So let's just figure out one more point over here and one more point would be, if we want all of these

00:03:44.215 --> 00:03:47.556
three to be equal to zero, would be the point (2,3,4).

00:03:52.310 --> 00:03:57.667
Because this would be zero, zero, zero. So that's also sitting the plane and that sits on the magenta

00:03:57.667 --> 00:03:59.026
line right over there.

00:03:59.026 --> 00:04:03.461
So let's use these three points to figure out two vectors on the plane that aren't multiples of each

00:04:03.461 --> 00:04:11.333
other, then we can take their cross-product to actually figure out a normal vector to the blue plane.

00:04:11.333 --> 00:04:16.788
So let's say the first vector, 'a', that sits on this plane, let's say it's the difference in the position

00:04:16.788 --> 00:04:21.526
vectors that specify these two points, and then we know that will be on the plane.

00:04:21.526 --> 00:04:37.362
And so that will be three minus one is 2i plus five minus two is 3j, plus seven minus three is 4k.

00:04:37.362 --> 00:04:44.014
So vector 'a' is actually going to sit on this green line, because both of these points are on this line,

00:04:44.014 --> 00:04:48.554
so it's going to sit on that line. If we put it on the plane or if we were to start it at one of those

00:04:48.554 --> 00:04:50.667
points it will sit on that line.

00:04:50.667 --> 00:04:55.636
And then we can do another vector, and it's essentially going between a point on the green line and a

00:04:55.636 --> 00:04:59.862
point on the purple line, but that's definitely going to be a vector on our blue plane.

00:04:59.862 --> 00:05:04.718
Let's go between these two points - that looks pretty straight-forward, so let's call vector b,

00:05:07.902 --> 00:05:21.480
so let's call vector b, let's call that, let's see, two minus one is 'i', three minus two is 'j', and then four minus three, that's just 1k.

00:05:21.480 --> 00:05:29.375
So this vector here is also sitting on the plane. So if I take the cross-product of 'a' and 'b' I am

00:05:29.375 --> 00:05:34.656
going to get a vector that is perpendicular to the plane, or a normal vector to the plane.

00:05:34.656 --> 00:05:43.493
So let's do that. So let's find what 'a' cross 'b' is. 'a' cross 'b' is equal to - and this is how I

00:05:43.493 --> 00:05:49.223
find it easiest - I just write 'i', 'j', 'k', this is really the definition of the cross product, or

00:05:49.223 --> 00:05:56.194
I guess one of them. And we write our first vector, we have 2, 3, 4, and then we have our second vector,

00:05:56.194 --> 00:06:02.968
which is just 1,1,1, and then this is going to be equal to, first we'll look at the 'i' component, so

00:06:02.968 --> 00:06:04.941
cross that row, that column out.

00:06:04.941 --> 00:06:13.029
three times one minus one times four, so that's just three minus four, so it's negative 'i', and then

00:06:13.029 --> 00:06:19.089
minus, we're going to have the 'j', so let we write up minus here, minus , let me just swap signs, we

00:06:19.089 --> 00:06:23.454
have positive, negative, positive, so 'j' , get rid of that column, that row,

00:06:23.454 --> 00:06:28.725
two times one which is two, minus one times four, so that's minus four, is negative 2,

00:06:28.725 --> 00:06:33.927
so we could have written negative 2 here, but the negatives cancel out, so it becomes plus 2j,

00:06:33.927 --> 00:06:39.778
and finally for the 'k', get rid of that row that column, two times one is two, minus one times three,

00:06:39.778 --> 00:06:44.352
is two minus three, which is negative one, so it's negative k.

00:06:44.352 --> 00:06:52.270
So this right here is a normal vector to the plane.

00:06:52.270 --> 00:07:00.389
So if we want to find the equation for that plane, we've done it multiple times, we'd just have to take

00:07:00.389 --> 00:07:13.609
the dot product of that normal vector and any arbitrary vector on that that specified that we can specify

00:07:13.609 --> 00:07:17.859
with with an arbitrary x y and z, and we've done this multiple times in multiple videos.

00:07:17.859 --> 00:07:29.097
If this is any point, x, y, z, that sits on the plane, then the vector - let me draw the vector, let's

00:07:29.097 --> 00:07:34.740
say we go to this point right over here, so this vector right over here is going to be

00:07:34.740 --> 00:07:39.639
- let me draw it the other way actually, so this vector right over here, let's say we're going between

00:07:39.639 --> 00:07:53.014
this point and x,y,z, this vector right here is going to be x minus three i, plus y minus five j, plus

00:07:53.014 --> 00:07:56.369
z minus 7 k.

00:07:56.385 --> 00:08:01.119
That's what this vector is, it sits on the plane, assuming x,y and z sit on the plane.

00:08:01.119 --> 00:08:06.458
So if we take the dot product of this and the normal vector that has got to be equal to zero,

00:08:06.458 --> 00:08:09.430
because it sits on the plane, and then we'll have our equation.

00:08:09.430 --> 00:08:23.806
So let's take 'n' dot that over there, so n dot x minus three i, plus y minus five j, plus z minus seven k.

00:08:23.806 --> 00:08:28.540
If any of this is confusing to you, I've gone into a little bit more depth in previous videos,

00:08:28.540 --> 00:08:32.348
especially in the linear algebra playlist where I talk about constructing the equation of a plane given

00:08:32.348 --> 00:08:38.177
a point on the plane and a normal vector, and even how you find that normal vector,

00:08:38.177 --> 00:08:40.720
so you might want to watch those if you want some review there.

00:08:40.720 --> 00:08:46.698
But these are going to be equal to zero, so when you take the dot product, n, our normal vector is this,

00:08:46.698 --> 00:08:51.087
so we just take the x term, which is negative one, times this x term right over here,

00:08:51.087 --> 00:08:58.749
so negative one times this is just three minus x, and then plus this y component times this y component,

00:08:58.749 --> 00:09:05.785
so it's two times this, so it's plus two y minus ten, and then finally the z component -

00:09:05.785 --> 00:09:15.979
negative one times this. So this is plus seven minus z is equal to zero, and what do we get?

00:09:15.979 --> 00:09:27.496
So we have our negative x, plus two y, minus z, and then is equal to, let's subtract three from both sides

00:09:27.496 --> 00:09:35.251
So if we take it out there it will be minus three. If we add ten to both sides, so then you have

00:09:35.251 --> 00:09:43.401
a plus ten over here, and then we subtract seven from both sides, this becomes a minus seven

00:09:43.401 --> 00:09:50.689
So then on the right-hand-side, negative three plus ten minus seven, well that's just going to be zero!

00:09:50.689 --> 00:10:04.276
And just like that we have the equation for this blue plane over here - the plane that contains these two lines.

00:10:04.276 --> 00:10:08.511
Now remember what we said at the beginning of the video - these two planes are parallel, so the ratio

00:10:08.511 --> 00:10:14.655
of the coefficients on the 'x' terms, the 'y' term and the 'z' term has got to be the same.

00:10:14.655 --> 00:10:20.739
So this one has a positive 2, that has a negative 2, this is just to simplify it, so it looks very similar

00:10:20.739 --> 00:10:25.476
to each other. Let's multiple this equation right here, both sides, by negative one. And then we're going

00:10:25.476 --> 00:10:34.698
to get x minus two y plus z is equal to zero; so this is a completely valid, another alternate way of

00:10:34.698 --> 00:10:39.129
expressing the same plane. And what like about this about this is that it looks very similar to this -

00:10:39.129 --> 00:10:46.304
at least the ratio of the x, y's and z's, negative two y, negative two y; one z, one z, and remember

00:10:46.304 --> 00:10:47.860
the ratios have to be the same.

00:10:47.860 --> 00:10:52.945
So here we have a one-to-one ratio between the z-coefficient and the z-coefficient, the y-coefficient

00:10:52.945 --> 00:10:56.335
and the y-coefficient, so it's also going to be for the x coefficient.

00:10:56.335 --> 00:11:01.885
So here we know if this is going to be parallel to the blue plane we know that 'a' has got to be equal

00:11:01.885 --> 00:11:03.301
to one.

00:11:03.301 --> 00:11:18.342
So this is x minus two y plus z is equal to d. So now let's figure out the actual distance between these

00:11:18.342 --> 00:11:24.942
two planes. So what we can do is we can take a point on this blue plane - and we have several examples

00:11:24.942 --> 00:11:30.469
of points on the blue plane, and find the distance between that point and this plane over here.

00:11:30.469 --> 00:11:35.600
And actually I just finished doing some videos on how to find the distance between a point and a plane,

00:11:35.600 --> 00:11:39.780
so I'm just going to use that formula (if you want it to be proved go watch that video - it's actually

00:11:39.780 --> 00:11:47.127
pretty interesting proof I think) but the distance between let's say this point, (1,2,3), and this plane

00:11:47.127 --> 00:11:52.179
over here, so this distance right here is going to be in the direction of the normal,

00:11:52.179 --> 00:12:08.364
the distance is going to be, you literally just evaluate this, let me do it this way:

00:12:08.364 --> 00:12:14.331
you literally just put in this point for the x, y and z, and then you subtract the d in the numerator,

00:12:14.331 --> 00:12:20.229
and we saw that as the formula for finding the distance. So it's literally going to be one, one,

00:12:20.229 --> 00:12:23.062
(I'm actually using this point right over here)

00:12:23.062 --> 00:12:27.218
So it's going to be one, one, because we just have one x so it's just going to be one minus two times

00:12:27.218 --> 00:12:39.432
two, one minus four (that's two times two), plus three, minus d, over here the d is just d, so we're

00:12:39.432 --> 00:12:46.235
just going to write minus d, just like that, all of that over what is essentially the magnitude of the

00:12:46.235 --> 00:12:50.601
normal vector, and we saw in several videos that's just the square of the coefficients on each of these

00:12:50.601 --> 00:12:54.409
terms right here, and taking the sum of those, taking the square root,

00:12:54.409 --> 00:13:05.267
so it's going to be equal to: 1 squared, plus negative 2 squared which is four, plus one squared which is one.

00:13:05.413 --> 00:13:13.008
So this is going to simplify to the distance is equal to one minus four plus three is zero.

00:13:13.008 --> 00:13:21.200
So in the numerator we have negative d, all of that over the square root of one plus four plus one,

00:13:21.200 --> 00:13:30.949
so all over the square root of six. So they say the distance

00:13:30.949 --> 00:13:36.506
between this plane and this plane over here is square root of six. So they're saying the distance is

00:13:36.506 --> 00:13:44.953
equal to the square root of six, that's what this information right over here is, maybe I should do that

00:13:44.953 --> 00:13:47.466
in another colour.

00:13:49.343 --> 00:13:55.431
The distance between the two planes is going to be the square root of six, and so then if we solve for d,

00:13:55.431 --> 00:14:01.352
multiple both sides of this equation times the square root of six, you get six is equal to negative d,

00:14:01.352 --> 00:14:07.203
or d is equal to negative six.

00:14:07.203 --> 00:14:12.079
Now, what they care about is the absolute value of d, or the absolute distance, so this would be kinda

00:14:12.079 --> 00:14:15.818
the signed distance - it specifies whether we're above or below the plane.

00:14:15.818 --> 00:14:19.696
Since we're below the plane we got a negative number - I just happened to draw it right, if we above

00:14:19.696 --> 00:14:22.366
the plane we would get a positive number.

00:14:22.366 --> 00:14:27.730
So this distance is negative six, the absolute value of it, the absolute value of d which is the same

00:14:27.730 --> 00:14:31.724
as the absolute value of negative six, is equal to six.

00:14:31.724 --> 00:14:38.759
So take any point on this blue plane and you look for the closest point on the orange plane, and they

00:14:38.759 --> 00:14:44.402
will be exactly six apart. Anyway, hopefully you found that interesting.


Point distance to plane

WEBVTT
Kind: captions
Language: en

00:00:00.424 --> 00:00:01.840
What I want to do
in this video is

00:00:01.840 --> 00:00:04.370
start with some point
that's not on the plane,

00:00:04.370 --> 00:00:06.420
or maybe not necessarily
on the plane.

00:00:06.420 --> 00:00:09.890
So let me draw a
point right over here.

00:00:09.890 --> 00:00:13.460
And let's say the coordinates
of that point are x 0 x sub 0,

00:00:13.460 --> 00:00:16.500
y sub 0, and z sub 0.

00:00:16.500 --> 00:00:19.100
Or it could be specified
as a position vector.

00:00:19.100 --> 00:00:21.570
I could draw the position
vector like this.

00:00:21.570 --> 00:00:24.930
So the position vector-- let
me draw a better dotted lines.

00:00:24.930 --> 00:00:32.670
The position vector for this
could be x0i plus y0j plus z0k.

00:00:32.670 --> 00:00:35.280
It specifies this
coordinate right over here.

00:00:35.280 --> 00:00:37.930
What I want to do
is find the distance

00:00:37.930 --> 00:00:39.400
between this point
and the plane.

00:00:39.400 --> 00:00:41.400
And obviously, there could
be a lot of distance.

00:00:41.400 --> 00:00:43.490
I could find the distance
between this point

00:00:43.490 --> 00:00:45.530
and that point, and this
point and this point,

00:00:45.530 --> 00:00:47.000
and this point this point.

00:00:47.000 --> 00:00:48.791
And when I say I want
to find the distance,

00:00:48.791 --> 00:00:50.360
I want to find the
minimum distance.

00:00:50.360 --> 00:00:53.130
And you're actually going to
get the minimum distance when

00:00:53.130 --> 00:00:56.930
you go the perpendicular
distance to the plane,

00:00:56.930 --> 00:00:58.640
or the normal
distance to the plane.

00:00:58.640 --> 00:01:00.540
And we'll, hopefully,
see that visually

00:01:00.540 --> 00:01:03.650
as we try to figure out how
to calculate the distance.

00:01:03.650 --> 00:01:06.040
So the first thing we can
do is, let's just construct

00:01:06.040 --> 00:01:08.060
a vector between
this point that's

00:01:08.060 --> 00:01:10.350
off the plane and some
point that's on the plane.

00:01:10.350 --> 00:01:12.600
And we already have a point
from the last video that's

00:01:12.600 --> 00:01:15.480
on the plane, this x
sub p, y sub p, z sub p.

00:01:15.480 --> 00:01:17.570
So let's construct
a vector here.

00:01:17.570 --> 00:01:21.080
Let's construct this
orange vector that

00:01:21.080 --> 00:01:23.650
starts on the plane, it's
tail is on the plane,

00:01:23.650 --> 00:01:25.560
and it goes off the plane.

00:01:25.560 --> 00:01:27.440
I want to do that in orange.

00:01:27.440 --> 00:01:32.950
It goes off the plane to
this vector, to this position

00:01:32.950 --> 00:01:34.880
x0 y0 z0.

00:01:34.880 --> 00:01:40.350
So how could we specify this
vector, right over here?

00:01:40.350 --> 00:01:42.859
Well, that vector, let
me call that vector,

00:01:42.859 --> 00:01:45.150
well, I'll just call that
vector, what letters have I'm

00:01:45.150 --> 00:01:45.810
not used yet?

00:01:45.810 --> 00:01:48.090
Let me call that vector f.

00:01:48.090 --> 00:01:53.350
Vector f is just going to
be this yellow position

00:01:53.350 --> 00:01:56.260
vector, minus this
green position vector.

00:01:56.260 --> 00:01:59.215
So it's going to
be, this x component

00:01:59.215 --> 00:02:01.340
is going to be the difference
of the x-coordinates,

00:02:01.340 --> 00:02:03.923
it's y-coordinate is going the
difference of the y-coordinate.

00:02:03.923 --> 00:02:08.370
S So it's going to
be x0 minus x sub p.

00:02:08.370 --> 00:02:11.730
I subtracted the
x-coordinates, i.

00:02:11.730 --> 00:02:19.790
Plus y0 minus ypj plus-- we'll
go to the next line-- plus z0

00:02:19.790 --> 00:02:23.600
minus zp minus zpk.

00:02:23.600 --> 00:02:24.410
So fair enough.

00:02:24.410 --> 00:02:28.630
That's just some vector
that comes off of the plane

00:02:28.630 --> 00:02:29.990
and onto this point.

00:02:29.990 --> 00:02:34.440
But what we want to find
out is this distance.

00:02:34.440 --> 00:02:36.620
We want to find out
this distance in yellow,

00:02:36.620 --> 00:02:39.590
the distance that if I were
take a normal off of the plane

00:02:39.590 --> 00:02:41.190
and go straight to
the point, that's

00:02:41.190 --> 00:02:42.390
going to be the
shortest distance.

00:02:42.390 --> 00:02:44.230
And actually, you can
see it visually now.

00:02:44.230 --> 00:02:47.170
Because if look at-- we can
actually form a right triangle

00:02:47.170 --> 00:02:50.970
here-- so this base of the right
triangle is along the plane.

00:02:50.970 --> 00:02:52.970
This side is normal
to the plane.

00:02:52.970 --> 00:02:54.970
So this is a right angle.

00:02:54.970 --> 00:02:58.620
And you can see, if I take
any point, any other point

00:02:58.620 --> 00:03:03.560
on the plane, it will form a
hypotenuse on a right triangle.

00:03:03.560 --> 00:03:06.130
And obviously the shortest
side here, or the shortest

00:03:06.130 --> 00:03:07.720
way to get to the
plane, is going

00:03:07.720 --> 00:03:11.080
to be this distance, right here,
as opposed to the hypotenuse.

00:03:11.080 --> 00:03:14.900
This side will always be
shorter than that side.

00:03:14.900 --> 00:03:17.190
So given that we know
this vector here,

00:03:17.190 --> 00:03:21.500
how can we figure
out this length here?

00:03:21.500 --> 00:03:24.450
How can we figure out
this length here in blue?

00:03:24.450 --> 00:03:27.550
Well, we could figure out
the magnitude of this vector.

00:03:27.550 --> 00:03:30.439
So the length of
this side right here

00:03:30.439 --> 00:03:32.230
is going to be the
magnitude of the vector,

00:03:32.230 --> 00:03:34.615
so it's going to be the
magnitude of the vector f.

00:03:34.615 --> 00:03:35.990
That'll just give
us this length.

00:03:35.990 --> 00:03:37.661
But we want this blue length.

00:03:37.661 --> 00:03:38.910
Well, we could think about it.

00:03:38.910 --> 00:03:41.770
If this was some angle-- I know
the writing is getting small.

00:03:41.770 --> 00:03:45.370
If this was some angle theta, we
could use some pretty straight

00:03:45.370 --> 00:03:49.530
up, pretty straightforward
trigonometry.

00:03:49.530 --> 00:03:51.900
If the distance
under question is d,

00:03:51.900 --> 00:03:59.232
you could say cosine of theta
is equal to the adjacent side

00:03:59.232 --> 00:04:00.065
over the hypotenuse.

00:04:00.065 --> 00:04:03.990
Or is is equal to d-- d
is the adjacent side-- is

00:04:03.990 --> 00:04:06.400
equal to d over the hypotenuse.

00:04:06.400 --> 00:04:09.070
Well, the hypotenuse is the
magnitude of this vector.

00:04:09.070 --> 00:04:12.000
It's the magnitude
of the vector f.

00:04:12.000 --> 00:04:15.320
Or we could say the
magnitude of the vector

00:04:15.320 --> 00:04:18.660
f times the cosine of
theta-- I'm just multiplying

00:04:18.660 --> 00:04:25.290
both sides times the magnitude
of vector x-- f is equal to d.

00:04:25.290 --> 00:04:30.112
But still you might say, OK,
well Sal, we know what f is.

00:04:30.112 --> 00:04:31.070
We can figure that out.

00:04:31.070 --> 00:04:33.400
We can figure out its magnitude.

00:04:33.400 --> 00:04:35.370
But we don't know what theta is.

00:04:35.370 --> 00:04:37.490
How do we figure out what theta?

00:04:37.490 --> 00:04:40.020
And to do that, let's just
think about it a little bit.

00:04:40.020 --> 00:04:44.020
This angle, this angle of
theta, is the same angle.

00:04:44.020 --> 00:04:46.150
So this distance here
isn't necessarily

00:04:46.150 --> 00:04:49.110
the same as the length
of the normal vector.

00:04:49.110 --> 00:04:52.170
But it's definitely going
in the same direction.

00:04:52.170 --> 00:04:54.560
So this angle here, is
really the same thing

00:04:54.560 --> 00:04:59.150
as the angle between this
vector and the normal vector.

00:04:59.150 --> 00:05:02.420
And so you might remember
from earlier linear algebra,

00:05:02.420 --> 00:05:05.380
when we talk about the dot
product of two vectors,

00:05:05.380 --> 00:05:06.960
it involves something
with the cosine

00:05:06.960 --> 00:05:08.450
of the angle between them.

00:05:08.450 --> 00:05:11.920
And to make that fresh
in your mind, let's

00:05:11.920 --> 00:05:15.270
multiply and divide both sides.

00:05:15.270 --> 00:05:17.660
Let me multiply and divide
the left side of this equation

00:05:17.660 --> 00:05:19.724
by the magnitude of
the normal vector.

00:05:19.724 --> 00:05:21.390
So I'm obviously not
changing its value.

00:05:21.390 --> 00:05:23.716
I'm multiplying and
dividing by the same number.

00:05:23.716 --> 00:05:26.340
So I'm going to multiply by the
magnitude of the normal vector.

00:05:26.340 --> 00:05:31.750
And I'm going to divide by the
magnitude of the normal vector.

00:05:31.750 --> 00:05:33.930
So I'm just essentially
multiplying by 1.

00:05:33.930 --> 00:05:35.590
So I have not changed this.

00:05:35.590 --> 00:05:38.110
But when you do it in
this, it might ring a bell.

00:05:38.110 --> 00:05:42.450
This expression up here,
this expression right here,

00:05:42.450 --> 00:05:48.370
is the dot product of the
normal vector and this vector

00:05:48.370 --> 00:05:49.840
right here, f.

00:05:49.840 --> 00:05:52.200
So this right here
is the dot product.

00:05:52.200 --> 00:05:56.580
This is n dot f, up there.

00:05:56.580 --> 00:05:58.930
It's equal to the product
of their magnitudes

00:05:58.930 --> 00:06:01.330
times the cosine of
the angle between them.

00:06:01.330 --> 00:06:03.640
So the distance, that shortest
distance we care about,

00:06:03.640 --> 00:06:07.410
is a dot product between this
vector, the normal vector,

00:06:07.410 --> 00:06:17.342
divided by the magnitude
of the normal vector.

00:06:17.342 --> 00:06:18.050
So let's do that.

00:06:18.050 --> 00:06:21.330
Let's take the dot product
between the normal and this.

00:06:21.330 --> 00:06:23.260
And we already figured
out, in the last video,

00:06:23.260 --> 00:06:25.744
the normal vector, if you
have the equation of a plane,

00:06:25.744 --> 00:06:27.660
the normal vector is
literally, its components

00:06:27.660 --> 00:06:29.850
are just the coefficients
on the x, y, and z terms.

00:06:29.850 --> 00:06:32.390
So this is a normal
vector right over here.

00:06:32.390 --> 00:06:35.590
So let's literally
take the dot product.

00:06:35.590 --> 00:06:47.120
So n dot f is going to be
equal to A times x0 minus xp.

00:06:47.120 --> 00:06:48.520
I'll do that in pink.

00:06:48.520 --> 00:06:53.950
So it'll be Ax0 minus Axp.

00:06:53.950 --> 00:06:57.380
And then plus B times
the y component here.

00:06:57.380 --> 00:07:00.190
So plus By0.

00:07:00.190 --> 00:07:03.910
I'm just distributing
the B, minus Byp.

00:07:03.910 --> 00:07:09.190
And then plus-- I'll
do another color here,

00:07:09.190 --> 00:07:13.820
that's too close of a color--
plus C times the z component.

00:07:13.820 --> 00:07:18.205
So plus Cz0 minus Czp.

00:07:22.160 --> 00:07:25.040
And all of that over the
magnitude of the normal vector.

00:07:25.040 --> 00:07:28.160
So what's the magnitude of
the normal vector going to be?

00:07:28.160 --> 00:07:30.640
It's just the square
root of the normal vector

00:07:30.640 --> 00:07:32.070
dotted with itself.

00:07:32.070 --> 00:07:34.664
So it's just each of these
guys squared added to themself,

00:07:34.664 --> 00:07:36.080
and you're taking
the square root.

00:07:36.080 --> 00:07:40.160
So it's the square
root-- maybe I

00:07:40.160 --> 00:07:45.400
can do a nicer looking radical
sign than that-- of A squared

00:07:45.400 --> 00:07:50.720
plus B squared plus C squared.

00:07:50.720 --> 00:07:52.480
Now, what is this up
here simplified to?

00:07:52.480 --> 00:07:53.890
Let me just rewrite this.

00:07:53.890 --> 00:07:56.750
So this is the
distance in question.

00:07:56.750 --> 00:07:59.230
This right here is
equal to the distance.

00:07:59.230 --> 00:08:01.130
But let's see if
we can simplify it.

00:08:01.130 --> 00:08:04.280
So first, we can take all
of the terms with the x0.

00:08:04.280 --> 00:08:06.790
These involve the point
that sits off the plane.

00:08:06.790 --> 00:08:09.750
Remember, x0, y0, z0
sat off the plane.

00:08:09.750 --> 00:08:17.880
So this is Ax0
plus By0 plus Cz0.

00:08:17.880 --> 00:08:19.830
And then what are
these terms equal to?

00:08:19.830 --> 00:08:20.810
What are these terms?

00:08:24.380 --> 00:08:29.330
Negative Axp minus
Byp minus Czp?

00:08:29.330 --> 00:08:32.320
Well, if you remember
here, D in the equation

00:08:32.320 --> 00:08:34.169
of in the equation
of a plane, D,

00:08:34.169 --> 00:08:35.549
when we started
in the last video

00:08:35.549 --> 00:08:38.506
when we tried to figure out
what the normal to a plane is,

00:08:38.506 --> 00:08:42.860
D is-- if this point
xp sits on the plane--

00:08:42.860 --> 00:08:49.860
D is Axp plus Byp plus Czp.

00:08:49.860 --> 00:08:51.770
Or another way you
could say it is,

00:08:51.770 --> 00:08:57.182
negative D would be
negative A-- and it's just

00:08:57.182 --> 00:08:59.640
the difference between lowercase
and uppercase here, right?

00:08:59.640 --> 00:09:02.410
We're saying that lowercase is
the same as this uppercase A.

00:09:02.410 --> 00:09:10.052
So it's negative Axp
minus Byp minus Czp.

00:09:10.052 --> 00:09:12.010
I'm just using what we
got from the last video.

00:09:12.010 --> 00:09:15.701
This is what D is so negative
D will be this business.

00:09:15.701 --> 00:09:17.450
And that's exactly
what we have over here.

00:09:17.450 --> 00:09:21.920
We have negative Axp
negative Byp negative Czp.

00:09:21.920 --> 00:09:25.460
So all of this term,
this term, and this term

00:09:25.460 --> 00:09:30.500
simplifies to a minus D. And
remember, this negative capital

00:09:30.500 --> 00:09:35.480
D, this is the D from the
equation of the plane, not

00:09:35.480 --> 00:09:36.680
the distance d.

00:09:36.680 --> 00:09:39.404
So this is the numerator
of our distance.

00:09:39.404 --> 00:09:41.070
And then the denominator
of our distance

00:09:41.070 --> 00:09:47.470
is just the square root of A
squared plus B squared plus C

00:09:47.470 --> 00:09:47.995
squared.

00:09:47.995 --> 00:09:48.620
And we're done.

00:09:48.620 --> 00:09:52.090
This tells us the distance
between any point and a plane.

00:09:52.090 --> 00:09:54.280
And this is a pretty
intuitive formula here.

00:09:54.280 --> 00:09:57.680
Because all we're
doing, if I give

00:09:57.680 --> 00:09:59.770
you-- let me give
you an example.

00:09:59.770 --> 00:10:01.406
Let's say I have the plane.

00:10:01.406 --> 00:10:09.490
If I have the plane 1x minus
2y plus 3z is equal to 5.

00:10:09.490 --> 00:10:10.900
So that's some plane.

00:10:10.900 --> 00:10:14.250
And let me pick some point
that's not on the plane.

00:10:14.250 --> 00:10:18.640
So let's say I have the point,
I don't know, let me say I have

00:10:18.640 --> 00:10:21.789
the 2, 2, 3.

00:10:21.789 --> 00:10:23.580
And let me make sure
it's not on the plane.

00:10:23.580 --> 00:10:28.260
So it's 2 minus 6 is
negative-- yeah, so this won't.

00:10:28.260 --> 00:10:30.110
Let me just pick a random 1.

00:10:30.110 --> 00:10:31.740
So this definitely
is not on the plane,

00:10:31.740 --> 00:10:35.830
because we have
2 minus 6 plus 3.

00:10:35.830 --> 00:10:38.230
That gives us negative
1, which is not 5.

00:10:38.230 --> 00:10:40.290
So this is definitely
not on the plane.

00:10:40.290 --> 00:10:43.050
We can find the distance
between this point and the plane

00:10:43.050 --> 00:10:45.100
using the formula
we just derived.

00:10:45.100 --> 00:10:53.935
We literally just evaluate at--
so this will just be 1 times 2.

00:10:53.935 --> 00:10:55.060
Let me use that same color.

00:10:55.060 --> 00:11:03.100
1 times 2 minus 2
times-- I'm going

00:11:03.100 --> 00:11:09.050
to fill it in-- plus 3
times something, minus 5.

00:11:09.050 --> 00:11:11.590
All of that over, and I
haven't put these guys in.

00:11:11.590 --> 00:11:12.650
Let me do that right now.

00:11:12.650 --> 00:11:19.150
So 1 times 2 minus 2
times 3 plus 3 times 1.

00:11:19.150 --> 00:11:21.430
This 1 minus 5, you're
kind of bringing it over

00:11:21.430 --> 00:11:22.750
to the left hand side.

00:11:22.750 --> 00:11:27.910
All of that over
the square root of 1

00:11:27.910 --> 00:11:30.720
squared, which is
1, plus negative 2

00:11:30.720 --> 00:11:35.860
squared, which is 4, plus
3 squared, which is 9.

00:11:35.860 --> 00:11:42.120
So it's going to be equal to,
let's see, this is 2 minus 6,

00:11:42.120 --> 00:11:43.520
or negative 6.

00:11:43.520 --> 00:11:45.660
And then you have plus 3.

00:11:45.660 --> 00:11:47.570
And then minus 5.

00:11:47.570 --> 00:11:48.650
So this is what?

00:11:48.650 --> 00:11:50.100
This is 5.

00:11:50.100 --> 00:11:52.770
2 plus 3 is 5 minus 5.

00:11:52.770 --> 00:11:53.880
So those cancel out.

00:11:53.880 --> 00:11:55.490
So this is negative 6.

00:11:55.490 --> 00:11:59.360
So it's equal to negative
6 over the square root of 5

00:11:59.360 --> 00:12:00.950
plus 9 is 14.

00:12:00.950 --> 00:12:03.800
Over the square root of 14.

00:12:03.800 --> 00:12:05.960
And you're done.

00:12:05.960 --> 00:12:07.710
So hopefully, you
find that useful.

00:12:07.710 --> 00:12:11.780
And hopefully, we can apply this
in the other example problems.


Normal vector from plane equation

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:02.510
What I want to do in
this video is make sure

00:00:02.510 --> 00:00:04.480
that we're good at
picking out what

00:00:04.480 --> 00:00:06.890
the normal vector
to a plane is, if we

00:00:06.890 --> 00:00:09.780
are given the
equation for a plane.

00:00:09.780 --> 00:00:12.680
So to understand that, let's
just start off with some plane

00:00:12.680 --> 00:00:13.660
here.

00:00:13.660 --> 00:00:15.790
Let's just start off--
so this is a plane,

00:00:15.790 --> 00:00:17.290
I'm drawing part
of it, obviously it

00:00:17.290 --> 00:00:18.620
keeps going in every direction.

00:00:18.620 --> 00:00:20.850
So let's say that is our plane.

00:00:20.850 --> 00:00:23.680
And let's say that this is a
normal vector to the plane.

00:00:23.680 --> 00:00:25.720
So that is our normal
vector to the plane.

00:00:25.720 --> 00:00:32.570
It's given by ai
plus bj plus ck.

00:00:32.570 --> 00:00:36.700
So that is our normal
vector to the plane.

00:00:36.700 --> 00:00:38.520
So it's perpendicular.

00:00:38.520 --> 00:00:40.910
It's perpendicular
to every other vector

00:00:40.910 --> 00:00:42.210
that's on the plane.

00:00:42.210 --> 00:00:44.460
And let's say we have
some point on the plane.

00:00:44.460 --> 00:00:46.000
We have some point.

00:00:46.000 --> 00:00:48.150
It's the point x sub p.

00:00:48.150 --> 00:00:49.462
I'll say p for plane.

00:00:49.462 --> 00:00:50.670
So it's a point on the plane.

00:00:50.670 --> 00:00:54.470
Xp yp zp.

00:00:54.470 --> 00:00:55.880
If we pick the origin.

00:00:55.880 --> 00:00:58.680
So let's say that
our axes are here.

00:00:58.680 --> 00:01:02.140
So let me draw our
coordinate axes.

00:01:02.140 --> 00:01:04.569
So let's say our coordinate
axes look like that.

00:01:04.569 --> 00:01:06.260
This is our z-axis.

00:01:06.260 --> 00:01:08.660
This is, let's say
that's a y-axis.

00:01:08.660 --> 00:01:11.900
And let's say that
this is our x-axis.

00:01:11.900 --> 00:01:14.760
Let's say this is our
x-axis coming out like this.

00:01:14.760 --> 00:01:16.070
This is our x-axis.

00:01:16.070 --> 00:01:18.490
You can specify this
is a position vector.

00:01:18.490 --> 00:01:19.880
There is a position vector.

00:01:19.880 --> 00:01:20.985
Let me draw it like this.

00:01:23.590 --> 00:01:26.460
Then it would be behind the
plane, right over there.

00:01:26.460 --> 00:01:28.200
You have a position vector.

00:01:28.200 --> 00:01:36.550
That position vector would
be xpi plus ypj plus zpk.

00:01:36.550 --> 00:01:39.480
It specifies this
coordinate, right here,

00:01:39.480 --> 00:01:41.167
that sits on the plane.

00:01:41.167 --> 00:01:42.500
Let me just call that something.

00:01:42.500 --> 00:01:44.400
Let me call that
position vector,

00:01:44.400 --> 00:01:49.940
I don't know-- let
me call that p1.

00:01:49.940 --> 00:01:52.390
So this is a point on the plane.

00:01:52.390 --> 00:01:56.810
So it's p-- it is p1
and it is equal to this.

00:01:56.810 --> 00:02:01.294
Now, we could take another
point on the plane.

00:02:01.294 --> 00:02:02.960
This is a particular
point of the plane.

00:02:02.960 --> 00:02:05.730
Let's say we just say, any
other point on the plane, xyz.

00:02:05.730 --> 00:02:08.490
But we're saying that
xyz sits on the plane.

00:02:08.490 --> 00:02:12.080
So let's say we take this
point right over here, xyz.

00:02:12.080 --> 00:02:15.170
That clearly, same logic, can
be specified by another position

00:02:15.170 --> 00:02:16.360
vector.

00:02:16.360 --> 00:02:19.180
We could have a position
vector that looks like this.

00:02:19.180 --> 00:02:20.570
And dotted line.

00:02:20.570 --> 00:02:22.950
It's going under the
plane right over here.

00:02:22.950 --> 00:02:24.610
And this position
vector, I don't know,

00:02:24.610 --> 00:02:30.130
let me just call it p, instead
of that particular, that P1.

00:02:30.130 --> 00:02:36.009
This would just be
xi plus yj plus zk.

00:02:36.009 --> 00:02:37.800
Now, the whole reason
why I did this set up

00:02:37.800 --> 00:02:42.180
is because, given some
particular point that I know

00:02:42.180 --> 00:02:45.480
is on the plane, and any other
xyz that is on the plane,

00:02:45.480 --> 00:02:48.570
I can find-- I can construct--
a vector that is definitely

00:02:48.570 --> 00:02:49.242
on the plane.

00:02:49.242 --> 00:02:50.700
And we've done this
before, when we

00:02:50.700 --> 00:02:53.826
tried to figure out what the
equations of a plane are.

00:02:53.826 --> 00:02:55.450
A vector that's
definitely on the plane

00:02:55.450 --> 00:02:57.440
is going to be the difference
of these two vectors.

00:02:57.440 --> 00:02:58.500
And I'll do that in blue.

00:02:58.500 --> 00:03:02.080
So if you take the yellow
vector, minus the green vector.

00:03:02.080 --> 00:03:05.450
We take this position,
you'll get the vector

00:03:05.450 --> 00:03:07.000
that if you view
it that way, that

00:03:07.000 --> 00:03:08.240
connects this point
in that point.

00:03:08.240 --> 00:03:09.690
Although you can
shift the vector.

00:03:09.690 --> 00:03:13.920
But you'll get a vector that
definitely lies along the plane

00:03:13.920 --> 00:03:15.490
So if you start
one of these points

00:03:15.490 --> 00:03:17.520
it will definitely
lie along the plane.

00:03:17.520 --> 00:03:19.870
So the vector will
look like this.

00:03:19.870 --> 00:03:23.280
And it would be lying
along our plane.

00:03:23.280 --> 00:03:25.250
So this vector lies
along our plan.

00:03:25.250 --> 00:03:28.780
And that vector is p minus p1.

00:03:28.780 --> 00:03:32.130
This is the vector p minus p1.

00:03:32.130 --> 00:03:34.880
It's this position vector minus
that position vector, gives you

00:03:34.880 --> 00:03:35.380
this one.

00:03:35.380 --> 00:03:37.390
Or another way to view
it is this green position

00:03:37.390 --> 00:03:40.190
vector plus this blue vector
that sits on the plane

00:03:40.190 --> 00:03:43.060
will clearly equal this
yellow vector, right?

00:03:43.060 --> 00:03:43.945
Heads to tails.

00:03:43.945 --> 00:03:45.455
It clearly equals it.

00:03:45.455 --> 00:03:47.080
And the whole reason
why did that is we

00:03:47.080 --> 00:03:50.190
can now take the dot product,
between this blue thing

00:03:50.190 --> 00:03:51.410
and this magenta thing.

00:03:51.410 --> 00:03:52.780
And we've done it before.

00:03:52.780 --> 00:03:55.860
And they have to be equal to 0,
because this lies on the plane.

00:03:55.860 --> 00:03:58.040
This is perpendicular
to everything

00:03:58.040 --> 00:03:59.870
that sits on the
plane and it equals 0.

00:03:59.870 --> 00:04:02.790
And so we will get the
equation for the plane.

00:04:02.790 --> 00:04:04.420
But before I do
that, let me make

00:04:04.420 --> 00:04:08.150
sure we know what the components
of this blue vector are.

00:04:08.150 --> 00:04:11.082
So p minus p1, that's
the blue vector.

00:04:11.082 --> 00:04:13.290
You're just going to subtract
each of the components.

00:04:13.290 --> 00:04:16.140
So it's going to be x minus xp.

00:04:16.140 --> 00:04:26.930
It's going to be x minus
xpi plus y minus ypj plus z

00:04:26.930 --> 00:04:30.030
minus zpk.

00:04:30.030 --> 00:04:32.310
And we just said,
this is in the plane.

00:04:32.310 --> 00:04:34.432
And this is, this
right, the normal vector

00:04:34.432 --> 00:04:35.390
is normal to the plane.

00:04:35.390 --> 00:04:38.740
You take their dot product--
it's going to be equal to zero.

00:04:38.740 --> 00:04:48.540
So n dot this vector is
going to be equal to 0.

00:04:48.540 --> 00:04:52.400
But it's also equal to this
a times this expression.

00:04:52.400 --> 00:04:53.560
I'll do it right over here.

00:04:53.560 --> 00:04:56.180
So these-- find some good color.

00:04:56.180 --> 00:05:05.910
So a times that, which is ax
minus axp plus b times that.

00:05:05.910 --> 00:05:11.420
So that is plus by minus byp.

00:05:11.420 --> 00:05:14.110
And then-- let me make
sure I have enough colors--

00:05:14.110 --> 00:05:18.110
and then it's going to
be plus that times that.

00:05:18.110 --> 00:05:23.920
So that's plus cz minus czp.

00:05:23.920 --> 00:05:26.840
And all of this is equal to 0.

00:05:26.840 --> 00:05:31.330
Now what I'm going to do is,
I'm going to rewrite this.

00:05:31.330 --> 00:05:34.341
So we have all of these
terms I'm looking for, right?

00:05:34.341 --> 00:05:34.840
Color.

00:05:34.840 --> 00:05:37.060
We have all of the x terms-- ax.

00:05:37.060 --> 00:05:39.440
Remember, this is any
x that's on the plane,

00:05:39.440 --> 00:05:40.990
will satisfy this.

00:05:40.990 --> 00:05:44.330
So ax, by and cz.

00:05:44.330 --> 00:05:46.180
Let me leave that on
the right hand side.

00:05:46.180 --> 00:05:52.754
So we have ax plus by
plus cz is equal to--

00:05:52.754 --> 00:05:54.170
and what I want
to do is I'm going

00:05:54.170 --> 00:05:56.330
to subtract each of
these from both sides.

00:05:56.330 --> 00:05:58.860
Another way is, I'm going
to move them all over.

00:05:58.860 --> 00:06:00.740
Let me do it-- let me
not do too many things.

00:06:00.740 --> 00:06:03.130
I'm going to move them
over to the left hand side.

00:06:03.130 --> 00:06:06.010
So I'm going to add
positive axp to both sides.

00:06:06.010 --> 00:06:08.740
That's equivalent of
subtracting negative axp.

00:06:08.740 --> 00:06:13.230
So this is going
to be positive axp.

00:06:13.230 --> 00:06:16.980
And then we're going to have
positive byp plus-- I'll

00:06:16.980 --> 00:06:23.670
do that same green-- plus byp,
and then finally plus czp.

00:06:23.670 --> 00:06:26.839
Plus czp is going
to be equal to that.

00:06:26.839 --> 00:06:29.130
Now, the whole reason why
did this-- and I've done this

00:06:29.130 --> 00:06:32.310
in previous videos, where we're
trying to find the formula,

00:06:32.310 --> 00:06:34.060
or trying to find the
equation of a plane,

00:06:34.060 --> 00:06:36.340
is now we say, hey, if
you have a normal vector,

00:06:36.340 --> 00:06:39.470
and if you're given a point
on the plane-- where it's

00:06:39.470 --> 00:06:42.410
in this case is
xp yp zp-- we now

00:06:42.410 --> 00:06:44.939
have a very quick way of
figuring out the equation.

00:06:44.939 --> 00:06:46.230
But I want to go the other way.

00:06:46.230 --> 00:06:48.040
I want you to be
able to, if I were

00:06:48.040 --> 00:06:57.510
to give you a equation for
plane, where I were to say, ax

00:06:57.510 --> 00:07:02.200
plus by plus cz, is equal to d.

00:07:02.200 --> 00:07:05.000
So this is the general
equation for a plane.

00:07:05.000 --> 00:07:06.460
If I were to give
you this, I want

00:07:06.460 --> 00:07:08.840
to be able to figure out the
normal vector very quickly.

00:07:08.840 --> 00:07:10.180
So how could you do that?

00:07:10.180 --> 00:07:13.470
Well, this ax plus by
plus cz is completely

00:07:13.470 --> 00:07:15.839
analogous to this part
right up over here.

00:07:15.839 --> 00:07:18.130
Let me rewrite all this over
here, so it becomes clear.

00:07:18.130 --> 00:07:23.540
This part is ax
plus by plus cz is

00:07:23.540 --> 00:07:28.360
equal to all of this stuff
on the left hand side.

00:07:28.360 --> 00:07:32.110
So let me copy and paste it.

00:07:32.110 --> 00:07:35.290
So I just essentially
flipped this expression.

00:07:35.290 --> 00:07:41.300
But now you see this, all of
this, this a has to be this a.

00:07:41.300 --> 00:07:42.510
This b has to be this b.

00:07:42.510 --> 00:07:44.190
This c has to be this thing.

00:07:44.190 --> 00:07:45.490
And then the d is all of this.

00:07:45.490 --> 00:07:47.099
And this is just
going to be a number.

00:07:47.099 --> 00:07:48.890
This is just going to
be a number, assuming

00:07:48.890 --> 00:07:50.200
you knew what the
normal vector is,

00:07:50.200 --> 00:07:51.824
what your a's, b's
and c's are, and you

00:07:51.824 --> 00:07:53.150
know a particular value.

00:07:53.150 --> 00:07:55.351
So this is what d is.

00:07:55.351 --> 00:07:57.600
So this is how you could get
the equation for a plane.

00:07:57.600 --> 00:07:59.433
Now if I were to give
you equation or plane,

00:07:59.433 --> 00:08:00.550
what is the normal vector?

00:08:00.550 --> 00:08:01.424
Well, we just saw it.

00:08:01.424 --> 00:08:03.830
The normal vector, this a
corresponds to that a, this b

00:08:03.830 --> 00:08:07.280
corresponds to that b, that
c corresponds to that c.

00:08:07.280 --> 00:08:09.460
The normal vector to
this plane we started off

00:08:09.460 --> 00:08:12.240
with, it has the
component a, b, and c.

00:08:12.240 --> 00:08:16.110
So if you're given
equation for plane here,

00:08:16.110 --> 00:08:20.830
the normal vector to this
plane right over here,

00:08:20.830 --> 00:08:28.340
is going to be ai
plus bj plus ck.

00:08:28.340 --> 00:08:29.950
So it's a very easy thing to do.

00:08:29.950 --> 00:08:32.190
If I were to give you
the equation of a plane--

00:08:32.190 --> 00:08:34.100
let me give you a
particular example.

00:08:34.100 --> 00:08:37.620
If I were to tell you that
I have some plane in three

00:08:37.620 --> 00:08:40.830
dimensions-- let's say it's
negative 3, although it'll

00:08:40.830 --> 00:08:42.320
work for more dimensions.

00:08:42.320 --> 00:08:47.850
Let's say I have negative 3
x plus the square root of 2

00:08:47.850 --> 00:08:51.860
y-- let me put it this
way-- minus, or let's say,

00:08:51.860 --> 00:08:56.052
plus 7 z is equal to pi.

00:08:56.052 --> 00:08:58.010
So you have this crazy--
I mean it's not crazy.

00:08:58.010 --> 00:08:59.620
It's just a plane
in three dimensions.

00:08:59.620 --> 00:09:02.970
And I say what is a normal
vector to this plane?

00:09:02.970 --> 00:09:05.755
You literally can just pick
out these coefficients,

00:09:05.755 --> 00:09:08.460
and you say, a normal
vector to this plane

00:09:08.460 --> 00:09:13.580
is negative 3i plus
the square root of 2

00:09:13.580 --> 00:09:17.870
plus 2 square root
of 2 j plus 7 k.

00:09:17.870 --> 00:09:19.687
And you could ignore
the d part there.

00:09:19.687 --> 00:09:21.270
And the reason why
you can ignore that

00:09:21.270 --> 00:09:23.170
is that will just
shift the plane,

00:09:23.170 --> 00:09:26.320
but it won't fundamentally
change how the plane is tilted.

00:09:26.320 --> 00:09:32.060
So a this normal vector, will
also be normal if this was e,

00:09:32.060 --> 00:09:35.880
or if this was 100, it would be
normal to all of those planes,

00:09:35.880 --> 00:09:37.630
because all those
planes are just shifted,

00:09:37.630 --> 00:09:40.456
but they all have
the same inclination.

00:09:40.456 --> 00:09:42.580
So they would all kind of
point the same direction.

00:09:42.580 --> 00:09:45.709
And so the normal vectors would
point in the same direction.

00:09:45.709 --> 00:09:47.500
So hopefully you found
that vaguely useful.

00:09:47.500 --> 00:09:50.450
We'll now build on this
to find the distance

00:09:50.450 --> 00:09:53.540
between any point in three
dimensions, and some plane.

00:09:53.540 --> 00:09:56.739
The shortest distance that
we can get to that plane.