IIT JEE Complex Root Probability (part 1)

WEBVTT
Kind: captions
Language: en

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Let omega be a complex cube
root of unity with omega

00:00:04.760 --> 00:00:06.730
not being equal to 1.

00:00:06.730 --> 00:00:08.510
A fair die is
thrown three times.

00:00:08.510 --> 00:00:11.680
If r1, r2, r3 are the
numbers obtained on the die,

00:00:11.680 --> 00:00:14.480
then the probability
that omega to the r1 plus

00:00:14.480 --> 00:00:17.940
omega to the r2 plus
omega to the r3 equals 0

00:00:17.940 --> 00:00:20.496
is-- it's a fascinating problem.

00:00:20.496 --> 00:00:22.120
So let's just start
with the first part

00:00:22.120 --> 00:00:24.286
and then see if some patterns
fall into place for us

00:00:24.286 --> 00:00:26.810
to be able to tackle
the second part.

00:00:26.810 --> 00:00:30.460
So omega is a complex
cube root of unity,

00:00:30.460 --> 00:00:32.890
which is just a fancy way of
saying it's a complex cube

00:00:32.890 --> 00:00:35.940
root of 1, with omega
not being equal to 1.

00:00:35.940 --> 00:00:40.840
So let's just set
up an equation here.

00:00:40.840 --> 00:00:43.240
Let's just think about all
of the cube roots of 1.

00:00:43.240 --> 00:00:46.800
So if we say-- that's analogous
to figuring out all of the x's

00:00:46.800 --> 00:00:50.780
that satisfy x to the
third is equal to 1.

00:00:50.780 --> 00:00:54.480
Or we can subtract 1 from
both sides of this equation

00:00:54.480 --> 00:00:58.920
and write this as x to the
third minus 1 is equal to zero.

00:00:58.920 --> 00:01:02.180
All of the roots of
this equation right

00:01:02.180 --> 00:01:05.269
here will be cube roots of 1.

00:01:05.269 --> 00:01:08.067
They will satisfy
this over here.

00:01:08.067 --> 00:01:10.400
This is the same thing as--
I could write it over here--

00:01:10.400 --> 00:01:15.030
x is equal to the
cube root of 1.

00:01:15.030 --> 00:01:17.000
These are all
equivalent statements.

00:01:17.000 --> 00:01:18.490
So how do we do that?

00:01:18.490 --> 00:01:22.120
Well, we know what
one of the roots are.

00:01:22.120 --> 00:01:27.206
We know that x
equals 1 is a root.

00:01:27.206 --> 00:01:28.830
Or another way we
could think about it,

00:01:28.830 --> 00:01:32.010
we know that x minus
1 is a factor here.

00:01:32.010 --> 00:01:33.420
We know that this
can be factored

00:01:33.420 --> 00:01:38.380
as x minus 1 times something,
something something squared

00:01:38.380 --> 00:01:39.680
plus something x.

00:01:39.680 --> 00:01:41.780
Some quadratic,
some-- let me write

00:01:41.780 --> 00:01:49.460
here-- some quadratic right
here is equal to zero.

00:01:49.460 --> 00:01:50.704
We can factor this.

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And to figure out
what the quadratic is,

00:01:52.370 --> 00:01:54.380
we just have to do a
little bit of polynomial,

00:01:54.380 --> 00:01:56.680
or I guess we could call
it algebraic long division.

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We just have to divide
x minus 1 into this,

00:01:59.200 --> 00:02:01.190
and then we'll get
what this quadratic is.

00:02:01.190 --> 00:02:02.750
So let's do that.

00:02:02.750 --> 00:02:12.380
Let us divide x minus 1
into x to the third minus 1.

00:02:12.380 --> 00:02:13.480
And I'll do it in place.

00:02:13.480 --> 00:02:15.980
So this is the third degree
place. x to the third,

00:02:15.980 --> 00:02:18.410
I'll leave space for
the x squared place,

00:02:18.410 --> 00:02:20.970
leave space for the
x to the first place,

00:02:20.970 --> 00:02:24.430
and then we have x to the
zero, which is the ones place.

00:02:24.430 --> 00:02:25.450
So that's negative 1.

00:02:25.450 --> 00:02:27.075
So I just wrote x to
the third minus 1,

00:02:27.075 --> 00:02:28.460
but I wrote it
out this so I have

00:02:28.460 --> 00:02:31.520
space to keep our
counting pretty clean.

00:02:31.520 --> 00:02:34.230
So we just do algebraic
long division here.

00:02:34.230 --> 00:02:36.411
So x minus 1 goes into
x to the third minus 1,

00:02:36.411 --> 00:02:38.160
you just look at the
highest degree terms.

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x goes into x to the
third x squared times.

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It goes x squared times.

00:02:44.350 --> 00:02:47.570
x squared times x
is x to the third.

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x squared times negative
1 is minus x squared.

00:02:51.770 --> 00:02:53.520
Now, we want to subtract
the magenta stuff

00:02:53.520 --> 00:02:56.110
from the orange stuff, or we
could add the negative of it.

00:02:56.110 --> 00:02:58.150
So let's just multiply
this times negative 1.

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So positive-- sorry, we're
multiplying times negative 1.

00:03:01.240 --> 00:03:03.740
So this is a negative, and then
this will become a positive.

00:03:03.740 --> 00:03:05.230
We're subtracting
what I originally

00:03:05.230 --> 00:03:07.355
had written from that, but
I just made it negative.

00:03:07.355 --> 00:03:08.140
And now we'll add.

00:03:08.140 --> 00:03:11.000
And so this will become--
these cancel out.

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We get x squared minus 1.

00:03:16.590 --> 00:03:18.470
Just algebraic long division.

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Once again, how many
times does x minus 1

00:03:20.850 --> 00:03:22.300
go into x squared minus 1?

00:03:22.300 --> 00:03:24.100
Look at the highest
degree terms.

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x goes into x squared--
I'll just arbitrarily switch

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colors-- x goes into
x squared x times.

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So this is plus x.

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x times x is x squared.

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x times negative
1 is negative x.

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Same drill.

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Subtract this from up there,
or we could add the negative.

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So this becomes negative,
this becomes positive.

00:03:44.820 --> 00:03:46.400
These guys cancel out.

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You get x-- bring
this down-- x minus 1.

00:03:50.760 --> 00:03:55.240
x minus 1 goes into x
minus 1 exactly one time.

00:03:55.240 --> 00:03:57.040
It goes exactly one time.

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1 times x minus 1 is x minus 1.

00:03:59.920 --> 00:04:01.670
And you get a remainder of zero.

00:04:01.670 --> 00:04:03.760
So it divided evenly,
which makes sense.

00:04:03.760 --> 00:04:05.820
Because this is one of
the factors of that.

00:04:05.820 --> 00:04:07.800
We know that this is a root.

00:04:07.800 --> 00:04:11.120
So if we wanted to
completely factor this,

00:04:11.120 --> 00:04:16.529
we could write this as x
minus 1 times this thing

00:04:16.529 --> 00:04:20.120
over here, times the
quadratic, times x

00:04:20.120 --> 00:04:25.340
squared plus x plus
1 is equal to 0.

00:04:25.340 --> 00:04:29.600
So if we wanted to find
the non-one roots of this

00:04:29.600 --> 00:04:33.150
over here, we just have to find
the roots of this thing right

00:04:33.150 --> 00:04:33.670
over here.

00:04:33.670 --> 00:04:34.700
Whatever the roots
are of this are

00:04:34.700 --> 00:04:37.460
going to make this expression
equal to 0, which would make

00:04:37.460 --> 00:04:39.280
this entire
expression equal to 0,

00:04:39.280 --> 00:04:41.000
make this entire
expression equal zero.

00:04:41.000 --> 00:04:43.276
So we just have to
find this guy's roots.

00:04:43.276 --> 00:04:45.650
And I can already guess that
they're going to be complex.

00:04:45.650 --> 00:04:47.460
So don't even
attempt to factor it.

00:04:47.460 --> 00:04:49.575
You could attempt to
factor it if you like just

00:04:49.575 --> 00:04:50.950
by looking at it,
but we're going

00:04:50.950 --> 00:04:52.570
to use the quadratic formula.

00:04:52.570 --> 00:04:55.620
So the roots here
for this part--

00:04:55.620 --> 00:04:58.410
we say x is equal to negative b.

00:04:58.410 --> 00:05:03.510
b is 1, so negative 1 plus
or minus the square root of b

00:05:03.510 --> 00:05:10.889
squared, b is 1, so 1
minus 4 times a times c.

00:05:10.889 --> 00:05:11.930
Well, a and c are both 1.

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So it's just minus 4.

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All of that over 2
times a. a is just 1.

00:05:17.870 --> 00:05:19.290
So all of that over 2.

00:05:19.290 --> 00:05:22.940
So x is equal to
negative 1 plus or minus

00:05:22.940 --> 00:05:27.930
the square root of
negative 3-- we were right,

00:05:27.930 --> 00:05:31.460
this is going to be a
complex root-- over 2.

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Now, what's the square
root of negative 3?

00:05:34.874 --> 00:05:37.040
We could write this-- the
square root of negative 3,

00:05:37.040 --> 00:05:39.950
this is the same thing as
the square root of negative 1

00:05:39.950 --> 00:05:42.160
times the square root of 3.

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And this we know from
studying imaginary numbers

00:05:45.530 --> 00:05:46.420
and complex numbers.

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This is i.

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That right there is i.

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So the other roots--
and there are two here,

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because we're plus
square root of negative 3

00:05:56.334 --> 00:05:58.250
and subtracting the
square root of negative 3.

00:05:58.250 --> 00:06:02.150
So the other roots are x
could be equal to negative 1

00:06:02.150 --> 00:06:08.060
plus or minus the square
root of 3 times i.

00:06:08.060 --> 00:06:11.860
And I'll put i here, just
so it's nice and clear.

00:06:11.860 --> 00:06:12.840
That's not a vector.

00:06:12.840 --> 00:06:14.070
Just like that.

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i.

00:06:14.790 --> 00:06:19.200
All of that over 2.

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So what we just did
so far is we were

00:06:21.930 --> 00:06:26.762
able-- we now know
all three roots of 1.

00:06:26.762 --> 00:06:28.970
Especially if we're thinking
about the complex roots.

00:06:28.970 --> 00:06:30.610
1 is one of the roots.

00:06:30.610 --> 00:06:33.100
The other roots are negative
1 plus the square root

00:06:33.100 --> 00:06:37.850
of 3i over 2, and negative 1
minus the square root of 3i

00:06:37.850 --> 00:06:39.020
over 2.

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And so they're saying that
omega is one of these two roots.

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We either take the
positive or the negative.

00:06:44.960 --> 00:06:46.459
And it looks like
it doesn't matter.

00:06:46.459 --> 00:06:50.620
It says let omega be a
complex cube root of unity.

00:06:50.620 --> 00:06:52.440
So it could either one of these.

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So let's just pick one of them.

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So let's just say omega
is equal to negative 1

00:07:01.190 --> 00:07:07.100
minus the square root
of 3i-- and I'll just

00:07:07.100 --> 00:07:08.691
distribute the divided by 2.

00:07:08.691 --> 00:07:10.190
So we could say we
could call this--

00:07:10.190 --> 00:07:11.960
let me write it this way.

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It's negative 1/2 minus
the square root of 3/2i.

00:07:18.640 --> 00:07:20.570
I'm just picking-- once
again, not a vector.

00:07:20.570 --> 00:07:22.710
My brain keeps wanting me to
put a little hat on top of it.

00:07:22.710 --> 00:07:23.370
It's not a vector.

00:07:23.370 --> 00:07:24.620
It's just an imaginary number.

00:07:24.620 --> 00:07:25.180
It's i.

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This is actually
a complex number,

00:07:26.596 --> 00:07:29.150
because we have a real
part and an imaginary part.

00:07:29.150 --> 00:07:32.590
So I'm just going to
pick that to be my omega.

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It's one of the non-one
complex roots, third roots,

00:07:36.830 --> 00:07:39.460
or I guess we could say-- what
was the actual wording they

00:07:39.460 --> 00:07:43.440
said-- one of the
complex cube roots of 1,

00:07:43.440 --> 00:07:45.540
of unity, that is not 1.

00:07:45.540 --> 00:07:48.370
So let's just let
omega be equal to that.

00:07:48.370 --> 00:07:50.970
So this first statement
can translate into this.

00:07:50.970 --> 00:07:52.720
And then a fair die
is thrown three times.

00:07:52.720 --> 00:07:55.500
If r1, r2, and r3 are the
numbers obtained on the die,

00:07:55.500 --> 00:07:57.926
then the probability
that omega to the r--

00:07:57.926 --> 00:07:59.550
So let's think about
this a little bit.

00:07:59.550 --> 00:08:02.946
Because we're going to be taking
omega to different powers.

00:08:02.946 --> 00:08:04.320
So let's think
about what happens

00:08:04.320 --> 00:08:07.060
when we take this omega
to different powers.

00:08:07.060 --> 00:08:11.950
So this right here is clearly
omega to the first power,

00:08:11.950 --> 00:08:13.470
the way we've defined it.

00:08:13.470 --> 00:08:16.000
Let's think about
what omega squared is.

00:08:16.000 --> 00:08:21.500
So omega squared is
equal to negative 1/2

00:08:21.500 --> 00:08:25.120
minus square root of 3 over
2i-- and this will give us

00:08:25.120 --> 00:08:28.600
good practice multiplying
complex numbers-- times

00:08:28.600 --> 00:08:31.960
negative 1/2 minus times--
well, it's the same thing.

00:08:31.960 --> 00:08:33.409
We're squaring it.

00:08:33.409 --> 00:08:35.700
And so this is going to be
equal to negative 1/2 times

00:08:35.700 --> 00:08:39.750
negative 1/2 is positive 1/4.

00:08:39.750 --> 00:08:45.640
And then you have negative
1/2 times negative 3/2i.

00:08:45.640 --> 00:08:47.820
The negatives cancel out.

00:08:47.820 --> 00:08:49.540
The square root of
3/2 I should say.

00:08:49.540 --> 00:08:51.520
So negative 1/2
times this is going

00:08:51.520 --> 00:08:54.410
to be positive
square root of 3/4i.

00:08:58.630 --> 00:09:00.535
Then we could multiply
this times this.

00:09:00.535 --> 00:09:02.410
It's going to give us
the same thing as that.

00:09:02.410 --> 00:09:04.440
So the negatives cancel out.

00:09:04.440 --> 00:09:08.040
And so we have plus the
square root of 3/4i.

00:09:11.710 --> 00:09:13.900
And then we're going to
want to multiply the two

00:09:13.900 --> 00:09:15.390
imaginary terms.

00:09:15.390 --> 00:09:17.187
So the negatives cancel out.

00:09:17.187 --> 00:09:19.520
And so you have square root
of 3 times square root of 3,

00:09:19.520 --> 00:09:21.550
which is just 3.

00:09:21.550 --> 00:09:25.870
And then you have 2
times 2, which is 4.

00:09:25.870 --> 00:09:29.370
And then you have i times
i, which is negative 1.

00:09:29.370 --> 00:09:30.710
So let me just put a negative 1.

00:09:30.710 --> 00:09:32.670
So times negative 1 over here.

00:09:32.670 --> 00:09:34.560
So let's simplify this.

00:09:34.560 --> 00:09:42.410
This gives us 1/4 plus
square root of 3/4i

00:09:42.410 --> 00:09:44.130
plus square root of 3/4i.

00:09:44.130 --> 00:09:46.570
That's 2 square roots
of 3/4i or square root

00:09:46.570 --> 00:09:51.120
of-- so that's the same
thing as square root of 3/2i.

00:09:51.120 --> 00:09:52.743
And then minus 3/4.

00:09:55.900 --> 00:09:57.500
And we can simplify
this even more.

00:09:57.500 --> 00:10:02.700
The real parts, we can add
1/4 minus 3/4 is negative 2/4,

00:10:02.700 --> 00:10:03.770
or negative 1/2.

00:10:03.770 --> 00:10:10.754
So this is negative 1/2 plus
the square root of 3/2i.

00:10:10.754 --> 00:10:11.670
So that's interesting.

00:10:11.670 --> 00:10:16.220
When I squared omega,
I got another one

00:10:16.220 --> 00:10:18.780
of the complex cube roots of 1.

00:10:18.780 --> 00:10:20.440
I actually got its conjugate.

00:10:20.440 --> 00:10:22.650
And if any of y'all who've
studied complex analysis

00:10:22.650 --> 00:10:24.150
and think about
the complex plane,

00:10:24.150 --> 00:10:25.983
it should be pretty
clear why this happened.

00:10:25.983 --> 00:10:27.990
But if you don't, don't
worry too much about it.

00:10:27.990 --> 00:10:31.200
Now what I want to do
is I want to take omega

00:10:31.200 --> 00:10:33.500
to the third power, just
to really understand

00:10:33.500 --> 00:10:35.640
what's happening with omega.

00:10:35.640 --> 00:10:37.490
And if you've studied
complex analysis,

00:10:37.490 --> 00:10:39.220
this part would be
quicker for you.

00:10:39.220 --> 00:10:40.150
You would know it.

00:10:40.150 --> 00:10:41.736
So what's omega to
the third power?

00:10:41.736 --> 00:10:43.110
Omega to the third
power is going

00:10:43.110 --> 00:10:45.990
to be omega squared times omega.

00:10:45.990 --> 00:10:48.970
Omega squared is just
that right over there.

00:10:48.970 --> 00:10:50.120
We just calculated it.

00:10:50.120 --> 00:10:56.992
It's negative 1/2 plus
square root of 3/2i.

00:10:56.992 --> 00:10:59.200
And we're going to have to
multiply that times omega.

00:10:59.200 --> 00:11:00.920
Omega was the conjugate
of this thing.

00:11:00.920 --> 00:11:03.590
So it's negative 1/2
minus-- conjugate just

00:11:03.590 --> 00:11:06.170
means opposite side
on the complex part.

00:11:06.170 --> 00:11:10.065
So minus square root of 3/2i.

00:11:10.065 --> 00:11:12.190
That's what we defined
omega to be right over here.

00:11:12.190 --> 00:11:13.970
This right over here is omega.

00:11:13.970 --> 00:11:15.160
That's omega.

00:11:15.160 --> 00:11:17.420
This right here
is omega squared.

00:11:17.420 --> 00:11:19.270
So let's multiply these.

00:11:19.270 --> 00:11:23.160
So first, negative
1/2 times negative 1/2

00:11:23.160 --> 00:11:27.410
gives us negative-- sorry,
gives us positive 1/4.

00:11:27.410 --> 00:11:28.250
Have to be careful.

00:11:28.250 --> 00:11:30.430
This gives us 1/4.

00:11:30.430 --> 00:11:35.326
And then if we
take negative 1/2--

00:11:35.326 --> 00:11:36.450
I shouldn't do it that way.

00:11:36.450 --> 00:11:41.930
Let's do negative 1/2 times
negative square root of 3/2i.

00:11:41.930 --> 00:11:47.320
Negatives cancel out, so we
get plus the square root of 3--

00:11:47.320 --> 00:11:49.310
let me make sure I have--
square root of 3/4i.

00:11:52.520 --> 00:11:54.675
And then we can multiply
this times this.

00:11:54.675 --> 00:11:55.800
It gives us the same thing.

00:11:55.800 --> 00:12:00.259
So it gives us plus-- let me
make sure I'm doing this right.

00:12:00.259 --> 00:12:02.300
Oh, no, it's not going to
give us the same thing.

00:12:02.300 --> 00:12:04.610
Now one is negative,
one is positive.

00:12:04.610 --> 00:12:07.690
So it's going to give
us-- be very careful.

00:12:07.690 --> 00:12:10.870
So negative 1/2 times
positive square root of 3/2i

00:12:10.870 --> 00:12:13.955
is negative square root of 3/4i.

00:12:17.200 --> 00:12:20.500
And then finally, multiply
the two imaginary parts.

00:12:20.500 --> 00:12:24.600
Multiply that-- I want to use
a different color-- multiply

00:12:24.600 --> 00:12:26.426
that times that.

00:12:26.426 --> 00:12:29.720
That gives us positive times a
negative gives us a negative.

00:12:29.720 --> 00:12:32.350
Square root of 3 times
square root of 3 is 3.

00:12:32.350 --> 00:12:34.400
2 times 2 is 4.

00:12:34.400 --> 00:12:37.220
i times i is negative 1.

00:12:37.220 --> 00:12:39.080
Or we can multiply
the negative 1 times

00:12:39.080 --> 00:12:41.090
this negative over here,
becomes a positive.

00:12:41.090 --> 00:12:42.150
So what do we get?

00:12:42.150 --> 00:12:43.940
These guys cancel out.

00:12:43.940 --> 00:12:45.385
We have 1/4 fourth plus 3/4.

00:12:45.385 --> 00:12:50.060
It is equal to 1.

00:12:50.060 --> 00:12:51.670
So this is interesting.

00:12:51.670 --> 00:12:53.490
This is interesting over here.

00:12:53.490 --> 00:12:57.580
So omega is equal to this
thing right over here.

00:12:57.580 --> 00:13:00.440
Omega squared is equal
to its conjugate.

00:13:00.440 --> 00:13:03.680
Omega squared is equal to its
conjugate right over here.

00:13:03.680 --> 00:13:05.450
We just swapped the
sign over there.

00:13:05.450 --> 00:13:07.040
And then omega to
the third is going

00:13:07.040 --> 00:13:08.200
to be equal to this thing.

00:13:08.200 --> 00:13:09.600
It's going to be equal to unity.

00:13:09.600 --> 00:13:11.200
Actually, we knew that.

00:13:11.200 --> 00:13:12.950
I shouldn't even have
to worry about this.

00:13:12.950 --> 00:13:14.270
I don't know why I even
went through that process.

00:13:14.270 --> 00:13:15.520
Well, it was good multiplying.

00:13:15.520 --> 00:13:17.207
We knew it was the
cube root of unity.

00:13:17.207 --> 00:13:19.040
So we knew if we took
it to the third power.

00:13:19.040 --> 00:13:21.880
So this was to some degree
just a waste of time.

00:13:21.880 --> 00:13:23.160
But what does this tell us?

00:13:23.160 --> 00:13:24.821
So we now kind of
understand what

00:13:24.821 --> 00:13:26.070
the different powers of omega.

00:13:26.070 --> 00:13:28.480
What will omega to
the fourth power be?

00:13:28.480 --> 00:13:30.279
What's going to be
this times omega?

00:13:30.279 --> 00:13:31.820
So it's just going
to be omega again.

00:13:31.820 --> 00:13:33.030
What's omega to the fifth power?

00:13:33.030 --> 00:13:34.050
It's going to be this.

00:13:34.050 --> 00:13:36.090
What's omega to the sixth power?

00:13:36.090 --> 00:13:37.465
It's going to be this again.

00:13:37.465 --> 00:13:38.840
So let's just use
that, and we're

00:13:38.840 --> 00:13:41.006
going to tackle that in the
next part of the problem

00:13:41.006 --> 00:13:42.850
in the next video.


IIT JEE Complex Root Probability (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:03.255
In the last video, we started
on this problem right here,

00:00:03.255 --> 00:00:06.950
where we said let omega be a
complex cube root of unity,

00:00:06.950 --> 00:00:08.810
and omega cannot be equal to 1.

00:00:08.810 --> 00:00:10.960
And so we figured out what
all of the complex cube

00:00:10.960 --> 00:00:12.500
roots of unity were.

00:00:12.500 --> 00:00:14.920
We figured out-- well, we
knew that 1 was one of them,

00:00:14.920 --> 00:00:19.484
and we used that to factor
out this third degree equation

00:00:19.484 --> 00:00:20.150
right over here.

00:00:20.150 --> 00:00:24.510
And we figured out the other
roots, cube roots, of unity

00:00:24.510 --> 00:00:28.660
were negative 1 plus or minus
the square root of 3i over 2.

00:00:28.660 --> 00:00:32.600
And we said, look, the problem
said let omega be one of the,

00:00:32.600 --> 00:00:34.700
or be a complex
cube root of unity.

00:00:34.700 --> 00:00:37.230
So I just picked it to
be the negative version.

00:00:37.230 --> 00:00:40.700
So I said let omega be
this thing right over here.

00:00:40.700 --> 00:00:43.850
Now, just to kind of explore
the space a little bit,

00:00:43.850 --> 00:00:45.380
we said, OK, what's
omega squared?

00:00:45.380 --> 00:00:47.963
And we figured out that it was
actually the conjugate of this.

00:00:47.963 --> 00:00:49.250
It's the plus version of this.

00:00:49.250 --> 00:00:52.810
So we got omega squared
right over here.

00:00:52.810 --> 00:00:54.649
And then I kind of
wasted your time

00:00:54.649 --> 00:00:56.690
a little bit because we
know what omega cubed is.

00:00:56.690 --> 00:00:58.580
We know that omega is
a cube root of unity,

00:00:58.580 --> 00:01:00.460
so omega cubed must be unity.

00:01:00.460 --> 00:01:03.852
But it wasn't harmful, I guess,
to go through the process.

00:01:03.852 --> 00:01:05.810
It shows you sometimes
my brain gets into a rut

00:01:05.810 --> 00:01:08.570
and just does stuff that
it doesn't have to do.

00:01:08.570 --> 00:01:10.990
But we figured out-- we
multiplied this times omega

00:01:10.990 --> 00:01:14.240
to show that, hey, it is
definitely equal to 1.

00:01:14.240 --> 00:01:16.730
So what we were able to
do is set up a situation,

00:01:16.730 --> 00:01:18.850
and we will use this to
think about the next part

00:01:18.850 --> 00:01:19.900
of the problem.

00:01:19.900 --> 00:01:23.840
So omega is equal
to negative 1/2.

00:01:23.840 --> 00:01:26.030
Or maybe I should say
omega to the first power

00:01:26.030 --> 00:01:31.590
is equal to negative 1/2 minus
the square root of 3 over 2i.

00:01:31.590 --> 00:01:35.480
Omega to the second
power-- let me

00:01:35.480 --> 00:01:37.840
write it over here--
omega to the second power

00:01:37.840 --> 00:01:44.950
is equal to negative 1 plus
the square root of 3 over 2i.

00:01:44.950 --> 00:01:47.310
And then omega to
the third power,

00:01:47.310 --> 00:01:49.814
and that was pretty
straightforward, is equal to 1.

00:01:49.814 --> 00:01:51.480
And I touched on this
in the last video.

00:01:51.480 --> 00:01:53.304
What is omega to
the fourth power?

00:01:53.304 --> 00:01:54.720
Omega to the fourth
power is going

00:01:54.720 --> 00:01:56.710
to be omega to the
third times omega.

00:01:56.710 --> 00:01:58.230
So it's going to be omega again.

00:01:58.230 --> 00:02:00.391
So this is also
going to be-- let

00:02:00.391 --> 00:02:01.890
me scroll to the
left a little bit--

00:02:01.890 --> 00:02:05.070
this is also going to be
omega to the fourth power.

00:02:05.070 --> 00:02:06.950
Omega to the fourth
power is the same thing

00:02:06.950 --> 00:02:09.520
as a omega to the first
power, which is this.

00:02:09.520 --> 00:02:11.100
Now, what's to the fifth power?

00:02:11.100 --> 00:02:13.450
Well, it's going to be omega
times the fourth power,

00:02:13.450 --> 00:02:15.170
so it's omega times this.

00:02:15.170 --> 00:02:17.560
Well, omega times this is
the same thing as omega

00:02:17.560 --> 00:02:18.990
squared, so it's
going to be that.

00:02:18.990 --> 00:02:22.420
So this is omega
to the fifth power.

00:02:22.420 --> 00:02:24.180
And what's omega
to the sixth power?

00:02:24.180 --> 00:02:26.980
Well, that's just the same
thing as taking omega cubed

00:02:26.980 --> 00:02:29.400
and squaring it, so this
is also going to be 1.

00:02:29.400 --> 00:02:30.860
So this is omega to the sixth.

00:02:30.860 --> 00:02:34.036
Or you could just use,
it's this times omega

00:02:34.036 --> 00:02:35.660
because it's to the
fifth power, and we

00:02:35.660 --> 00:02:38.622
saw that that's also equal to 1.

00:02:38.622 --> 00:02:40.580
And the reason why I went
up to the sixth power

00:02:40.580 --> 00:02:43.030
is because, if you'll
remember the old problem,

00:02:43.030 --> 00:02:44.640
we're rolling a die.

00:02:44.640 --> 00:02:46.211
And on that die--
I'm assuming it's

00:02:46.211 --> 00:02:47.710
a normal six-sided
die-- we're going

00:02:47.710 --> 00:02:49.780
to get values between 1 and 6.

00:02:49.780 --> 00:02:52.630
And we're going to take omega
to the different powers,

00:02:52.630 --> 00:02:54.552
and we're going to
see if it equals to 0.

00:02:54.552 --> 00:02:56.010
And we want to find
the probability

00:02:56.010 --> 00:02:57.640
of this being equal to 0.

00:02:57.640 --> 00:02:59.400
So let me write this down.

00:02:59.400 --> 00:03:05.400
We want to find the probability
that omega-- and we're just

00:03:05.400 --> 00:03:07.400
using the omega that we
picked because they said

00:03:07.400 --> 00:03:09.595
it's one of the complex
roots that isn't 1--

00:03:09.595 --> 00:03:11.470
so we want to find the
probability that omega

00:03:11.470 --> 00:03:16.430
to the first die of the roll
plus omega to the second

00:03:16.430 --> 00:03:20.230
die of the roll plus omega
to the third die of the roll,

00:03:20.230 --> 00:03:24.560
that when you take their
sum, that that is equal to 0.

00:03:24.560 --> 00:03:26.740
This is what we
need to figure out.

00:03:26.740 --> 00:03:28.550
So to figure this
out, let's just

00:03:28.550 --> 00:03:32.040
figure out what combinations
of powers of omega

00:03:32.040 --> 00:03:34.540
will even add up to 0.

00:03:34.540 --> 00:03:38.900
How can we even
add them up to 0?

00:03:38.900 --> 00:03:40.826
I mean, they all have
to cancel out some way.

00:03:40.826 --> 00:03:42.200
And if you look
at them, it looks

00:03:42.200 --> 00:03:44.900
pretty interesting of how
they might cancel out.

00:03:44.900 --> 00:03:47.229
If I take one version
of this, the only way

00:03:47.229 --> 00:03:48.645
that they really
can cancel out is

00:03:48.645 --> 00:03:52.650
if I take one version of
this, and add it to this,

00:03:52.650 --> 00:03:54.480
and then add it to this.

00:03:54.480 --> 00:03:56.210
Let me show you.

00:03:56.210 --> 00:04:02.800
If I take negative 1/2 minus
the square root of 3 over 2i,

00:04:02.800 --> 00:04:08.780
and I add it to this, and I
add it to this, negative 1/2

00:04:08.780 --> 00:04:14.090
plus the square root of 3 over
2i, and then I add it to 1,

00:04:14.090 --> 00:04:18.380
and then I add it to this,
and then I add it to this,

00:04:18.380 --> 00:04:19.790
what do I get?

00:04:19.790 --> 00:04:22.350
Well, you're going
to have this guy

00:04:22.350 --> 00:04:24.380
and this guy are
going to cancel out.

00:04:24.380 --> 00:04:27.040
Negative 1/2 plus
negative 1/2 are

00:04:27.040 --> 00:04:30.560
going to be equal to negative 1.

00:04:30.560 --> 00:04:34.710
You add negative 1 to 1,
it's going to be equal to 0.

00:04:34.710 --> 00:04:38.160
So essentially, what
it was asking us,

00:04:38.160 --> 00:04:42.200
what's the probability that I
get, for each of these terms,

00:04:42.200 --> 00:04:46.870
that I get one each of each
of these powers of omega?

00:04:46.870 --> 00:04:49.730
Or another way of thinking
about this, what's

00:04:49.730 --> 00:04:52.320
the probability-- so if
we think of it this way.

00:04:55.175 --> 00:04:56.300
Let's think of it this way.

00:04:56.300 --> 00:05:02.340
r1 could be equal to-- we're
going to get omega here,

00:05:02.340 --> 00:05:04.690
which is the same thing
as omega the fourth power.

00:05:04.690 --> 00:05:07.055
So r1 could be 1 or 4.

00:05:09.485 --> 00:05:11.860
And this is in the situation
where the first one gives us

00:05:11.860 --> 00:05:12.360
omega.

00:05:12.360 --> 00:05:14.680
The second one is omega
squared, or the second one

00:05:14.680 --> 00:05:15.990
gives us this value.

00:05:15.990 --> 00:05:18.210
And then the third
one gives us 1.

00:05:18.210 --> 00:05:20.930
So r2 would be 5 or 2.

00:05:23.700 --> 00:05:31.895
And then r3 would
be equal to 3 or 6.

00:05:34.480 --> 00:05:36.852
Now, I want to be very clear.

00:05:36.852 --> 00:05:38.600
We could swap these around.

00:05:38.600 --> 00:05:42.775
There's actually six ways
that you could-- instead

00:05:42.775 --> 00:05:45.764
of this being the first roll,
this roll could be 1 or 4,

00:05:45.764 --> 00:05:47.180
and then this one
could be 5 or 2,

00:05:47.180 --> 00:05:48.800
and then this one
could be 3 or 6.

00:05:48.800 --> 00:05:50.770
So there's actually
six ways of doing this,

00:05:50.770 --> 00:05:51.686
I guess you could say.

00:05:51.686 --> 00:05:55.300
You could permute these six
different ways, or 3 times

00:05:55.300 --> 00:05:55.865
2 times 1.

00:05:55.865 --> 00:05:57.490
But we'll think about
that in a second.

00:05:57.490 --> 00:06:02.930
But assuming that we want this
way, where this first one is

00:06:02.930 --> 00:06:05.110
going to evaluate to
this, and the second one

00:06:05.110 --> 00:06:07.180
is going to evaluate to
this, and the third one

00:06:07.180 --> 00:06:08.880
is going to evaluate
to this, what's

00:06:08.880 --> 00:06:10.504
the probability
of that happening?

00:06:10.504 --> 00:06:12.920
And then we're going to multiply
that 6 by 6 because there

00:06:12.920 --> 00:06:17.470
are six ways to arrange
these terms right over here.

00:06:17.470 --> 00:06:18.270
So let's do that.

00:06:18.270 --> 00:06:20.120
So what's the probability
of that happening?

00:06:20.120 --> 00:06:24.120
Well, the probability
that r1 is a 1 or a 4,

00:06:24.120 --> 00:06:25.890
well, that's two
values out of six,

00:06:25.890 --> 00:06:27.920
so the probability is 1/3 there.

00:06:27.920 --> 00:06:30.330
The probability
that r2 is a 5 or 2,

00:06:30.330 --> 00:06:32.100
well, that's also
going to be 1/3.

00:06:32.100 --> 00:06:34.570
There's two values out
of a possible of six.

00:06:34.570 --> 00:06:37.165
The probability that
r3 is a 3 or a 6,

00:06:37.165 --> 00:06:38.860
only two possibilities there.

00:06:38.860 --> 00:06:42.950
So two out of six possible
faces of the die, so times 1/3.

00:06:42.950 --> 00:06:46.880
So the probability
that this first one is

00:06:46.880 --> 00:06:51.780
going to evaluate to this
value right over here,

00:06:51.780 --> 00:06:53.950
the second one is going
to evaluate to this value,

00:06:53.950 --> 00:06:56.140
and the third one is
going to evaluate to 1,

00:06:56.140 --> 00:07:00.000
is going to be 1/3 times 1/3
times 1/3, which is 1/27.

00:07:03.170 --> 00:07:07.665
And I touched on this
already, there's six ways.

00:07:07.665 --> 00:07:11.980
You could rearrange
this six ways.

00:07:11.980 --> 00:07:14.580
You have three terms, and you're
putting them in three places.

00:07:14.580 --> 00:07:18.360
So in the first place, you
could put three of the terms.

00:07:18.360 --> 00:07:20.330
In the second place,
you have two terms

00:07:20.330 --> 00:07:21.620
left that you could put.

00:07:21.620 --> 00:07:23.930
And in the last place, you
only have one term left.

00:07:23.930 --> 00:07:28.040
So there's 3 times 2 times 1
ways to arrange these things.

00:07:28.040 --> 00:07:32.030
We can arrange this 3 times
2 times 1 different ways,

00:07:32.030 --> 00:07:34.400
so there's six
different arrangements.

00:07:34.400 --> 00:07:37.850
The probability of each is 1/27.

00:07:37.850 --> 00:07:41.510
So the probability under
question, this thing over here,

00:07:41.510 --> 00:07:43.380
there's six arrangements.

00:07:43.380 --> 00:07:47.530
There's six arrangements of
getting these things to be

00:07:47.530 --> 00:07:50.320
added up in this way, and the
probability of each of them

00:07:50.320 --> 00:07:52.200
is 1/27.

00:07:52.200 --> 00:07:55.580
So it's equal to
6/27, and if you

00:07:55.580 --> 00:07:57.760
divide the numerator and
the denominator by 3,

00:07:57.760 --> 00:08:00.560
it's equal to 2/9.

00:08:00.560 --> 00:08:01.840
And we're done.

00:08:01.840 --> 00:08:04.930
It wasn't too bad, I think.

00:08:04.930 --> 00:08:07.250
The probability of getting
omega to the r1 plus

00:08:07.250 --> 00:08:11.060
omega to the r2 plus omega to
the r3, where r1 and r2 and r3

00:08:11.060 --> 00:08:13.230
are numbers obtained
from rolling

00:08:13.230 --> 00:08:16.280
a fair die, the probability--
when you add all of these,

00:08:16.280 --> 00:08:20.526
it's going to be
equal to 0-- is 2/9.

00:08:20.526 --> 00:08:23.820
And I thought that was
a pretty neat problem.


IIT JEE Perpendicular Plane (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.020
Where we had left off, we
had used the information

00:00:03.020 --> 00:00:05.679
from these two lines to
figure out two vectors,

00:00:05.679 --> 00:00:07.720
which are really just
almost, you could just say,

00:00:07.720 --> 00:00:11.040
two position vectors
lying in this white plane.

00:00:11.040 --> 00:00:14.720
And we took their cross
product to find a vector that's

00:00:14.720 --> 00:00:20.050
perpendicular, that's normal
to this entire white plane.

00:00:20.050 --> 00:00:21.490
And that was vector d there.

00:00:21.490 --> 00:00:27.214
And vector d will have to sit,
especially to visualize it,

00:00:27.214 --> 00:00:28.880
if you start the
vector-- you can always

00:00:28.880 --> 00:00:30.820
position a vector
anywhere you want--

00:00:30.820 --> 00:00:34.490
but if you start it at 0, it
will clearly sit in that plane.

00:00:34.490 --> 00:00:36.490
But since you could always
position it anywhere,

00:00:36.490 --> 00:00:38.740
you could say, OK,
let's start it at 0.

00:00:38.740 --> 00:00:41.190
So it will sit in
this yellow plane

00:00:41.190 --> 00:00:43.580
that we need to find
the equation for.

00:00:43.580 --> 00:00:45.180
And we know vector d sits in it.

00:00:45.180 --> 00:00:47.360
We already know that
vector a sits in it.

00:00:47.360 --> 00:00:51.320
We got that early on in the
first video on this problem.

00:00:51.320 --> 00:00:53.620
And so we could take the
cross product of d and a

00:00:53.620 --> 00:00:55.630
to find a normal vector to it.

00:00:55.630 --> 00:00:57.550
And then using
that normal vector,

00:00:57.550 --> 00:01:00.050
and then thinking about
what an arbitrary point,

00:01:00.050 --> 00:01:02.630
or what an arbitrary
vector on this plane

00:01:02.630 --> 00:01:05.080
might look like, we
can then figure out

00:01:05.080 --> 00:01:07.160
the equation for
this yellow plane.

00:01:07.160 --> 00:01:09.060
And we already touched
on it a little bit

00:01:09.060 --> 00:01:11.440
in the linear algebra
playlist, if any of this stuff

00:01:11.440 --> 00:01:12.364
makes you confused.

00:01:12.364 --> 00:01:14.280
And just a reminder, we
take the cross product

00:01:14.280 --> 00:01:17.157
of two vectors in three
dimensional space,

00:01:17.157 --> 00:01:19.740
it will give you a third vector
that is perpendicular to both.

00:01:19.740 --> 00:01:23.500
That's the whole tool that
we're using over here.

00:01:23.500 --> 00:01:26.960
So let's just take the
cross product of d and a.

00:01:26.960 --> 00:01:31.360
So this is going to be
equal to-- so for the i,

00:01:31.360 --> 00:01:36.430
we have 1 times 4, which
is 4, minus 3 times 10.

00:01:36.430 --> 00:01:39.640
So 4 minus 30 is negative 26.

00:01:39.640 --> 00:01:42.300
So we have negative 26i.

00:01:42.300 --> 00:01:45.690
And then we have
negative or minus for j.

00:01:45.690 --> 00:01:47.340
We have to swap signs for the j.

00:01:47.340 --> 00:01:52.540
And so we have negative
8 times 4 is negative 32.

00:01:52.540 --> 00:01:58.010
Negative 32 minus 2 times 10,
so minus 20, is negative 52.

00:01:58.010 --> 00:01:59.750
So we put a negative 52 here.

00:01:59.750 --> 00:02:02.680
Or we could just
say positive 52.

00:02:02.680 --> 00:02:05.120
Because it's minus
negative 52, which

00:02:05.120 --> 00:02:07.570
is the same thing as plus 52j.

00:02:07.570 --> 00:02:09.820
And finally, for the
k-- cross out that row,

00:02:09.820 --> 00:02:14.520
that column-- negative 8 times
3 is negative 24, minus 2 times

00:02:14.520 --> 00:02:15.020
1.

00:02:15.020 --> 00:02:17.270
So minus 2, it's negative 26.

00:02:17.270 --> 00:02:21.780
So minus 26k.

00:02:21.780 --> 00:02:25.980
So this normal vector
right here or an a vector

00:02:25.980 --> 00:02:27.940
that is normal to the
yellow plane, the plane

00:02:27.940 --> 00:02:31.570
that we have to find the
equation of, is this over here.

00:02:31.570 --> 00:02:34.200
And just because it kind of
pops out at me that all of these

00:02:34.200 --> 00:02:38.150
are divisible by 26,
I'm going to define

00:02:38.150 --> 00:02:39.067
another normal vector.

00:02:39.067 --> 00:02:41.400
I don't know, let's call it--
we could call it anything.

00:02:41.400 --> 00:02:46.230
Let me call it-- I don't
know, let's call it p.

00:02:46.230 --> 00:02:51.160
Or let's call it-- I don't
know, let's call it normal 1.

00:02:51.160 --> 00:02:53.990
I mean, just to show
it's a different vector.

00:02:53.990 --> 00:02:56.279
I'm just going to divide--
although essentially it's

00:02:56.279 --> 00:02:57.570
pointing in the same direction.

00:02:57.570 --> 00:02:59.778
It has the same direction,
just different magnitude--

00:02:59.778 --> 00:03:03.500
I'm just going to divide
all of these by negative 26.

00:03:03.500 --> 00:03:06.170
So if I divide all of
these by negative 26,

00:03:06.170 --> 00:03:07.400
I'm just scaling the vector.

00:03:07.400 --> 00:03:08.960
But it's still going to
go in the same direction.

00:03:08.960 --> 00:03:10.370
Because that's
what I care about.

00:03:10.370 --> 00:03:14.120
I care about just
finding any vector that

00:03:14.120 --> 00:03:19.817
is normal to this
yellow plane right here.

00:03:19.817 --> 00:03:20.900
I want to find any vector.

00:03:20.900 --> 00:03:21.764
I already found one.

00:03:21.764 --> 00:03:23.180
And now I'm just
going to scale it

00:03:23.180 --> 00:03:24.820
down so that it's
a simpler vector.

00:03:24.820 --> 00:03:27.140
So if I divide everything
by negative 26,

00:03:27.140 --> 00:03:36.760
I get i plus-- or I should
say, minus 2j plus k.

00:03:36.760 --> 00:03:40.400
So that might look something
like this right over there.

00:03:40.400 --> 00:03:43.900
So this is our n1 vector
that is normal to this.

00:03:43.900 --> 00:03:46.990
Now, just using this,
we can now figure out

00:03:46.990 --> 00:03:49.380
the equation of this
plane right here.

00:03:49.380 --> 00:03:51.170
So let's just think
about it a little bit.

00:03:51.170 --> 00:03:54.040
Let's just say I have a
point, or I could even

00:03:54.040 --> 00:03:56.070
say a position vector, x, y, z.

00:03:56.070 --> 00:03:58.990
And I know x, y, z
sits on this plane.

00:03:58.990 --> 00:04:01.880
So x, y, z.

00:04:01.880 --> 00:04:03.700
It sits on this plane.

00:04:03.700 --> 00:04:06.200
And we know that the point
0, 0, 0 is on the plane.

00:04:06.200 --> 00:04:12.200
So we also know that
the position vector

00:04:12.200 --> 00:04:19.491
xi plus yj plus zk have to
sit on this yellow plane

00:04:19.491 --> 00:04:19.990
in question.

00:04:19.990 --> 00:04:24.080
I'm assuming that x, y, z
is a point on this plane.

00:04:24.080 --> 00:04:26.724
I could have put
little 0's here just

00:04:26.724 --> 00:04:28.140
to make sure that--
you know, this

00:04:28.140 --> 00:04:30.190
could be a particular
point on this plane.

00:04:30.190 --> 00:04:31.950
I'm saying this is
just any point that I'm

00:04:31.950 --> 00:04:34.260
defining to be on this plane.

00:04:34.260 --> 00:04:37.730
Now, this position vector
will then be on this plane.

00:04:37.730 --> 00:04:42.060
And since this vector is normal
to this position vector right

00:04:42.060 --> 00:04:44.549
here, or another way to
say it, it's perpendicular,

00:04:44.549 --> 00:04:45.340
or it's orthogonal.

00:04:49.170 --> 00:04:52.290
If I were to take the dot
product, if I were to take that

00:04:52.290 --> 00:04:55.894
and then dot it with this
vector right over here,

00:04:55.894 --> 00:04:57.810
and dot it with that
vector right over there--

00:04:57.810 --> 00:05:06.360
so I would take the dot product
of that and this-- yj plus zk.

00:05:06.360 --> 00:05:08.400
If I take the dot product
of these two things,

00:05:08.400 --> 00:05:12.540
it has to be equal to 0.

00:05:12.540 --> 00:05:15.740
If I take the dot product
of two vectors that

00:05:15.740 --> 00:05:17.894
are orthogonal to each
other-- and by definition,

00:05:17.894 --> 00:05:18.810
these two vectors are.

00:05:18.810 --> 00:05:22.560
I'm saying that this
guy is in the plane.

00:05:22.560 --> 00:05:25.680
This vector, we
already figured out,

00:05:25.680 --> 00:05:27.110
is perpendicular to the plane.

00:05:27.110 --> 00:05:28.620
Which means it's
perpendicular, it's

00:05:28.620 --> 00:05:30.730
orthogonal to
everything in the plane.

00:05:30.730 --> 00:05:34.490
So if we take their dot
product, it will be equal to 0.

00:05:34.490 --> 00:05:35.830
So what does that give us?

00:05:35.830 --> 00:05:40.070
So if we take their dot
product, we get x times 1.

00:05:40.070 --> 00:05:42.650
So if we take-- let me
just go term by term.

00:05:42.650 --> 00:05:46.340
So if we take x times--
the coefficient there is 1.

00:05:46.340 --> 00:05:53.670
So we have x plus
y times negative 2.

00:05:53.670 --> 00:05:55.870
So I should really
call that-- well, I

00:05:55.870 --> 00:05:58.930
could just call
that negative 2y.

00:05:58.930 --> 00:06:04.010
And then finally,
I have z times 1.

00:06:04.010 --> 00:06:05.530
There's a 1 in front of this k.

00:06:05.530 --> 00:06:06.920
It's implicitly there.

00:06:06.920 --> 00:06:09.960
So z times 1.

00:06:09.960 --> 00:06:11.020
So plus z.

00:06:11.020 --> 00:06:12.280
That's the dot product.

00:06:12.280 --> 00:06:15.042
And that has to be equal to 0.

00:06:15.042 --> 00:06:17.000
Or another way to write
it, just to simplify it

00:06:17.000 --> 00:06:23.480
a little bit, is that x minus
2y plus z is equal to 0.

00:06:23.480 --> 00:06:26.474
And that is the equation
of our plane in question.

00:06:26.474 --> 00:06:28.140
Or I should actually
write it in yellow,

00:06:28.140 --> 00:06:29.590
because it's the yellow plane.

00:06:29.590 --> 00:06:32.719
x minus 2y plus z is equal to 0.

00:06:32.719 --> 00:06:35.260
And I know this might have been
a little daunting or a little

00:06:35.260 --> 00:06:37.612
confusing, and if you actually
had to do this in exam,

00:06:37.612 --> 00:06:39.570
you wouldn't have to do
all of the explanation.

00:06:39.570 --> 00:06:42.050
But just to visualize
what we did,

00:06:42.050 --> 00:06:45.410
we used these two lines--
you can kind of view

00:06:45.410 --> 00:06:47.787
the vectors we used as kind
of the slopes of these lines

00:06:47.787 --> 00:06:48.620
in three dimensions.

00:06:48.620 --> 00:06:51.000
Don't worry about it
if that confuses you.

00:06:51.000 --> 00:06:54.410
But we used the vectors
in-- both of these

00:06:54.410 --> 00:06:56.020
specify a different vector.

00:06:56.020 --> 00:06:57.770
You could keep scaling
them into whatever.

00:06:57.770 --> 00:06:59.520
But we just took
their cross product

00:06:59.520 --> 00:07:03.340
to find a vector that is normal
to this entire white plane.

00:07:03.340 --> 00:07:05.810
And it would have to
sit in the yellow plane.

00:07:05.810 --> 00:07:07.429
Now, if it sits on
the yellow plane,

00:07:07.429 --> 00:07:09.970
we could take the cross product
of that with the first vector

00:07:09.970 --> 00:07:12.090
that we found that was
in the yellow plane.

00:07:12.090 --> 00:07:14.486
A vector that sat
on that line, or you

00:07:14.486 --> 00:07:15.860
could almost view
it as the slope

00:07:15.860 --> 00:07:17.809
of that line in
three dimensions.

00:07:17.809 --> 00:07:19.600
And if you take the
cross product of that--

00:07:19.600 --> 00:07:20.600
it's not exactly the slope.

00:07:20.600 --> 00:07:21.840
I don't want you
all to nit-pick me.

00:07:21.840 --> 00:07:22.710
So no, it's not
exactly the slope.

00:07:22.710 --> 00:07:24.390
But a vector in
three dimensions is

00:07:24.390 --> 00:07:26.030
kind of specifying
a direction, which

00:07:26.030 --> 00:07:29.510
is analogous to a slope
in two dimensions.

00:07:29.510 --> 00:07:31.930
But if we take the cross product
of these two vectors now,

00:07:31.930 --> 00:07:33.280
we'll get a normal vector.

00:07:33.280 --> 00:07:34.950
That's what we started
this video doing.

00:07:34.950 --> 00:07:38.050
I scaled back this normal vector
just to simplify the math.

00:07:38.050 --> 00:07:40.300
And I said, OK, if I
have this normal vector,

00:07:40.300 --> 00:07:42.890
and any arbitrary vector
on this plane, when

00:07:42.890 --> 00:07:45.510
I take their dot
product it'll be 0.

00:07:45.510 --> 00:07:46.930
And so what that
allowed us to do

00:07:46.930 --> 00:07:51.640
is figure out what are the
constraints on that x, y and z.

00:07:51.640 --> 00:07:53.630
Any x, y and z that's
on the plane, when

00:07:53.630 --> 00:07:56.160
I take the dot product
of that position vector

00:07:56.160 --> 00:07:58.480
with the normal vector,
it has to be equal to 0.

00:07:58.480 --> 00:08:02.300
So that gave us the actual
constraints for the plane.

00:08:02.300 --> 00:08:04.910
Anyway, hopefully
you found that fun.


IIT JEE Perpendicular Planes (Part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.330
We define the equation
of the plane containing

00:00:03.330 --> 00:00:06.400
the straight line-- They give us
the equation right over here--

00:00:06.400 --> 00:00:08.700
And this plane has to be
perpendicular to the plane

00:00:08.700 --> 00:00:11.040
containing the straight lines.

00:00:11.040 --> 00:00:13.590
This thing, right over
here, x/3 is equal to y/4,

00:00:13.590 --> 00:00:15.140
is equal to z/2.

00:00:15.140 --> 00:00:20.880
And also it has the line x/4 is
equal to y/2, is equal to z/3.

00:00:20.880 --> 00:00:25.230
Now just to visualize
this properly,

00:00:25.230 --> 00:00:28.522
let me draw the planes.

00:00:28.522 --> 00:00:29.980
The first plane in
question that we

00:00:29.980 --> 00:00:34.270
need to find the equation
of-- Let me draw it.

00:00:34.270 --> 00:00:36.960
So we need to find the
equation of this plane.

00:00:36.960 --> 00:00:39.750
And I'm going to try to do
this whole problem using vector

00:00:39.750 --> 00:00:41.840
notation, because
it's easier in my mind

00:00:41.840 --> 00:00:45.150
to manipulate vectors in a way
that you can find perpendicular

00:00:45.150 --> 00:00:46.750
vectors and
equations for planes.

00:00:46.750 --> 00:00:49.070
There's other ways that
you could do this problem.

00:00:49.070 --> 00:00:51.920
So we've defined the equation
of this plane right here.

00:00:51.920 --> 00:00:54.210
We need to find the equation.

00:00:54.210 --> 00:00:57.550
Now we know this plane
contains this line,

00:00:57.550 --> 00:00:59.650
and this line, if we
look at right over here,

00:00:59.650 --> 00:01:01.920
it contains the point 0, 0, 0.

00:01:01.920 --> 00:01:04.890
If x and y and z are
equal to 0, then all sides

00:01:04.890 --> 00:01:07.020
of these equations are 0.

00:01:07.020 --> 00:01:09.400
So it contains
the point 0, 0, 0.

00:01:11.990 --> 00:01:15.000
Or you could even view it as it
contains the position vector 0,

00:01:15.000 --> 00:01:16.430
0, 0.

00:01:16.430 --> 00:01:20.030
And it also contains
the point 2, 3, 4.

00:01:20.030 --> 00:01:23.290
Because if x is 2, y is 3,
and z is 4, then all of these

00:01:23.290 --> 00:01:24.830
are going to be equal to 1.

00:01:24.830 --> 00:01:26.650
So let me draw that
right over here.

00:01:26.650 --> 00:01:29.470
So it also contains the
point, or the position vector,

00:01:29.470 --> 00:01:34.060
we could say 2, 3, 4.

00:01:34.060 --> 00:01:36.120
So if we wanted to
just draw this line,

00:01:36.120 --> 00:01:42.200
to point specify the line,
this line would look like that.

00:01:42.200 --> 00:01:43.970
But if we wanted to
think about a vector

00:01:43.970 --> 00:01:46.710
that this plane
contains, it would

00:01:46.710 --> 00:01:50.640
be the vector that goes
from 0, 0, 0 to 2, 3, 4.

00:01:50.640 --> 00:01:52.970
So it would be this
vector right here, which

00:01:52.970 --> 00:01:55.319
really is the position
vector 2, 3, 4.

00:01:55.319 --> 00:01:57.110
And I'm going to write
an ijk notation just

00:01:57.110 --> 00:01:59.070
to make it clear that
this is a vector.

00:01:59.070 --> 00:02:02.110
So this vector that lies
in the plane in question

00:02:02.110 --> 00:02:12.089
is the vector 2i-- I'll write it
over here-- 2i plus 3j plus 4k.

00:02:12.089 --> 00:02:13.630
I just wrote in unit
vector notation.

00:02:13.630 --> 00:02:16.361
I'll say a, I'll
call this vector a.

00:02:16.361 --> 00:02:17.830
Vector a is equal to that.

00:02:17.830 --> 00:02:19.600
And I'll write a over here.

00:02:19.600 --> 00:02:24.544
Now we need to figure out
the equation of this line.

00:02:24.544 --> 00:02:26.460
And the best way to do
it, or at least the way

00:02:26.460 --> 00:02:31.570
my brain is telling
me to do it, if I

00:02:31.570 --> 00:02:33.570
can find another
vector in this plane,

00:02:33.570 --> 00:02:35.880
then I can find a normal
vector to this plane.

00:02:35.880 --> 00:02:38.860
And then using a normal vector
and some arbitrary other vector

00:02:38.860 --> 00:02:42.050
on this plane, I can then figure
out the equation of the line.

00:02:42.050 --> 00:02:45.390
Because the normal vector dotted
that arbitrary other vector

00:02:45.390 --> 00:02:46.720
has to be equal to 0.

00:02:46.720 --> 00:02:48.780
Now if none of that
makes any sense, maybe

00:02:48.780 --> 00:02:52.510
as we go through the problem
it will make more sense.

00:02:52.510 --> 00:02:57.080
So what I'm going to do is use
these other lines, or really

00:02:57.080 --> 00:02:59.460
the other vectors,
that are implicitly

00:02:59.460 --> 00:03:03.590
defined by these
lines to figure out

00:03:03.590 --> 00:03:06.350
another vector in
this plane over here.

00:03:06.350 --> 00:03:08.930
So let me draw what I'm talking
about, that might be easier.

00:03:08.930 --> 00:03:10.730
So let me draw this other plane.

00:03:10.730 --> 00:03:12.810
And I'll draw it in white.

00:03:12.810 --> 00:03:15.420
And this other plane
also contains 0, 0, 0.

00:03:15.420 --> 00:03:18.530
Or actually, both of these
lines over here contain 0, 0, 0.

00:03:18.530 --> 00:03:21.030
So let me draw it over here.

00:03:21.030 --> 00:03:25.080
So this other plane, I
think what I just said,

00:03:25.080 --> 00:03:27.506
if I do it step by step, it
will make a lot more sense.

00:03:27.506 --> 00:03:29.630
So the other plane, that's
when it comes vertically

00:03:29.630 --> 00:03:35.070
above this plane, and it goes
vertically below the plane

00:03:35.070 --> 00:03:36.110
as well.

00:03:36.110 --> 00:03:37.840
And we see the
plane that contains

00:03:37.840 --> 00:03:40.141
these two lines is
perpendicular to this.

00:03:40.141 --> 00:03:41.640
They're perpendicular
to each other,

00:03:41.640 --> 00:03:44.410
and we need to find the
equation of the yellow plane.

00:03:44.410 --> 00:03:48.750
Now this blue line over here
contains the point 0, 0, 0

00:03:48.750 --> 00:03:50.862
and the point 3, 4, 2.

00:03:50.862 --> 00:03:52.820
Same logic that we've
got to get to this point.

00:03:52.820 --> 00:03:55.690
If this is 3, this is 4,
this is 2, they all equal 1.

00:03:55.690 --> 00:03:57.650
The equation holds.

00:03:57.650 --> 00:04:00.360
So let me just draw
some arbitrary place.

00:04:00.360 --> 00:04:02.430
So 3, 4, 2.

00:04:02.430 --> 00:04:04.330
And I want to be
clear, I haven't even

00:04:04.330 --> 00:04:05.780
defined the coordinate axis.

00:04:05.780 --> 00:04:08.370
We don't know what angle
we're looking at this from.

00:04:08.370 --> 00:04:10.250
This is just for
visualization purposes.

00:04:10.250 --> 00:04:12.166
So this line right over
here, I could draw it,

00:04:12.166 --> 00:04:14.140
would look something like this.

00:04:14.140 --> 00:04:16.600
It would look
something like that.

00:04:16.600 --> 00:04:19.420
It sits on that white plane.

00:04:19.420 --> 00:04:22.070
We can do the same thing
for this magenta line.

00:04:22.070 --> 00:04:24.850
The magenta line also
contains 0, 0, 0,

00:04:24.850 --> 00:04:28.286
but it contains the
point 4, 2, and 3.

00:04:28.286 --> 00:04:29.660
So let me just
draw it over here.

00:04:34.170 --> 00:04:37.222
So that's the magenta line.

00:04:37.222 --> 00:04:39.430
And you wouldn't even see
it if this was transparent.

00:04:39.430 --> 00:04:42.640
You would see the line, it would
just keep going down like that.

00:04:42.640 --> 00:04:44.810
Now, I'm going to
attempt to explain

00:04:44.810 --> 00:04:47.190
how I'm going to tackle
this problem again.

00:04:47.190 --> 00:04:50.280
I'm going to use vectors
defined, that are implicitly

00:04:50.280 --> 00:04:52.887
defined, by this blue line
and this magenta line.

00:04:52.887 --> 00:04:55.220
I'm going to take the cross
product of those two vectors

00:04:55.220 --> 00:05:01.070
to find a vector that is normal
to this entire white plane.

00:05:01.070 --> 00:05:04.890
And that vector has to sit
in this perpendicular plane.

00:05:04.890 --> 00:05:08.190
And then I can use that vector
and vector a to figure out

00:05:08.190 --> 00:05:10.010
a normal vector for this plane.

00:05:10.010 --> 00:05:12.840
And then I'll be able to figure
out the equation of the plane.

00:05:12.840 --> 00:05:15.290
Let me just do it, instead
of keep trying to--

00:05:15.290 --> 00:05:16.330
So what's a vector?

00:05:16.330 --> 00:05:19.160
What are two vectors that
sit in this white plane?

00:05:19.160 --> 00:05:22.130
Well, the point 0,0, 0, or the
position vector is 0, 0, 0.

00:05:22.130 --> 00:05:26.080
And the position vector
4,2,3 is in this plane.

00:05:26.080 --> 00:05:32.060
So the vector, I'll
call that b, the vector

00:05:32.060 --> 00:05:41.340
b is equal to 4i plus 2j plus
3k is also in this white plane.

00:05:41.340 --> 00:05:45.490
And you really, I'm just
subtracting this position

00:05:45.490 --> 00:05:49.377
vector from that position vector
to get this position vector,

00:05:49.377 --> 00:05:50.710
which clearly sits in the plane.

00:05:50.710 --> 00:05:53.812
This vector starts at the
origin goes to 4, 2, 3.

00:05:53.812 --> 00:05:55.520
This plane contains
the origin is clearly

00:05:55.520 --> 00:05:56.870
sitting in that plane.

00:05:56.870 --> 00:06:00.985
And by that same logic I
could figure out a vector.

00:06:00.985 --> 00:06:02.360
It's actually the
position vector

00:06:02.360 --> 00:06:05.120
itself sits on this plane.

00:06:05.120 --> 00:06:08.170
So this is the vector
I'll call that c

00:06:08.170 --> 00:06:13.019
is equal to 3i plus 4j plus 2k.

00:06:13.019 --> 00:06:15.060
The fact that everything
is at the origin kind of

00:06:15.060 --> 00:06:15.643
simplifies it.

00:06:15.643 --> 00:06:20.700
The position vectors
themselves sit on the plane.

00:06:20.700 --> 00:06:22.309
If this wasn't
centered at the origin

00:06:22.309 --> 00:06:23.600
we wouldn't be able to do that.

00:06:23.600 --> 00:06:25.930
We would have to subtract
between these two vectors,

00:06:25.930 --> 00:06:28.000
or between two other
points to actually get

00:06:28.000 --> 00:06:29.370
vectors on the plane.

00:06:29.370 --> 00:06:31.600
But with that out
of the way, I could

00:06:31.600 --> 00:06:39.550
take the cross
product of b and c,

00:06:39.550 --> 00:06:44.980
and that will give
us some vector that

00:06:44.980 --> 00:06:47.900
is perpendicular
to both b and c.

00:06:47.900 --> 00:06:49.550
Which means that
it's perpendicular,

00:06:49.550 --> 00:06:52.810
or it's normal to this
entire white plane, which

00:06:52.810 --> 00:06:56.320
means it must sit in
this yellow plane.

00:06:56.320 --> 00:06:58.240
So I need to find,
I want to find,

00:06:58.240 --> 00:07:02.850
a normal vector that's in the
yellow plane, just like that.

00:07:02.850 --> 00:07:04.280
Let's call that vector d.

00:07:04.280 --> 00:07:06.940
So the vector d is going
to be equal to this.

00:07:06.940 --> 00:07:10.740
And so the cross product of b
and c, this is how I remember,

00:07:10.740 --> 00:07:12.800
is the determinant
of you take ijk.

00:07:15.276 --> 00:07:17.150
I'm going to explain
this in a lot more depth

00:07:17.150 --> 00:07:19.700
in the linear algebra playlist.

00:07:19.700 --> 00:07:24.060
But it's determinate of that,
The next row is vector b.

00:07:24.060 --> 00:07:27.610
So it's 4, 2 and 3.

00:07:27.610 --> 00:07:31.730
And then the next
row is the vector c.

00:07:31.730 --> 00:07:33.790
So 3, 4, and 2.

00:07:37.650 --> 00:07:42.840
And so this is going to be
equal to-- let's think about it.

00:07:42.840 --> 00:07:45.640
So for i you cross out
that column, that row,

00:07:45.640 --> 00:07:49.090
it's 2 times 2 minus 3 times 4.

00:07:49.090 --> 00:07:51.500
So that's 4 minus 12.

00:07:51.500 --> 00:07:53.130
So it's negative 8.

00:07:53.130 --> 00:07:57.862
Negative 8 times
i minus-- remember

00:07:57.862 --> 00:07:58.820
you have to swap signs.

00:07:58.820 --> 00:08:01.800
You go plus, minus, plus when
you're taking the determinant--

00:08:01.800 --> 00:08:05.700
so you ignore the j,
so it's 4 times 2.

00:08:05.700 --> 00:08:12.070
4 times 2 is 8, minus 3
times 3, which is negative 1.

00:08:12.070 --> 00:08:14.090
So this would be
a negative 1 here,

00:08:14.090 --> 00:08:15.960
but I already have a
negative out there.

00:08:15.960 --> 00:08:23.410
So it's going to be plus
1j and then finally, the k.

00:08:23.410 --> 00:08:26.010
You cross out that row, that
column, 4 times 4 is 16.

00:08:30.214 --> 00:08:33.819
16 minus 6 is 10.

00:08:33.819 --> 00:08:36.940
So plus 10.

00:08:36.940 --> 00:08:38.740
Let me make sure
I got that right.

00:08:38.740 --> 00:08:41.030
So the first one, I
always have to make sure.

00:08:41.030 --> 00:08:45.810
So i got 4 minus
12 is negative 8.

00:08:45.810 --> 00:08:51.130
Then j, cross that out, 4 times
2 is 8, minus 9 is negative 1.

00:08:51.130 --> 00:08:53.250
But the j has a negative,
so that's right.

00:08:53.250 --> 00:08:55.600
And then k, 4 times 4 is 16.

00:08:55.600 --> 00:09:00.080
16 minus 6 is 10.

00:09:00.080 --> 00:09:01.056
So that looks right.

00:09:01.056 --> 00:09:04.230
So plus 10k.

00:09:04.230 --> 00:09:06.906
So this vector right here,
this is the vector d.

00:09:06.906 --> 00:09:07.780
This is the vector d.

00:09:07.780 --> 00:09:09.540
I just figured it out.

00:09:09.540 --> 00:09:13.700
Now, if we take the cross
product of d and vector a,

00:09:13.700 --> 00:09:15.350
the vector that
was-- you can kind of

00:09:15.350 --> 00:09:19.350
view it as lying along
this line over here--

00:09:19.350 --> 00:09:22.100
then we will get a vector
that is normal to this plane.

00:09:27.190 --> 00:09:30.190
We could call that
vector n for normal.

00:09:30.190 --> 00:09:32.860
That is normal to
the yellow plane.

00:09:32.860 --> 00:09:35.600
And it would actually
sit in the white plane.

00:09:35.600 --> 00:09:36.554
So let's figure that.

00:09:36.554 --> 00:09:37.970
We'll see in a few
steps, probably

00:09:37.970 --> 00:09:40.240
in the next video, why
that is actually useful.

00:09:43.960 --> 00:09:53.570
So let's find what d cross-- so
if I take the cross product-- d

00:09:53.570 --> 00:09:56.655
cross a is going to be
equal to our normal vector.

00:09:59.420 --> 00:10:02.030
Which is equal to
the same thing, ijk.

00:10:05.400 --> 00:10:10.860
We're going to take d,
which is negative 8, 1, 10.

00:10:10.860 --> 00:10:19.182
And take the cross of that and
the vector a, which is 2, 3, 4,

00:10:19.182 --> 00:10:20.640
and we will get--
and actually I'll

00:10:20.640 --> 00:10:22.010
continue this in the next video.

00:10:22.010 --> 00:10:25.610
Just so that I don't make
these videos too long.


IIT JEE trigonometry problem 1

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.440
- [Instructor] If the angles
A, B and C of a triangle

00:00:03.440 --> 00:00:05.068
are in an arithmetic progression,

00:00:05.068 --> 00:00:08.500
and if a, b and c, lowercase, a, b and c,

00:00:08.500 --> 00:00:12.380
denote the lengths of the
sides opposite to the capital,

00:00:12.380 --> 00:00:14.960
the angles A, capital A, capital B

00:00:14.960 --> 00:00:16.460
and capital C respectively,

00:00:16.460 --> 00:00:20.030
then what is the value of this
expression right over here?

00:00:20.030 --> 00:00:22.890
So let's see if we can
work our way through this.

00:00:22.890 --> 00:00:24.390
So let's just draw the triangle

00:00:24.390 --> 00:00:25.760
just so we have a visualization

00:00:25.760 --> 00:00:27.283
of what all of the letters represent.

00:00:27.283 --> 00:00:29.980
So we have the angles A, B, and C,

00:00:29.980 --> 00:00:32.490
so let me just draw them like this.

00:00:32.490 --> 00:00:37.070
So we have the angles A, B, and C.

00:00:37.070 --> 00:00:39.940
And then the sides opposite
them are the lowercase version.

00:00:39.940 --> 00:00:43.039
So the side opposite to
capital A is lowercase a,

00:00:43.039 --> 00:00:46.760
the side opposite to
capital B is lowercase b,

00:00:46.760 --> 00:00:51.760
and the side opposite to the
angle capital C is lowercase c.

00:00:51.780 --> 00:00:53.754
Now the first piece of
information they tell us

00:00:53.754 --> 00:00:57.840
is that the angles capital
A, capital B, and capital C

00:00:57.840 --> 00:01:01.007
of the triangle are in
arithmetic progression.

00:01:01.007 --> 00:01:06.007
Arithmetic progression, very fancy word,

00:01:06.040 --> 00:01:08.150
but on arithmetic progression is,

00:01:08.150 --> 00:01:09.963
is a series of numbers that are separated

00:01:09.963 --> 00:01:11.570
by the same amount.

00:01:11.570 --> 00:01:13.391
So let me give you some examples.

00:01:13.391 --> 00:01:16.540
So one, two, three, that's
an arithmetic progression.

00:01:16.540 --> 00:01:20.100
Two, four, six arithmetic progression.

00:01:20.100 --> 00:01:22.150
Were separated by two every time.

00:01:22.150 --> 00:01:26.830
I could do 10, 20, 30, also
an arithmetic progression.

00:01:26.830 --> 00:01:28.380
These are all arithmetic progression.

00:01:28.380 --> 00:01:31.850
So all they're saying is, is
to go from angle A to angle B,

00:01:31.850 --> 00:01:34.220
however much that is it's
the same amount to go

00:01:34.220 --> 00:01:36.330
from angle B to angle C.

00:01:36.330 --> 00:01:39.090
So let's see what that tells us,

00:01:39.090 --> 00:01:40.500
what that tells us about,

00:01:40.500 --> 00:01:42.160
or maybe tells us maybe it doesn't tell us

00:01:42.160 --> 00:01:43.980
anything about those angles.

00:01:43.980 --> 00:01:48.290
So if we could say we have angle A

00:01:48.290 --> 00:01:51.350
and then we have the notion angle B.

00:01:51.350 --> 00:01:56.064
So we could say that B is
equal to A plus some constant.

00:01:56.064 --> 00:01:57.190
We don't know what that is.

00:01:57.190 --> 00:01:58.870
It could go up by one, it could go by two,

00:01:58.870 --> 00:01:59.703
it could go by 10.

00:01:59.703 --> 00:02:00.990
We don't know what it is.

00:02:00.990 --> 00:02:03.150
So A plus N.

00:02:03.150 --> 00:02:07.188
And then C would be equal to B plus N,

00:02:07.188 --> 00:02:09.380
which is the same thing,

00:02:09.380 --> 00:02:11.960
which is the same thing as B is A plus N.

00:02:11.960 --> 00:02:16.960
So this is A plus N plus N,
which is equal to A plus 2N.

00:02:19.368 --> 00:02:20.930
So what does that do?

00:02:20.930 --> 00:02:23.750
The other thing we know about
three angles in a triangle

00:02:23.750 --> 00:02:25.927
is that they have to
add up to 180 degrees.

00:02:25.927 --> 00:02:30.927
So this, this, and this have
to add up to 180 degrees.

00:02:31.880 --> 00:02:32.713
Let's try it out.

00:02:32.713 --> 00:02:36.040
So we have A plus A plus N plus A plus 2N,

00:02:36.040 --> 00:02:41.040
plus A plus 2N is going to
be equal to 180 degrees.

00:02:42.960 --> 00:02:45.350
We have one, two, three A's here.

00:02:45.350 --> 00:02:48.730
So we get 3A plus we have
one N and then another 2N,

00:02:48.730 --> 00:02:53.470
3A plus 3N is equal to 180 degrees.

00:02:53.470 --> 00:02:56.120
Or you can divide both sides by three.

00:02:56.120 --> 00:03:00.770
And you get A plus N is to, A
plus N is equal to 60 degrees.

00:03:03.630 --> 00:03:05.602
So what does that tell us?

00:03:05.602 --> 00:03:08.730
Well, A could still be
anything 'cause if N is one,

00:03:08.730 --> 00:03:09.774
then A is 59.

00:03:09.774 --> 00:03:12.854
If N is 10, then A is gonna be 50.

00:03:12.854 --> 00:03:17.380
So it doesn't give us much
information about the angle A.

00:03:17.380 --> 00:03:20.760
But if you look up here, do
you see an A plus N anywhere?

00:03:20.760 --> 00:03:21.950
Well, you see it right over here.

00:03:21.950 --> 00:03:23.680
B is equal to A plus N.

00:03:23.680 --> 00:03:26.570
And we just figured out that A plus N

00:03:26.570 --> 00:03:28.463
has to be equal to 60 degrees.

00:03:28.463 --> 00:03:30.710
So using this first piece of information,

00:03:30.710 --> 00:03:33.530
we were able to come up with
something pretty tangible,

00:03:33.530 --> 00:03:37.550
B must be equal to 60 degrees.

00:03:37.550 --> 00:03:39.230
And you could try it out
with a bunch of numbers.

00:03:39.230 --> 00:03:43.670
These could be 59, 60 and 61.

00:03:43.670 --> 00:03:45.490
That's an arithmetic progression.

00:03:45.490 --> 00:03:48.300
And once again, B is the
middle one, right over here.

00:03:48.300 --> 00:03:51.980
These could be 50, 60, and 70,

00:03:51.980 --> 00:03:55.210
could be 40, 60, and 80.

00:03:55.210 --> 00:03:57.540
But no matter what the
arithmetic progression is

00:03:57.540 --> 00:04:00.410
in order for these three
angles to add up to 180,

00:04:00.410 --> 00:04:05.410
the middle one has to
be equal to 60 degrees.

00:04:05.563 --> 00:04:08.810
We're doing pretty well so far.

00:04:08.810 --> 00:04:11.950
So let's see what we can do at
the next part of the problem.

00:04:11.950 --> 00:04:13.640
And I'm trying to save
some screen real estate

00:04:13.640 --> 00:04:14.950
right over here.

00:04:14.950 --> 00:04:17.220
Okay, so they want us
to figure out the value

00:04:17.220 --> 00:04:18.053
of the expression, a over
c sin of 2C capital C

00:04:18.053 --> 00:04:22.900
plus c over a sin of two capital A.

00:04:22.900 --> 00:04:24.940
So let's let me just write it down.

00:04:24.940 --> 00:04:29.940
So we have, I'll do it in,
I'll do it in blue, a over c ,

00:04:30.640 --> 00:04:35.640
a over c sine of two times capital C plus

00:04:36.730 --> 00:04:41.730
c over a sine of two times capital A,

00:04:42.400 --> 00:04:43.510
what's that going to be equal to?

00:04:43.510 --> 00:04:46.550
So whenever you see stuff like
this, you've got a two here,

00:04:46.550 --> 00:04:47.920
two here,

00:04:47.920 --> 00:04:49.220
frankly the best things you do is

00:04:49.220 --> 00:04:50.390
just to experiment with your

00:04:50.390 --> 00:04:53.010
trigonometric identities and
see if anything pops out at

00:04:53.010 --> 00:04:54.150
you that might be useful.

00:04:54.150 --> 00:04:55.700
And a little bit of a clue here.

00:04:55.700 --> 00:04:57.440
The first part of the problem helped us

00:04:57.440 --> 00:04:58.597
figure out what B is.

00:04:58.597 --> 00:05:01.560
It helped us figure out what B is,

00:05:01.560 --> 00:05:03.440
but right now the
expression has no B in it.

00:05:03.440 --> 00:05:06.130
So right now this information
seems kind of useless,

00:05:06.130 --> 00:05:09.430
but if we could put this
somehow in terms of B,

00:05:09.430 --> 00:05:10.263
then we'll have we'll,

00:05:10.263 --> 00:05:13.420
we'll be making progress because
we know information about

00:05:13.420 --> 00:05:15.730
angle B. So let's see what we can do.

00:05:15.730 --> 00:05:20.070
So the first thing I would
use is well sine of two A,

00:05:20.070 --> 00:05:23.930
let me just rewrite each of
these. So sine of, I should say,

00:05:23.930 --> 00:05:25.760
sine of two times anything.

00:05:25.760 --> 00:05:28.254
That's just the same thing
as I think this is called the

00:05:28.254 --> 00:05:32.860
double angle formula. So this
is, although I might be wrong,

00:05:32.860 --> 00:05:34.430
I always forget the actual names of them,

00:05:34.430 --> 00:05:36.150
but sine of two times something is

00:05:36.150 --> 00:05:39.840
two sine of that something times

00:05:39.840 --> 00:05:42.970
the cosine of that . Times
a cosine of that something,

00:05:42.970 --> 00:05:44.870
and you'll see that in any
trigonometric book on the

00:05:44.870 --> 00:05:47.260
inside cover, even a
lot of calculus books,

00:05:47.260 --> 00:05:49.190
let's do that for the, the
same thing right over here.

00:05:49.190 --> 00:05:54.190
So sine of two A over here
is going to be two sine of A

00:05:55.170 --> 00:05:56.430
cosine of A,

00:05:56.430 --> 00:05:58.550
that's just a standard
trigonometric identity.

00:05:58.550 --> 00:06:00.400
And we in the trigonometric playlist,

00:06:00.400 --> 00:06:03.240
we proved that identity. I
think we do it multiple times.

00:06:03.240 --> 00:06:05.090
And then out in front, we
have our coefficients still.

00:06:05.090 --> 00:06:10.090
We have a over c times this
plus c over a times, this.

00:06:11.670 --> 00:06:12.980
Now is there anything we can do?

00:06:12.980 --> 00:06:14.740
And remember in the back of our mind,

00:06:14.740 --> 00:06:17.120
we should be thinking of how
can we use this information

00:06:17.120 --> 00:06:18.314
that B is equal to 60.

00:06:18.314 --> 00:06:22.306
So if we can somehow put this
in the form to get a B here.

00:06:22.306 --> 00:06:24.960
And when I think about how do
you get to a B here, I think,

00:06:24.960 --> 00:06:27.080
well, you know, we have a triangle here.

00:06:27.080 --> 00:06:29.446
So the things that relate
the sides of a triangle when,

00:06:29.446 --> 00:06:31.460
especially when it's not a right triangle,

00:06:31.460 --> 00:06:33.220
is we're really going to deal
with the law of sines and the

00:06:33.220 --> 00:06:35.520
law of cosines and the law of sines.

00:06:35.520 --> 00:06:38.200
Let me just rewrite it over
here just for our reference.

00:06:38.200 --> 00:06:40.900
So the law of sines would say, sine of A,

00:06:40.900 --> 00:06:45.330
over a is equal to sine of B over b,

00:06:45.330 --> 00:06:49.650
which is equal to sine of C over c.

00:06:49.650 --> 00:06:51.640
And it looks like we
might be able to use that.

00:06:51.640 --> 00:06:53.660
And let me just write the
law cosines here just in case

00:06:53.660 --> 00:06:57.970
it's useful in the future. So
the law of cosine C squared,

00:06:57.970 --> 00:06:59.990
it's really the Pythagorean
theorem with a little adjustment

00:06:59.990 --> 00:07:01.570
for the fact that it's
not a right triangle.

00:07:01.570 --> 00:07:06.480
So C squared is equal to a
squared plus b squared minus

00:07:06.480 --> 00:07:11.480
two a b cosine cosine of C of capital C.

00:07:12.600 --> 00:07:14.410
So there's law of sines
and law of cosines.

00:07:14.410 --> 00:07:16.500
Let's see if we can
somehow use both of these

00:07:16.500 --> 00:07:20.600
to put these in terms of B,
which we have information about.

00:07:20.600 --> 00:07:23.510
Well, the first thing
is I could rewrite this,

00:07:23.510 --> 00:07:25.180
so this has sine of C over c,

00:07:25.180 --> 00:07:27.890
and this is a sine of A
over a ,so let me do that.

00:07:27.890 --> 00:07:29.880
So I have, let me do this.

00:07:29.880 --> 00:07:34.340
So I have the two a, I
have two a cosine of C,

00:07:34.340 --> 00:07:35.950
let me write that separately.

00:07:35.950 --> 00:07:40.950
So I have two a cosine
cosine of capital C,

00:07:41.840 --> 00:07:45.810
and then times sine of C over c ,

00:07:45.810 --> 00:07:50.300
times I'll do that in white, sine of C.

00:07:50.300 --> 00:07:51.800
That's a capital C,

00:07:51.800 --> 00:07:54.980
sine of capital C over lowercase c that's

00:07:54.980 --> 00:07:57.100
that term and that term right over there.

00:07:57.100 --> 00:08:00.780
And then to that I'm adding
,to that I'm adding imma do

00:08:00.780 --> 00:08:03.410
the same thing over
here. I have two times,

00:08:03.410 --> 00:08:07.090
I'm going to separate these guys out.

00:08:07.090 --> 00:08:08.550
Actually. Now I want to do the sines.

00:08:08.550 --> 00:08:09.930
So let me separate.

00:08:09.930 --> 00:08:13.730
I'm going to separate
this guy and this guy out.

00:08:13.730 --> 00:08:18.287
And so I get plus two c cosine of A

00:08:20.470 --> 00:08:23.807
times, sine of capital A over lowercase a

00:08:23.807 --> 00:08:28.807
times sine of capital
A over a lowercase a .

00:08:28.810 --> 00:08:30.490
Now what did this do for me?

00:08:30.490 --> 00:08:31.323
Well, look at the laws,

00:08:31.323 --> 00:08:33.160
look at the lot of sines right over there.

00:08:33.160 --> 00:08:35.990
I have sine of C over c
that's that over there.

00:08:35.990 --> 00:08:38.490
And then I have sine of A
over a that's that over there,

00:08:38.490 --> 00:08:40.160
capital A over lowercase a.

00:08:40.160 --> 00:08:42.370
They're both equal to sine of B over b.

00:08:42.370 --> 00:08:44.810
So we're making progress.
We, we ha we've relayed

00:08:44.810 --> 00:08:47.350
we started introducing B into
the equation and that's where

00:08:47.350 --> 00:08:48.183
the expression,

00:08:48.183 --> 00:08:50.840
and that's what we actually
have information about.

00:08:50.840 --> 00:08:55.040
So this could be rewritten
as sine of B over b.

00:08:55.040 --> 00:08:59.131
So this is the same thing
as sine of capital B over

00:08:59.131 --> 00:09:00.630
lowercase b.

00:09:00.630 --> 00:09:04.376
And this is the same thing
as sine of capital B over

00:09:04.376 --> 00:09:08.189
lowercase b. And they're
both being multiplied,

00:09:08.189 --> 00:09:11.120
or both of these terms
are multiplying are,

00:09:11.120 --> 00:09:12.900
are being multiplied by

00:09:12.900 --> 00:09:16.960
that two a cosine of capital C times that,

00:09:16.960 --> 00:09:19.990
and then plus two c,

00:09:19.990 --> 00:09:24.230
the lowercase c cosine
of capital A times that.

00:09:24.230 --> 00:09:27.410
so we can factor out the sine of B over b.

00:09:27.410 --> 00:09:30.050
Let's do that. Let's factor it out.

00:09:30.050 --> 00:09:35.050
So this is the same thing as
this is the same thing as two,

00:09:35.150 --> 00:09:39.060
a two a and I already have a
sense of what the next step is.

00:09:39.060 --> 00:09:44.060
I'm leave a little space here,
two a times cosine of C plus

00:09:44.642 --> 00:09:46.210
this, and these are being multiplied.

00:09:46.210 --> 00:09:50.370
I left some space there,
plus two c, two lowercase c

00:09:50.370 --> 00:09:55.370
times the cosine times the
cosine of A and all of this,

00:09:55.660 --> 00:10:00.660
all of this times, the
sine, the sine of B over b.

00:10:00.747 --> 00:10:02.940
And we know what we already
know that B is 60 degrees.

00:10:02.940 --> 00:10:06.760
So we can evaluate this
pretty, pretty easily,

00:10:06.760 --> 00:10:07.593
but let's just,

00:10:07.593 --> 00:10:10.460
let's just continue to see
if we can somehow somehow put

00:10:11.392 --> 00:10:13.360
this right over here in terms of B.

00:10:13.360 --> 00:10:15.685
Well, if you look over here,
we have two a cosine of C

00:10:15.685 --> 00:10:17.700
two c cosine of A,

00:10:17.700 --> 00:10:21.470
it looks it's starting to
look pretty darn close,

00:10:21.470 --> 00:10:22.590
pretty darn close.

00:10:22.590 --> 00:10:25.680
Each of these terms look
pretty darn close to this part,

00:10:25.680 --> 00:10:28.310
to this part of the law
of cosines over there.

00:10:28.310 --> 00:10:29.143
And actually let's,

00:10:29.143 --> 00:10:32.700
let's solve for that part of
the law of cosines to see what

00:10:32.700 --> 00:10:34.550
we could do. So if you add,

00:10:34.550 --> 00:10:37.540
if you add two a b cosine C to both sides,

00:10:37.540 --> 00:10:42.540
you get two a b cosine of
capital C plus c squared is

00:10:44.360 --> 00:10:47.730
equal to a squared plus b squared.

00:10:47.730 --> 00:10:50.060
Or if you subtract c
squared from both sides,

00:10:50.060 --> 00:10:54.330
you get two a b cosine of capital C

00:10:54.330 --> 00:10:59.330
is equal to a squared plus
b squared minus c squared.

00:10:59.800 --> 00:11:00.970
And this is interesting.

00:11:00.970 --> 00:11:03.900
I know we can switch around
the letters later on,

00:11:03.900 --> 00:11:06.220
but this looks pretty darn
close to this. So what if,

00:11:06.220 --> 00:11:08.360
and this looks pretty darn close
to this, except we're here.

00:11:08.360 --> 00:11:10.317
We're dealing with an a instead
of a C we've just switched

00:11:10.317 --> 00:11:13.700
the letters around and we
could rewrite this. Actually,

00:11:13.700 --> 00:11:15.410
let me rewrite it just for fun.

00:11:15.410 --> 00:11:20.410
I could rewrite this over here
as two two c b, not rewrite

00:11:21.910 --> 00:11:24.900
I can swap the letters
times the cosine of A.

00:11:26.500 --> 00:11:28.100
Here I'm swapping the A's and the C's

00:11:28.100 --> 00:11:33.070
is equal to c squared plus
b squared minus a squared.

00:11:33.070 --> 00:11:34.700
There's nothing unique about the side C,

00:11:34.700 --> 00:11:37.380
I can do this with all
of the sides. So here,

00:11:37.380 --> 00:11:40.640
when you have a big C here,
you have an a and b out front,

00:11:40.640 --> 00:11:41.473
and then you have the,

00:11:41.473 --> 00:11:44.010
a squared plus b squared
minus the small c squared.

00:11:44.010 --> 00:11:44.944
If you have a big A,

00:11:44.944 --> 00:11:46.760
then you're gonna have
the c b in the front,

00:11:46.760 --> 00:11:49.547
and then you're subtracting
the, a squared right over here.

00:11:49.547 --> 00:11:53.230
And this is useful because
this term right over here,

00:11:53.230 --> 00:11:54.160
this term right over here,

00:11:54.160 --> 00:11:55.960
it looks almost like this term over here.

00:11:55.960 --> 00:11:59.380
If we just have to, if we
could just multiply this by b.

00:11:59.380 --> 00:12:01.760
So let's do that. We can
multiply that by b in fact,

00:12:01.760 --> 00:12:05.580
let's multiply this whole
numerator, this whole term by b.

00:12:05.580 --> 00:12:08.170
If we multiply this whole
term by B what do we get?

00:12:08.170 --> 00:12:11.140
We get a b there, we get a b there.

00:12:11.140 --> 00:12:13.400
And of course you can't
just arbitrarily multiply an

00:12:13.400 --> 00:12:15.149
expression by b that'll change its value.

00:12:15.149 --> 00:12:17.970
So what we can do is
multiply the expression by b,

00:12:17.970 --> 00:12:20.598
which we just did. We
distributed the b across here,

00:12:20.598 --> 00:12:24.410
but then we'll also divide
by b where at which so I'll

00:12:24.410 --> 00:12:25.243
divide by b

00:12:25.243 --> 00:12:27.485
that's the equivalent of
multiplying denominator,

00:12:27.485 --> 00:12:30.350
the denominator there not b squared.

00:12:30.350 --> 00:12:32.250
That's the equivalent of
multiplying the denominator there

00:12:32.250 --> 00:12:34.200
by b that's the same
thing as dividing by b.

00:12:34.200 --> 00:12:36.020
We've multiplied by b , divided by b,

00:12:36.020 --> 00:12:37.478
or that's the same thing

00:12:37.478 --> 00:12:40.220
as just turning this into b squared.

00:12:40.220 --> 00:12:42.750
Now, what does this give us?

00:12:42.750 --> 00:12:45.800
Well, we have this term right over here.

00:12:45.800 --> 00:12:48.900
This term right over here is
now the exact same thing as

00:12:48.900 --> 00:12:50.060
that over there.

00:12:50.060 --> 00:12:55.060
So it is now a squared plus
b squared minus c squared.

00:12:55.190 --> 00:12:59.560
And then this term right over
here is now the exact same

00:12:59.560 --> 00:13:00.940
thing as this thing over here,

00:13:00.940 --> 00:13:03.024
which is the same thing as
that we're using the law of

00:13:03.024 --> 00:13:08.024
cosines. So this is plus
c squared plus b squared,

00:13:08.750 --> 00:13:11.170
minus a squared.

00:13:11.170 --> 00:13:12.470
And then all of that times,

00:13:12.470 --> 00:13:17.059
this sine of B sine of
capital B over b squared.

00:13:17.059 --> 00:13:19.150
Now, what does this
give us the, we have an,

00:13:19.150 --> 00:13:20.810
a squared and a negative a squared

00:13:20.810 --> 00:13:22.770
things are starting to simplify,

00:13:22.770 --> 00:13:25.380
a squared negative a squared.

00:13:25.380 --> 00:13:29.400
We have a negative c squared
and a positive c squared.

00:13:29.400 --> 00:13:30.380
So what are we left with?

00:13:30.380 --> 00:13:32.640
We're just left with two b squared.

00:13:32.640 --> 00:13:37.210
So our whole expression has
simplified to two b squared sine

00:13:37.210 --> 00:13:42.210
of B sine of capital B
over lowercase b squared.

00:13:43.200 --> 00:13:46.720
These cancel out. So our whole
expression simplifies to,

00:13:46.720 --> 00:13:49.830
two sine of B.

00:13:49.830 --> 00:13:52.540
And from, from the get
go, we knew what B was.

00:13:52.540 --> 00:13:53.870
We know it's 60 degrees.

00:13:53.870 --> 00:13:58.870
So this is equal to two
times the sine of 60 degrees.

00:13:59.060 --> 00:14:01.990
And if you don't have the
sine of 60 degrees memorize,

00:14:01.990 --> 00:14:05.390
you can always just break
out a 30, 60, 90 triangle.

00:14:05.390 --> 00:14:06.940
So let me draw.

00:14:06.940 --> 00:14:09.010
This is a right triangle, right over here.

00:14:09.010 --> 00:14:12.150
This is 60 degrees
hypotenuse has length one.

00:14:12.150 --> 00:14:14.160
We're dealing with the unit circle.

00:14:14.160 --> 00:14:16.090
The, the side is 30 degrees.

00:14:16.090 --> 00:14:16.923
The side opposite.

00:14:16.923 --> 00:14:19.760
The 30 degrees is one
half the side opposite.

00:14:19.760 --> 00:14:21.703
The 60 degrees is squared
of three times that so it's

00:14:21.703 --> 00:14:23.110
squared three over two.

00:14:23.110 --> 00:14:25.560
You can even use the Pythagorean
theorem to figure out

00:14:25.560 --> 00:14:26.470
once you know one of them,

00:14:26.470 --> 00:14:27.860
you could figure out the other one.

00:14:27.860 --> 00:14:31.560
So it's the sine of sine is
opposite over hypotenuse.

00:14:31.560 --> 00:14:33.186
So square to three over two over one,

00:14:33.186 --> 00:14:34.483
or it's just square to three over two.

00:14:34.483 --> 00:14:38.290
So this is equal to two, two times.

00:14:38.290 --> 00:14:39.260
There's a home stretch.

00:14:39.260 --> 00:14:43.690
It's very exciting square
root of three over two,

00:14:43.690 --> 00:14:44.720
these cancel out.

00:14:44.720 --> 00:14:49.720
So we are left with the
square root root of three.

00:14:49.840 --> 00:14:51.870
That's a, that's a pretty neat problem.

00:14:51.870 --> 00:14:52.880
And just in case, you're curious,

00:14:52.880 --> 00:14:55.480
this came from the 2010 IIT.

00:14:55.480 --> 00:15:00.480
IIT's are these hard to get
into engineering and science

00:15:00.910 --> 00:15:02.500
universities in India,

00:15:02.500 --> 00:15:05.050
and they give you this exam to
like hundreds of thousands of

00:15:05.050 --> 00:15:07.190
kids. And, you know, the
top, the top, I don't know,

00:15:07.190 --> 00:15:09.550
like 2000 actually get into
one of the IITs, but anyway,

00:15:09.550 --> 00:15:12.150
I just thought it was a
pretty, pretty neat problem.


Object image height and distance relationship

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.200
In the last video, I showed
that a bunch of triangles

00:00:03.200 --> 00:00:04.750
are similar to each
other to come up

00:00:04.750 --> 00:00:07.890
with the relationship
between the focal length,

00:00:07.890 --> 00:00:11.450
the distance of an object
from the convex lens,

00:00:11.450 --> 00:00:15.450
and the distance of the image
of an object from that convex

00:00:15.450 --> 00:00:15.950
lens.

00:00:15.950 --> 00:00:18.330
And I realized that there was
one extra low-hanging fruit

00:00:18.330 --> 00:00:20.560
based on all of the
geometry that I had done,

00:00:20.560 --> 00:00:23.110
another relationship that
might come in useful.

00:00:23.110 --> 00:00:26.370
And that's the relationship
between the size

00:00:26.370 --> 00:00:28.800
of the object-- or maybe
since it's an arrow here,

00:00:28.800 --> 00:00:31.670
we can call it the
height of the object,

00:00:31.670 --> 00:00:34.000
and the height of the image.

00:00:36.850 --> 00:00:39.370
And we really set up
everything already.

00:00:39.370 --> 00:00:44.640
We already figured out that
this triangle over here

00:00:44.640 --> 00:00:47.680
is similar to this
triangle down here.

00:00:47.680 --> 00:00:49.420
And we figured out
in the last video

00:00:49.420 --> 00:00:53.540
that this triangle over here
is similar to this triangle

00:00:53.540 --> 00:00:54.960
over here.

00:00:54.960 --> 00:00:56.430
And since these
two are similar, we

00:00:56.430 --> 00:00:59.920
could say that A is to
B-- we did this over here.

00:00:59.920 --> 00:01:00.710
I'll rewrite it.

00:01:00.710 --> 00:01:06.020
A is to B as-- and both
of those are the sides

00:01:06.020 --> 00:01:09.350
opposite the right angles of
these two similar triangles.

00:01:09.350 --> 00:01:12.660
So that's going to be the same
thing as the ratio of the sides

00:01:12.660 --> 00:01:18.650
opposite this yellow
angle right over here.

00:01:18.650 --> 00:01:21.230
So in this triangle over here,
since we started with A first,

00:01:21.230 --> 00:01:23.480
it's this height
right over here.

00:01:23.480 --> 00:01:25.230
Now what is this
height right over here?

00:01:25.230 --> 00:01:26.910
This is the height
of the object.

00:01:29.660 --> 00:01:31.590
So this is the
height of the object

00:01:31.590 --> 00:01:34.339
is to-- Now what is
this opposite side

00:01:34.339 --> 00:01:35.880
of this yellow angle
right over here?

00:01:35.880 --> 00:01:37.463
Well, this is the
height of the image.

00:01:39.980 --> 00:01:43.065
Or we know from the
last video the distance

00:01:43.065 --> 00:01:44.815
of the object to the
distance of the image

00:01:44.815 --> 00:01:46.870
is the same thing as A to B.

00:01:46.870 --> 00:01:49.750
So this is going to be
the same thing as this.

00:01:49.750 --> 00:01:52.010
So the ratio of the
distances is also

00:01:52.010 --> 00:01:54.430
the same thing as the
ratio of their heights.

00:01:54.430 --> 00:01:56.240
So let me write it this way.

00:01:56.240 --> 00:02:00.970
So the ratio of the distance
from the object to the lens,

00:02:00.970 --> 00:02:03.640
to the distance from
the image to the lens,

00:02:03.640 --> 00:02:06.590
is the same as the
ratio of the height

00:02:06.590 --> 00:02:09.770
of the object to the
height of an image,

00:02:09.770 --> 00:02:11.732
or to the image of that object.

00:02:11.732 --> 00:02:14.190
So I just wanted to do that
little low-hanging fruit there,

00:02:14.190 --> 00:02:19.020
since we set up all of
the mechanics already.

00:02:19.020 --> 00:02:21.520
Anyway, hopefully you
found that useful.