2010 IIT JEE Paper 1 Problem 50 Hyperbola Eccentricity

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Language: en

00:00:00.560 --> 00:00:04.550
The line 2x plus y equals 1
is tangent to the hyperbola

00:00:04.550 --> 00:00:07.670
x squared over a squared
minus y squared over b squared

00:00:07.670 --> 00:00:08.940
equals 1.

00:00:08.940 --> 00:00:12.570
If this line passes through
the point of intersection

00:00:12.570 --> 00:00:15.960
of the nearest directrix
and the x-axis,

00:00:15.960 --> 00:00:19.890
then the eccentricity of
the hyperbola is-- so let's

00:00:19.890 --> 00:00:22.140
just set this up so we know
what this is even asking.

00:00:22.140 --> 00:00:24.550
Let me draw the
hyperbola itself.

00:00:24.550 --> 00:00:27.390
So that is my x-axis.

00:00:27.390 --> 00:00:30.140
That is my y-axis.

00:00:30.140 --> 00:00:32.330
And this will be a
left/right opening hyperbola

00:00:32.330 --> 00:00:34.780
since the x squared
term here is positive.

00:00:34.780 --> 00:00:38.540
So it will open to
the right like this.

00:00:38.540 --> 00:00:40.000
It'll open to the
right like that.

00:00:40.000 --> 00:00:42.407
And it will open to
the left like-- I

00:00:42.407 --> 00:00:44.240
could draw it a little
bit neater than that.

00:00:44.240 --> 00:00:46.760
It'll open to the
left like this.

00:00:46.760 --> 00:00:49.450
So that's our hyperbola
right over there.

00:00:49.450 --> 00:00:51.880
And then this point of
intersection right over here,

00:00:51.880 --> 00:00:54.940
if you set y is equal to 0, x
would be equal to positive a.

00:00:54.940 --> 00:00:57.390
So this the point a comma 0.

00:00:57.390 --> 00:01:00.750
This right here is the
point negative a comma 0.

00:01:00.750 --> 00:01:04.532
x could be plus or minus
a when y is equal to 0.

00:01:04.532 --> 00:01:06.990
And just so we have some of a
terminology for the hyperbola

00:01:06.990 --> 00:01:08.480
out of the way,
the hyperbola will

00:01:08.480 --> 00:01:13.040
have two foci, two focus points.

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So let's say that this
is one focus point.

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Let me do this in another color.

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This is one focus
point right over here.

00:01:19.740 --> 00:01:23.740
It'll be on this major axis
of symmetry for the hyperbola.

00:01:23.740 --> 00:01:28.030
And let's call that point F.
Let's call that point F comma

00:01:28.030 --> 00:01:29.899
0, where F is the x-coordinate.

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And then we'll have
another one right

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over here at negative F comma 0.

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It'll also have two directrices.

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I don't know what the
plural of directrix is.

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But these are lines, and we'll
talk about the relationship

00:01:44.140 --> 00:01:47.340
between the hyperbola, the
foci, and the directrices

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in a second.

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But these would be lines.

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One of them would
sit right about here.

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Let me do this in another color.

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So you'd have a directrix
right over here.

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And you'd have another
directrix right over here.

00:02:01.300 --> 00:02:03.670
Let me draw it a little
straighter than that.

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For this type of left/right
opening hyperbola,

00:02:05.970 --> 00:02:10.639
the directrices would be
vertical right over there.

00:02:10.639 --> 00:02:14.950
And the relationship between
the directrix, the foci,

00:02:14.950 --> 00:02:18.800
and the eccentricity
is you take any point.

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You take any point
on the hyperbola,

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and the distance from that
point to its closest directrix--

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so for this point, it would
be this directrix right here.

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Unfortunately, this doesn't
look like a straight line,

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but it's supposed to
be a straight line.

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So this distance right over
here to that directrix--

00:02:35.950 --> 00:02:37.420
so this is the distance.

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Let me call it the
distance to the directrix.

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And let me call
this the distance--

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this is the distance to
the focus on this side,

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the distance to the focus.

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The eccentricity is the ratio of
this distance to that distance.

00:02:54.050 --> 00:02:57.770
So the ratio of the distance
to the focus over the distance

00:02:57.770 --> 00:03:02.170
to the directrix is equal
to the eccentricity.

00:03:02.170 --> 00:03:04.345
So at least now that we
have the setup of everything

00:03:04.345 --> 00:03:05.220
we need to deal with.

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Now let's actually try
to solve the problem.

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So they give us this line.

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The line is 2x plus
y is equal to 1.

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And they say it's
tangent to the hyperbola.

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And they say this line
passes through the point

00:03:18.400 --> 00:03:26.140
of intersection of the nearest
directrix and the x-axis.

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So it intersects the x-axis
where one of the directrices

00:03:31.090 --> 00:03:32.295
intersect the x-axis.

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And we know where this
intersects the x-axis.

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We can set y equal to 0.

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If y is equal to 0, we
get 2x is equal to 1,

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just the y disappears.

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You get x is equal to 1/2.

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So it intersects the x-axis
right over there at 1/2 comma

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0.

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And so we know from
this information

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right over here
that this directrix

00:03:55.470 --> 00:03:58.472
must be the line
x is equal to 1/2.

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And actually, we can draw the
rest of this line right here.

00:04:00.930 --> 00:04:02.950
Its y-intercept,
when x is equal to 0,

00:04:02.950 --> 00:04:05.700
y is going to be equal to 1.

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So that is the y-intercept.

00:04:08.290 --> 00:04:10.220
And so this line is
going to look something

00:04:10.220 --> 00:04:12.870
like-- let me do
it in a new color.

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The line will look
something like this,

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and it will be tangent
to the hyperbola.

00:04:22.100 --> 00:04:25.540
Now, our goal is to figure
out this eccentricity.

00:04:25.540 --> 00:04:28.590
And to do that, I'm just going
to lay out all of the formulas

00:04:28.590 --> 00:04:31.470
I know that we've proven-- and
I think we've proven all of them

00:04:31.470 --> 00:04:35.450
in previous Khan Academy
videos-- related to parabolas--

00:04:35.450 --> 00:04:38.690
I'm sorry, not parabolas--
hyperbolas, eccentricity,

00:04:38.690 --> 00:04:41.694
focal point, and
directrices, and then

00:04:41.694 --> 00:04:44.110
see if we can do a little bit
of fancy algebra to do this.

00:04:44.110 --> 00:04:46.568
And as you know, I don't like
to just regurgitate formulas.

00:04:46.568 --> 00:04:48.930
That's why we've taken
some pains to prove them.

00:04:48.930 --> 00:04:52.012
But if you're doing this for
especially the IIT exam, where

00:04:52.012 --> 00:04:53.470
you don't have a
lot of time, it is

00:04:53.470 --> 00:04:57.070
good to have them very
quickly at your disposal.

00:04:57.070 --> 00:04:59.580
So in the last video, we
came up with the relationship

00:04:59.580 --> 00:05:02.070
between a tangent
line and a hyperbola.

00:05:05.000 --> 00:05:07.750
We said if the tangent
line has the equation

00:05:07.750 --> 00:05:11.610
y is equal to mx
plus c-- and I use

00:05:11.610 --> 00:05:14.630
c here not to get confused
with the b in the hyperbola--

00:05:14.630 --> 00:05:17.570
we figured out a relationship
between the m, c,

00:05:17.570 --> 00:05:18.505
the a, and the b.

00:05:18.505 --> 00:05:24.190
And the relationship we showed
in the last video was c squared

00:05:24.190 --> 00:05:29.730
plus b squared is equal to
m squared a squared, where,

00:05:29.730 --> 00:05:32.130
once again, the m and
the c come from the line.

00:05:32.130 --> 00:05:34.250
It's the slope and the
y-intercept of the line.

00:05:34.250 --> 00:05:36.830
And the a and the b
come from the hyperbola.

00:05:36.830 --> 00:05:41.950
So this is one tool that
seems like it might be useful.

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The other tool--
and I proved this.

00:05:43.950 --> 00:05:46.890
I think if you do a search
on Khan Academy for hyperbola

00:05:46.890 --> 00:05:50.250
and proof, or foci,
hyperbola, and proof,

00:05:50.250 --> 00:05:55.760
you get this right here in
that the focal length-- this

00:05:55.760 --> 00:05:58.680
coordinate right over here,
the distance from the origin

00:05:58.680 --> 00:06:01.540
to the focus point
along the x-axis,

00:06:01.540 --> 00:06:03.760
or the x-coordinate of this
focal point over here--

00:06:03.760 --> 00:06:04.970
and you could actually
do it for either one

00:06:04.970 --> 00:06:06.344
because you're
squaring it, so it

00:06:06.344 --> 00:06:14.690
becomes positive-- it's equal
to a squared plus b squared.

00:06:14.690 --> 00:06:17.489
So this seems like it
might be useful in some way

00:06:17.489 --> 00:06:19.530
if we use all this
information to somehow come up

00:06:19.530 --> 00:06:20.990
with the eccentricity.

00:06:20.990 --> 00:06:23.440
And then the last
pieces of information

00:06:23.440 --> 00:06:26.950
that should be useful
for this problem are

00:06:26.950 --> 00:06:30.930
the ones that relate the
eccentricity of a hyperbola

00:06:30.930 --> 00:06:35.052
to its focal point, and
a, and its directrix.

00:06:35.052 --> 00:06:37.010
And so these are-- and
I won't prove them here.

00:06:37.010 --> 00:06:38.870
I think I might have proven
them in previous videos.

00:06:38.870 --> 00:06:40.703
If I haven't, that's
something for my to-do.

00:06:40.703 --> 00:06:42.549
But these are pretty well known.

00:06:42.549 --> 00:06:44.090
If you look it up
on Wikipedia, these

00:06:44.090 --> 00:06:47.190
are well-known expressions
for a hyperbola,

00:06:47.190 --> 00:06:58.310
is that the focal point is equal
to the eccentricity times a,

00:06:58.310 --> 00:07:00.570
and the eccentricity
is greater than 1.

00:07:00.570 --> 00:07:02.710
So this is clearly
greater than a.

00:07:02.710 --> 00:07:04.850
So you multiply something
larger than 1 times a,

00:07:04.850 --> 00:07:08.440
you get to the focal point
and also the directrix.

00:07:08.440 --> 00:07:10.760
And in this situation,
the directrix we already

00:07:10.760 --> 00:07:13.180
know is at 1/2, is
at x is equal to 1/2.

00:07:13.180 --> 00:07:15.180
But I'll just write
the formula here.

00:07:15.180 --> 00:07:17.810
The directrix distance,
so the distance

00:07:17.810 --> 00:07:22.780
from here to the directrix,
is equal to a divided

00:07:22.780 --> 00:07:24.380
by the eccentricity.

00:07:24.380 --> 00:07:27.240
And we already know, from the
first part of this problem,

00:07:27.240 --> 00:07:29.720
that this is at the
point x is equal to 1/2.

00:07:29.720 --> 00:07:32.900
So this is equal to 1/2.

00:07:32.900 --> 00:07:34.870
That's this distance right here.

00:07:34.870 --> 00:07:36.870
This is the line
x is equal to 1/2.

00:07:36.870 --> 00:07:40.815
And this is just a known
formula for hyperbolas,

00:07:40.815 --> 00:07:42.190
which I won't
prove in this video

00:07:42.190 --> 00:07:43.970
because then it will
just take forever.

00:07:43.970 --> 00:07:46.480
Now let's see if we can
use all of this information

00:07:46.480 --> 00:07:50.786
right over here to solve
for the eccentricity,

00:07:50.786 --> 00:07:52.410
to actually solve
for the eccentricity.

00:07:52.410 --> 00:07:54.760
And here, it should
hopefully just

00:07:54.760 --> 00:07:56.549
be a bunch of
algebraic manipulation.

00:07:56.549 --> 00:07:58.840
I just want to get rid of as
many variables as possible

00:07:58.840 --> 00:08:00.930
and then solve for E.

00:08:00.930 --> 00:08:04.000
So the first thing I can
do is I can substitute

00:08:04.000 --> 00:08:06.550
for F in this equation
right over here,

00:08:06.550 --> 00:08:12.680
and then I have an expression
in terms of E, a, and b.

00:08:12.680 --> 00:08:15.290
So if I substitute
for F here, F squared

00:08:15.290 --> 00:08:18.030
is going to be
eccentricity squared

00:08:18.030 --> 00:08:21.140
times a squared-- I'm just
squaring this over here--

00:08:21.140 --> 00:08:29.340
is going to be equal to
a squared plus b squared.

00:08:29.340 --> 00:08:31.510
And actually, let's
see if we can even

00:08:31.510 --> 00:08:33.580
substitute more using this
information over here.

00:08:33.580 --> 00:08:40.200
We know that a over the
eccentricity is equal to 1/2.

00:08:40.200 --> 00:08:42.679
Or we could say that if
we multiply both sides

00:08:42.679 --> 00:08:48.260
by the eccentricity, a is equal
to 1/2 times the eccentricity.

00:08:48.260 --> 00:08:50.760
So let's substitute that
into this over here.

00:08:50.760 --> 00:08:56.140
So then we get eccentricity
squared times a squared.

00:08:56.140 --> 00:08:59.210
Instead of writing a squared,
let's just square this thing

00:08:59.210 --> 00:09:00.670
because that's also a squared.

00:09:00.670 --> 00:09:03.910
So it's going to be times
the eccentricity squared

00:09:03.910 --> 00:09:07.060
over 4-- that's just a
squared; I'm just substituting

00:09:07.060 --> 00:09:10.190
it-- is equal to the
eccentricity squared

00:09:10.190 --> 00:09:13.960
over 4, once again, a
squared, plus b squared.

00:09:16.540 --> 00:09:21.940
And so this will simplify
to the eccentricity

00:09:21.940 --> 00:09:25.810
to the fourth power over 4
is equal to the eccentricity

00:09:25.810 --> 00:09:29.142
squared over 4 plus b squared.

00:09:29.142 --> 00:09:31.350
And I don't like having
these 4's in the denominator,

00:09:31.350 --> 00:09:34.140
so let's multiply
both sides by 4.

00:09:34.140 --> 00:09:37.090
We get the eccentricity
to the fourth power

00:09:37.090 --> 00:09:43.550
is equal to the eccentricity
squared plus 4b squared.

00:09:43.550 --> 00:09:46.020
So we have an expression
in terms of E and b.

00:09:46.020 --> 00:09:50.400
And so far, we used this
information, this information,

00:09:50.400 --> 00:09:53.270
and this information.

00:09:53.270 --> 00:09:56.640
Oh, actually, right, and this
information right over here.

00:09:56.640 --> 00:09:59.406
Now let's see if we can
incorporate this somehow.

00:09:59.406 --> 00:10:01.780
And this, that relates the
tangent line to the hyperbola,

00:10:01.780 --> 00:10:03.321
is useful because
it actually gave us

00:10:03.321 --> 00:10:04.720
the equation of
the tangent line.

00:10:04.720 --> 00:10:07.220
We actually know
that this equation

00:10:07.220 --> 00:10:09.690
right here, if we write
it in slope-intercept,

00:10:09.690 --> 00:10:12.270
the tangent line is y
is equal to-- subtract

00:10:12.270 --> 00:10:16.070
2x from both sides--
negative 2x plus 1.

00:10:16.070 --> 00:10:18.150
So m is negative 2.

00:10:18.150 --> 00:10:19.760
And normally, we
would call this b,

00:10:19.760 --> 00:10:21.551
but I don't want to
confuse it with this b.

00:10:21.551 --> 00:10:23.040
We call it c in this formula.

00:10:23.040 --> 00:10:25.450
Our y-intercept is 1.

00:10:25.450 --> 00:10:29.360
So this expression right
over here, let's rewrite it.

00:10:29.360 --> 00:10:32.430
So we have c
squared-- I'll do it

00:10:32.430 --> 00:10:34.630
in green-- c squared
is 1 squared.

00:10:34.630 --> 00:10:36.920
So we have 1 plus--
we don't know

00:10:36.920 --> 00:10:42.110
what b squared is-- so 1 plus b
squared is equal to m squared.

00:10:42.110 --> 00:10:43.890
We know that m is negative 2.

00:10:43.890 --> 00:10:48.480
Negative 2 squared is
4, 4 times a squared.

00:10:48.480 --> 00:10:49.990
Now, instead of
putting an a there,

00:10:49.990 --> 00:10:52.740
we already have an expression
in terms of E and b,

00:10:52.740 --> 00:10:54.760
so let's put this in
terms of our eccentricity.

00:10:54.760 --> 00:11:00.730
We know that a is the same thing
as 1/2 times the eccentricity,

00:11:00.730 --> 00:11:04.780
or so we can write this 4 times
the eccentricity-- actually,

00:11:04.780 --> 00:11:09.520
this is an epsilon, not an E.
4 times, we just square this.

00:11:09.520 --> 00:11:14.760
Eccentricity squared over
eccentricity squared over 4.

00:11:14.760 --> 00:11:16.830
I just squared this over here.

00:11:16.830 --> 00:11:18.840
And so these will
cancel out, and so we

00:11:18.840 --> 00:11:22.850
have another equation,
that 1 plus b squared

00:11:22.850 --> 00:11:26.330
is equal to the
eccentricity squared.

00:11:26.330 --> 00:11:28.121
Or we can subtract
1 from both sides.

00:11:28.121 --> 00:11:29.620
And I want to do
that so that we can

00:11:29.620 --> 00:11:31.610
substitute for b
squared up in this.

00:11:31.610 --> 00:11:34.469
We really only have two
formulas now or two equations,

00:11:34.469 --> 00:11:35.510
and we have two unknowns.

00:11:35.510 --> 00:11:38.051
So in theory, we should be able
to solve for the eccentricity

00:11:38.051 --> 00:11:38.750
now.

00:11:38.750 --> 00:11:40.150
So we have this.

00:11:40.150 --> 00:11:41.570
Let's just solve for b squared.

00:11:41.570 --> 00:11:44.760
So we get b squared is equal
to the eccentricity squared

00:11:44.760 --> 00:11:46.360
minus 1.

00:11:46.360 --> 00:11:49.330
And then we can substitute
this back in for b squared

00:11:49.330 --> 00:11:53.350
and see if we can actually
solve for the eccentricity.

00:11:53.350 --> 00:11:56.150
So then this
expression up here is--

00:11:56.150 --> 00:11:58.660
we have the eccentricity
to the fourth

00:11:58.660 --> 00:12:01.670
is equal to the
eccentricity squared

00:12:01.670 --> 00:12:06.460
plus 4 times b squared,
which is this, which

00:12:06.460 --> 00:12:11.800
is the eccentricity
squared minus 1.

00:12:11.800 --> 00:12:15.720
And now we have a fourth-degree
polynomial in eccentricity,

00:12:15.720 --> 00:12:17.620
but let's see if we
can still solve it.

00:12:17.620 --> 00:12:21.080
Hopefully, they're not going to
give us something too painful.

00:12:21.080 --> 00:12:23.370
So we have the
eccentricity to the fourth

00:12:23.370 --> 00:12:27.450
is equal to eccentricity
squared plus 4 times

00:12:27.450 --> 00:12:30.440
the eccentricity
squared minus 4.

00:12:30.440 --> 00:12:32.740
Now let's bring all of that
on to the left-hand side.

00:12:32.740 --> 00:12:35.070
So we're going to subtract
all of this from both sides.

00:12:35.070 --> 00:12:37.430
And we get-- all right,
this is a new color.

00:12:37.430 --> 00:12:39.970
The eccentricity to
the fourth power,

00:12:39.970 --> 00:12:42.506
these two over here--
actually, let me merge them.

00:12:42.506 --> 00:12:45.020
This is 1 times
eccentricity squared

00:12:45.020 --> 00:12:46.580
plus 4 times
eccentricity squared.

00:12:46.580 --> 00:12:50.030
So this is 5 epsilon
squared or 5 times

00:12:50.030 --> 00:12:51.410
the eccentricity squared.

00:12:51.410 --> 00:12:53.530
We're going to subtract
it from the other side.

00:12:53.530 --> 00:12:58.330
So this becomes minus 5 times
the eccentricity squared.

00:12:58.330 --> 00:13:00.730
And then we're going
to add 4 to both sides.

00:13:00.730 --> 00:13:05.110
So plus 4 is equal to 0.

00:13:05.110 --> 00:13:08.029
And now we just have to
solve for eccentricity.

00:13:08.029 --> 00:13:09.820
Now, at first, this
might seem complicated,

00:13:09.820 --> 00:13:14.290
but this is actually a
very factorable polynomial.

00:13:14.290 --> 00:13:17.320
Think of two numbers,
two factors of 4

00:13:17.320 --> 00:13:19.940
that add up to negative 5.

00:13:19.940 --> 00:13:23.089
And hopefully, negative 4
and negative 1 pop to mind.

00:13:23.089 --> 00:13:24.380
So we can actually factor this.

00:13:24.380 --> 00:13:26.770
No quadratic formula necessary.

00:13:26.770 --> 00:13:30.270
This is factorable
into epsilon squared

00:13:30.270 --> 00:13:37.040
minus 4 times epsilon
squared minus 1.

00:13:37.040 --> 00:13:38.860
Negative 4 times
negative 1 is positive 4.

00:13:38.860 --> 00:13:41.024
If you add the two,
you get negative 5.

00:13:41.024 --> 00:13:42.940
And if this confuses you
with the fourth power

00:13:42.940 --> 00:13:44.460
and the squared, you
could do a substitution.

00:13:44.460 --> 00:13:46.668
So you could say-- I don't
know-- some random letter.

00:13:46.668 --> 00:13:48.510
Call it some letter
we haven't used yet.

00:13:48.510 --> 00:13:52.770
Call it the letter-- we already
used F. Call it the letter g.

00:13:52.770 --> 00:13:55.680
You could say that g is
equal to epsilon squared,

00:13:55.680 --> 00:13:59.190
and then this would become
g squared minus 5g plus 4.

00:13:59.190 --> 00:14:02.980
And then this just becomes kind
of a traditional factorable

00:14:02.980 --> 00:14:04.030
quadratic.

00:14:04.030 --> 00:14:05.540
But this is equal to 0.

00:14:05.540 --> 00:14:09.980
And so there's going to be two
solutions over here, actually

00:14:09.980 --> 00:14:11.700
more than two solutions.

00:14:11.700 --> 00:14:15.390
So you have epsilon squared
minus 4 is equal to 0

00:14:15.390 --> 00:14:18.770
and epsilon squared
minus 1 is equal to 0.

00:14:18.770 --> 00:14:22.110
This one is epsilon squared
is equal to positive 4.

00:14:22.110 --> 00:14:24.529
This one is epsilon
squared is equal to 1.

00:14:24.529 --> 00:14:26.070
If we were to just
solve this and not

00:14:26.070 --> 00:14:28.550
think about what eccentricity
is trying to tell us,

00:14:28.550 --> 00:14:31.650
we would get plus or minus
2 here, eccentricity.

00:14:31.650 --> 00:14:35.400
And we would get eccentricity is
equal to plus or minus 1 here.

00:14:35.400 --> 00:14:37.840
And now the key realization
is the eccentricity

00:14:37.840 --> 00:14:41.140
for a hyperbola is
always greater than 1.

00:14:41.140 --> 00:14:43.670
So the only eccentricity
here that is greater than 1

00:14:43.670 --> 00:14:45.440
is positive 2.

00:14:45.440 --> 00:14:48.080
The other ones are either
1, or they're negative.

00:14:48.080 --> 00:14:50.920
So the only thing greater
than 1 is positive 2.

00:14:50.920 --> 00:14:57.800
So the eccentricity of our
hyperbola is positive 2.

00:14:57.800 --> 00:14:59.272
And we're done.


Tangent line hyperbola relationship (very optional)

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.580
After going through
many of these Indian

00:00:02.580 --> 00:00:05.230
Institutes of Technology
Joint Entrance Exam problems,

00:00:05.230 --> 00:00:07.730
I realize that there are a lot
of problems where they really

00:00:07.730 --> 00:00:09.264
just expect you
to know something.

00:00:09.264 --> 00:00:11.430
So that's what I'm going
to cover in this video, one

00:00:11.430 --> 00:00:13.513
of those things that they
just expect you to know.

00:00:13.513 --> 00:00:15.060
What we're going
to do is come up

00:00:15.060 --> 00:00:18.840
with the relationship
between a conic section--

00:00:18.840 --> 00:00:21.195
and a particular hyperbola
is what we'll focus on--

00:00:21.195 --> 00:00:22.287
and a tangent line.

00:00:22.287 --> 00:00:23.870
We've done this in
a previous problem.

00:00:23.870 --> 00:00:25.370
But that wasn't in
the general case.

00:00:25.370 --> 00:00:29.270
So let's say we have a left
right opening hyperbola.

00:00:29.270 --> 00:00:35.140
So it'll have the equation, x
squared over a squared minus

00:00:35.140 --> 00:00:40.730
y squared over b squared
is going to be equal to 1.

00:00:40.730 --> 00:00:42.920
And so if I were to
draw that hyperbola

00:00:42.920 --> 00:00:45.140
it would look
something like this.

00:00:45.140 --> 00:00:47.020
That's the x-axis.

00:00:47.020 --> 00:00:48.610
That's the y-axis.

00:00:48.610 --> 00:00:51.850
And then it opens to the right.

00:00:51.850 --> 00:00:53.760
I could draw a
better bottom half.

00:00:53.760 --> 00:00:55.150
It opens to the right.

00:00:55.150 --> 00:00:58.970
And it opens to the left.

00:00:58.970 --> 00:01:00.950
And in case you're
curious, this point right

00:01:00.950 --> 00:01:03.070
here, if you set
y is equal to 0,

00:01:03.070 --> 00:01:05.920
this point right
here is a comma 0.

00:01:05.920 --> 00:01:10.540
And this right here
is negative a comma 0.

00:01:10.540 --> 00:01:12.300
So what I want to
do is figure out

00:01:12.300 --> 00:01:16.740
a relationship between these
a's and b's and the equation

00:01:16.740 --> 00:01:17.970
of a tangent line.

00:01:17.970 --> 00:01:19.770
So let's say I have
a tangent line that

00:01:19.770 --> 00:01:21.100
looks something like this.

00:01:21.100 --> 00:01:24.530
Let's say it's tangent only at
that point right over there.

00:01:24.530 --> 00:01:27.840
And so it would look like-- let
me draw it a little bit better

00:01:27.840 --> 00:01:31.830
than that-- it would
look something like that.

00:01:31.830 --> 00:01:33.890
And let's say the equation
for this tangent line

00:01:33.890 --> 00:01:38.750
is y is equal to mx,
where m is the slope,

00:01:38.750 --> 00:01:42.110
plus-- instead of saying
b for the y-intercept.

00:01:42.110 --> 00:01:44.860
So normally, we would call
the y-intercept b for a line.

00:01:44.860 --> 00:01:48.910
We've already used the b here in
the equation for the hyperbola.

00:01:48.910 --> 00:01:50.390
So let me just call this c.

00:01:50.390 --> 00:01:52.750
So the c-- this is a
little unconventional.

00:01:52.750 --> 00:01:56.000
This is going to
be the y-intercept.

00:01:56.000 --> 00:01:58.960
So let's see if we can come
up with a relation between

00:01:58.960 --> 00:02:02.220
the m's, the c's,
the a's, and the b's.

00:02:02.220 --> 00:02:05.780
And we already used it in
one of the IIT problems.

00:02:05.780 --> 00:02:08.369
And I suspect that the
next one I'm going to do

00:02:08.369 --> 00:02:11.052
will also use this.

00:02:11.052 --> 00:02:13.135
If you all have seen a lot
of Khan Academy videos,

00:02:13.135 --> 00:02:14.510
you know that I always
like to prove things

00:02:14.510 --> 00:02:16.760
from first principles, because
in life, you can't just

00:02:16.760 --> 00:02:17.511
memorize formulas.

00:02:17.511 --> 00:02:19.010
You won't know where
they came from.

00:02:19.010 --> 00:02:20.300
You'll memorize them wrong.

00:02:20.300 --> 00:02:22.470
You won't understand
what they actually mean.

00:02:22.470 --> 00:02:25.820
But if you're going to
take the IIT JEE exam,

00:02:25.820 --> 00:02:29.650
I would recommend that
you-- I have an appreciation

00:02:29.650 --> 00:02:31.560
for how little time they
give these problems.

00:02:31.560 --> 00:02:33.680
And if you have to prove
from first principles,

00:02:33.680 --> 00:02:35.890
you won't be able to prove them.

00:02:35.890 --> 00:02:38.750
So let's just come
up with the relation.

00:02:38.750 --> 00:02:41.110
Let me stop talking
without drawing.

00:02:41.110 --> 00:02:43.120
So let's just see
where they intersect.

00:02:43.120 --> 00:02:46.430
And the whole insight
here is that they're only

00:02:46.430 --> 00:02:48.235
going to intersect in one point.

00:02:48.235 --> 00:02:52.070
So what I'm going to do
here is solve for y squared.

00:02:52.070 --> 00:02:55.810
So over here, we can multiply
both sides of this equation

00:02:55.810 --> 00:02:59.640
by-- let's multiply both
sides by negative b squared.

00:02:59.640 --> 00:03:04.070
So you get negative b
squared over a squared

00:03:04.070 --> 00:03:08.650
x squared plus y squared--
I multiply by negative

00:03:08.650 --> 00:03:15.520
b squared-- is equal
to negative b squared.

00:03:15.520 --> 00:03:18.720
And now, let's add this thing
to both sides of this equation.

00:03:18.720 --> 00:03:22.600
And we get y squared
is equal to b squared

00:03:22.600 --> 00:03:27.430
over a squared x
squared minus b squared.

00:03:27.430 --> 00:03:29.990
So I just rewrote the
equation for the hyperbola.

00:03:29.990 --> 00:03:32.515
And let's also write this
in terms of y squared.

00:03:32.515 --> 00:03:34.390
And then we can set them
equal to each other.

00:03:34.390 --> 00:03:39.880
So over here in this
greenish yellow-like color,

00:03:39.880 --> 00:03:42.320
if we square both
sides, we get y squared

00:03:42.320 --> 00:03:47.730
is equal to m squared x squared
plus 2 times the product

00:03:47.730 --> 00:03:48.970
of those both terms.

00:03:48.970 --> 00:03:56.424
So plus 2 mcx plus c squared.

00:03:56.424 --> 00:03:57.840
So in order for
them to intersect,

00:03:57.840 --> 00:04:01.110
they both have to be at the
same place at some x and y.

00:04:01.110 --> 00:04:04.267
So we can set this y squared
being equal to that y squared,

00:04:04.267 --> 00:04:05.600
and then try to solve for the x.

00:04:05.600 --> 00:04:07.558
Obviously, we won't be
able to solve for the x,

00:04:07.558 --> 00:04:09.280
because there's
so many variables.

00:04:09.280 --> 00:04:12.780
We can find a relationship
between this a, this b, this m,

00:04:12.780 --> 00:04:15.620
and this c, so there's only
one point of intersection,

00:04:15.620 --> 00:04:17.230
which by definition,
it would have

00:04:17.230 --> 00:04:19.100
to be at the tangent point.

00:04:19.100 --> 00:04:20.089
So let's do that.

00:04:20.089 --> 00:04:28.410
So we have m squared x squared
plus 2 mcx plus c squared

00:04:28.410 --> 00:04:34.810
is equal to b squared
over a squared x squared

00:04:34.810 --> 00:04:36.619
minus b squared.

00:04:36.619 --> 00:04:38.160
And I've done a very
similar exercise

00:04:38.160 --> 00:04:41.260
to this in a previous
IIT JEE video.

00:04:41.260 --> 00:04:43.990
But here, I just want to focus
just on the most general case

00:04:43.990 --> 00:04:47.760
so that we have something that
we can add to our tool kit.

00:04:47.760 --> 00:04:51.130
So let's write this in terms
of a quadratic equation in x.

00:04:51.130 --> 00:04:57.110
So if we subtract this from
both sides, we get m squared.

00:04:57.110 --> 00:04:59.400
We have m squared
minus b squared

00:04:59.400 --> 00:05:03.700
over a squared x squared.

00:05:07.070 --> 00:05:10.520
That's that term and that
term, right over there.

00:05:10.520 --> 00:05:13.840
And then my only just
first degree x term

00:05:13.840 --> 00:05:14.790
is right over here.

00:05:14.790 --> 00:05:23.360
So plus 2mcx-- I'll
write it in-- plus 2mcx.

00:05:23.360 --> 00:05:28.249
And then finally, plus,
we have a c squared,

00:05:28.249 --> 00:05:29.790
and then we're
actually going to have

00:05:29.790 --> 00:05:32.670
a plus another b
squared over there.

00:05:32.670 --> 00:05:43.120
So this is going to be plus
c squared plus b squared.

00:05:43.120 --> 00:05:47.175
Now, in order for this equation
to only have one solution--

00:05:47.175 --> 00:05:48.050
let me write it down.

00:05:48.050 --> 00:05:52.110
This is going to be equal to 0.

00:05:52.110 --> 00:05:55.110
In order for this thing
to only have one solution,

00:05:55.110 --> 00:05:57.190
the discriminant of this
quadratic equation--

00:05:57.190 --> 00:06:00.450
remember, when you do the
quadratic equation-- and these

00:06:00.450 --> 00:06:02.280
are completely
different, so negative b

00:06:02.280 --> 00:06:05.930
plus or minus the quadratic
formula b squared minus

00:06:05.930 --> 00:06:10.240
4ac over 2a-- you're
only going to have one

00:06:10.240 --> 00:06:14.340
solution if this
thing over here,

00:06:14.340 --> 00:06:18.290
if the discriminant over
there is equal to 0.

00:06:18.290 --> 00:06:20.110
If b squared minus
4ac is equal to 0,

00:06:20.110 --> 00:06:23.370
then you only have the
solution negative b over 2a.

00:06:23.370 --> 00:06:25.830
So in this situation,
for the tangent line,

00:06:25.830 --> 00:06:27.810
you can only have
one solution, one x

00:06:27.810 --> 00:06:30.660
that satisfies this equation.

00:06:30.660 --> 00:06:32.660
So b squared minus
4ac is going to have

00:06:32.660 --> 00:06:34.490
to equal 0 in the
quadratic formula.

00:06:34.490 --> 00:06:36.690
These are different
b's and a's and c's

00:06:36.690 --> 00:06:38.320
than the ones we're using here.

00:06:38.320 --> 00:06:42.890
Over here, our b is
that right over there,

00:06:42.890 --> 00:06:44.820
the coefficient on the x term.

00:06:44.820 --> 00:06:49.690
So that squared is
4m squared c squared.

00:06:49.690 --> 00:06:55.480
And we're going to subtract
from that minus 4 times a.

00:06:55.480 --> 00:06:58.260
a is all of this
business right over here.

00:06:58.260 --> 00:07:00.990
And just to simplify things,
let me make this plus,

00:07:00.990 --> 00:07:03.300
and then multiply this
times a negative sign.

00:07:03.300 --> 00:07:04.680
And it'll reverse the sign.

00:07:04.680 --> 00:07:09.164
So I'll put minus 4-- I'll
put a plus here-- times,

00:07:09.164 --> 00:07:11.830
instead of writing, since I have
to multiply this by negative 1,

00:07:11.830 --> 00:07:16.600
it'll now be b squared over
a squared minus m squared.

00:07:16.600 --> 00:07:18.942
And then the c is this
term right over here.

00:07:18.942 --> 00:07:21.030
This right here is c, 4ac.

00:07:21.030 --> 00:07:24.400
c squared plus b squared.

00:07:24.400 --> 00:07:28.810
And this thing is going to
equal 0 if this line is tangent,

00:07:28.810 --> 00:07:31.000
if we only have one solution.

00:07:31.000 --> 00:07:33.610
So the first thing that
we can do to simplify this

00:07:33.610 --> 00:07:39.150
is, we can divide both
sides of this equation by 4.

00:07:39.150 --> 00:07:40.690
And if we do that,
this becomes--

00:07:40.690 --> 00:07:44.840
I wanted to do that in
black-- this becomes a 1.

00:07:44.840 --> 00:07:45.977
So we can ignore that.

00:07:45.977 --> 00:07:47.060
And then this becomes a 1.

00:07:47.060 --> 00:07:50.290
So that simplified our
equation a good bit.

00:07:50.290 --> 00:07:52.794
And now, let's multiply the
second part right over here.

00:07:52.794 --> 00:07:54.960
So we have b squared over
a squared times c squared.

00:07:54.960 --> 00:07:59.780
That is b squared c
squared over a squared.

00:07:59.780 --> 00:08:01.550
And then b squared
a squared times

00:08:01.550 --> 00:08:06.440
b squared-- so plus b to
the fourth over a squared.

00:08:06.440 --> 00:08:09.190
Then you have negative m
squared times c squared.

00:08:09.190 --> 00:08:11.620
So let me do this
in another color.

00:08:11.620 --> 00:08:14.090
So then you have
for the m squared,

00:08:14.090 --> 00:08:17.230
negative m squared c squared.

00:08:17.230 --> 00:08:19.495
And then you have negative
m squared b squared.

00:08:24.500 --> 00:08:27.800
And of course, this is all
going to be equal to 0.

00:08:27.800 --> 00:08:30.910
And we have this m squared c
squared out front over here.

00:08:34.320 --> 00:08:38.480
And lucky for us, this
cancels with this.

00:08:38.480 --> 00:08:40.340
And what do we have left?

00:08:40.340 --> 00:08:44.430
Now, every term here is
divisible by b squared.

00:08:44.430 --> 00:08:46.450
So let's divide every
term by b squared.

00:08:46.450 --> 00:08:47.950
So this will just become a 1.

00:08:47.950 --> 00:08:50.230
This will become b
to the second power.

00:08:50.230 --> 00:08:53.660
And then this will
just become a 1.

00:08:53.660 --> 00:08:56.540
And then let's multiply
everything by a squared,

00:08:56.540 --> 00:08:58.280
just so we get rid
of the fractions.

00:08:58.280 --> 00:09:00.230
So when you multiply
everything by a squared,

00:09:00.230 --> 00:09:03.670
this term right over
here becomes a c squared.

00:09:03.670 --> 00:09:08.070
This term right over
here is just a b squared.

00:09:08.070 --> 00:09:10.562
And then all we have
left, and then we

00:09:10.562 --> 00:09:11.770
have this negative m squared.

00:09:11.770 --> 00:09:13.830
Remember, we're
multiplying by a squared.

00:09:13.830 --> 00:09:19.880
So minus a squared m
squared is equal to 0.

00:09:19.880 --> 00:09:22.260
Or we could add this to
both sides of the equation.

00:09:22.260 --> 00:09:32.470
We get c squared
plus b squared is

00:09:32.470 --> 00:09:34.880
equal to a squared m squared.

00:09:34.880 --> 00:09:36.730
And what's really
neat about this is,

00:09:36.730 --> 00:09:38.710
we now have a very
simple relationship.

00:09:42.590 --> 00:09:45.600
If we know the equation of
the line right over here,

00:09:45.600 --> 00:09:47.830
we would then, if we
know what m and c are,

00:09:47.830 --> 00:09:50.190
we then have an interesting
relationship for a and b.

00:09:50.190 --> 00:09:51.690
If we know what a
and b are, we have

00:09:51.690 --> 00:09:54.435
an interesting relationship
for the equation of the line.

00:09:54.435 --> 00:09:56.310
And maybe if we have a
few other constraints,

00:09:56.310 --> 00:09:57.920
we could actually
solve for them.

00:09:57.920 --> 00:09:59.420
But we'll actually
take this and use

00:09:59.420 --> 00:10:03.740
this in the next IIT
problem we're going to do.


IIT JEE Lagrange's Formula

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.580
If a and b are
vectors in space given

00:00:03.580 --> 00:00:07.230
by a is i minus 2j over
the square root of 5

00:00:07.230 --> 00:00:12.320
and b is 2i plus j plus 3k
over the square root of 14,

00:00:12.320 --> 00:00:16.460
then the value of-- so we have
2a plus b dotted with-- that's

00:00:16.460 --> 00:00:21.960
a dot product there-- a
cross b cross a minus 2b is.

00:00:21.960 --> 00:00:24.260
And if you-- and I actually
tried this out a little bit

00:00:24.260 --> 00:00:25.100
before this video.

00:00:25.100 --> 00:00:27.140
If you actually just
tried to first take

00:00:27.140 --> 00:00:30.080
the cross product of a
and b, which you can do,

00:00:30.080 --> 00:00:32.259
and then try to take the
cross product of that

00:00:32.259 --> 00:00:36.360
with a minus 2b, it's going
to be unbelievably hairy,

00:00:36.360 --> 00:00:38.632
unbelievably
computationally intensive.

00:00:38.632 --> 00:00:40.840
And you're probably going
to make a careless mistake,

00:00:40.840 --> 00:00:44.350
especially when you take the
cross product with a minus 2b.

00:00:44.350 --> 00:00:47.039
It's a really, really,
really, really hairy problem.

00:00:47.039 --> 00:00:49.080
And in general, taking
cross products tend to be,

00:00:49.080 --> 00:00:54.020
but even more so when the
actual vectors look like this.

00:00:54.020 --> 00:00:56.540
So the trick of this
problem is to try

00:00:56.540 --> 00:01:00.420
to do it without actually
having to take cross products.

00:01:00.420 --> 00:01:02.660
And to do that, we're
going to use a little trick

00:01:02.660 --> 00:01:04.800
called the triple
product expansion

00:01:04.800 --> 00:01:06.130
or Lagrange's formula.

00:01:06.130 --> 00:01:08.880
And I actually just uploaded
a video proving the triple

00:01:08.880 --> 00:01:11.220
product expansion for those
of y'all who don't like--

00:01:11.220 --> 00:01:12.630
and I'm one of those.

00:01:12.630 --> 00:01:15.591
I'm one of the y'all that
doesn't like just random

00:01:15.591 --> 00:01:16.090
formulas.

00:01:16.090 --> 00:01:17.600
I like to see
where it came from.

00:01:17.600 --> 00:01:19.920
But the triple product
expansion, it really

00:01:19.920 --> 00:01:23.120
allows us to rewrite a--
let me write it over here.

00:01:23.120 --> 00:01:28.440
It allows us to write
a cross b cross c.

00:01:28.440 --> 00:01:29.650
It allows us to write that.

00:01:29.650 --> 00:01:32.160
And I'm just speaking in general
terms of a, b, and c here,

00:01:32.160 --> 00:01:34.250
not these particular
a's and b's.

00:01:34.250 --> 00:01:37.694
It allows you to rewrite
that as the first vector.

00:01:37.694 --> 00:01:40.110
And when I talk about the first
vector and the first cross

00:01:40.110 --> 00:01:41.450
product, I'm talking
about the cross product

00:01:41.450 --> 00:01:42.510
you would have to take first.

00:01:42.510 --> 00:01:43.923
In this case, it would
be this one, where

00:01:43.923 --> 00:01:45.380
it's the cross
product on the right,

00:01:45.380 --> 00:01:46.280
I guess I should call it.

00:01:46.280 --> 00:01:47.450
Because there's
other ways that you

00:01:47.450 --> 00:01:50.010
could take the cross product
first where it's on the left.

00:01:50.010 --> 00:01:52.210
So you take this,
the first vector

00:01:52.210 --> 00:01:58.595
here, so it'll be b
scaled by a dot c--

00:01:58.595 --> 00:02:00.920
let me scroll to the
left a little bit--

00:02:00.920 --> 00:02:04.700
minus the second vector
in that second cross

00:02:04.700 --> 00:02:10.560
product minus c times a dot b.

00:02:10.560 --> 00:02:14.160
So when you have this kind of
triple product right over here,

00:02:14.160 --> 00:02:16.350
you're taking the cross
product of three vectors,

00:02:16.350 --> 00:02:17.387
you can simplify it.

00:02:17.387 --> 00:02:19.470
And it is a simplification
because the dot product

00:02:19.470 --> 00:02:23.669
is a much simpler operation to
compute than the cross product.

00:02:23.669 --> 00:02:25.460
So let's see how we
can apply it over here.

00:02:25.460 --> 00:02:28.410
It looks like we have a triple
product-looking thing here.

00:02:28.410 --> 00:02:32.290
We have the cross product of two
vectors crossed with a third.

00:02:32.290 --> 00:02:35.260
So it's a slightly different
thing than this over here.

00:02:35.260 --> 00:02:37.440
But what we can do is
we can swap it around

00:02:37.440 --> 00:02:38.590
so it's the same order.

00:02:38.590 --> 00:02:44.190
So this over here in green,
this is the exact same thing.

00:02:44.190 --> 00:02:46.934
If I just swap the order of
these two cross products,

00:02:46.934 --> 00:02:47.850
it'll be the negative.

00:02:47.850 --> 00:02:53.010
So this will be negative
a-- that's ugly parentheses.

00:02:53.010 --> 00:03:03.050
a minus 2b, so I'm putting
it first, cross a cross b.

00:03:03.050 --> 00:03:05.056
So all I did is I
rewrote this green part.

00:03:05.056 --> 00:03:06.930
When you swap the order
of the cross product,

00:03:06.930 --> 00:03:10.035
you put a negative out front.

00:03:10.035 --> 00:03:12.160
And actually, let me just
distribute this negative.

00:03:12.160 --> 00:03:13.321
So this is the same thing.

00:03:13.321 --> 00:03:14.820
If I multiply a
negative times this,

00:03:14.820 --> 00:03:16.520
it just swaps these two terms.

00:03:16.520 --> 00:03:24.627
So it's 2b minus
a cross a cross b.

00:03:24.627 --> 00:03:26.460
Now, the reason why I
like this form is it's

00:03:26.460 --> 00:03:28.670
exactly the form
that I have up here.

00:03:28.670 --> 00:03:30.520
It's exactly this form.

00:03:30.520 --> 00:03:34.010
And so I can use Lagrange's
formula or the triple product

00:03:34.010 --> 00:03:36.320
expansion.

00:03:36.320 --> 00:03:38.787
And so I'm just going to focus
on the green part right now.

00:03:38.787 --> 00:03:40.870
We can worry about this
thing out front that we're

00:03:40.870 --> 00:03:42.578
going to take the dot
product with later.

00:03:42.578 --> 00:03:45.870
Let's see if we can
simplify this green part.

00:03:45.870 --> 00:03:49.080
So this will be equal to
just using the triple product

00:03:49.080 --> 00:03:50.210
expansion.

00:03:50.210 --> 00:03:52.590
We're looking at our
second cross product.

00:03:52.590 --> 00:03:54.580
This is our second
cross product over here.

00:03:54.580 --> 00:03:56.740
So it's analogous to
that right over here.

00:03:56.740 --> 00:03:59.340
You take the first vector in
your second cross product.

00:03:59.340 --> 00:04:01.080
So that's the vector a.

00:04:01.080 --> 00:04:05.500
And I multiply that
times the dot product

00:04:05.500 --> 00:04:07.420
of this vector and that vector.

00:04:07.420 --> 00:04:17.325
So it's going to be 2b
minus a dotted with b,

00:04:17.325 --> 00:04:19.589
dotted with vector b.

00:04:19.589 --> 00:04:22.730
And then from that,
I'm just going

00:04:22.730 --> 00:04:24.385
to subtract the second vector.

00:04:27.290 --> 00:04:29.320
Let me do that in another color.

00:04:29.320 --> 00:04:32.356
I'm going to subtract b.

00:04:32.356 --> 00:04:34.800
So this is the same b over here.

00:04:34.800 --> 00:04:37.220
Maybe I should color code it.

00:04:37.220 --> 00:04:39.070
This a is that a there.

00:04:39.070 --> 00:04:42.300
And then this b is this b here.

00:04:42.300 --> 00:04:46.160
So b times the dot
product of-- so it's

00:04:46.160 --> 00:04:48.020
going to be other
than magenta again--

00:04:48.020 --> 00:04:55.610
the dot product of this guy,
2b minus a dotted with a.

00:04:55.610 --> 00:04:58.850
So the dot product of
the other two vectors.

00:04:58.850 --> 00:05:01.080
Now let's see if we
can simplify this.

00:05:01.080 --> 00:05:04.360
So what we have in
parentheses here,

00:05:04.360 --> 00:05:08.965
so this over here in
green, the dot product,

00:05:08.965 --> 00:05:11.950
that's just the distributive
property applied.

00:05:11.950 --> 00:05:16.790
So this is going to be
2b dotted with b again.

00:05:16.790 --> 00:05:19.160
Or you might already know
that if I take a vector

00:05:19.160 --> 00:05:21.110
and I take the dot
product with itself,

00:05:21.110 --> 00:05:24.940
that's the same thing
as its magnitude.

00:05:24.940 --> 00:05:27.620
That's the same thing
as the magnitude

00:05:27.620 --> 00:05:29.710
of the vector squared.

00:05:29.710 --> 00:05:31.890
So let's just write that down.

00:05:31.890 --> 00:05:34.690
So 2b dot b is going to
be-- actually, let me just

00:05:34.690 --> 00:05:39.210
write it as 2b dot b.

00:05:39.210 --> 00:05:43.420
And then we have minus a dot b.

00:05:45.889 --> 00:05:46.805
So I just distributed.

00:05:46.805 --> 00:05:48.690
I didn't actually
use that notation.

00:05:48.690 --> 00:05:51.110
And all of that's going
to be times the vector a.

00:05:54.670 --> 00:05:58.280
And then we have
minus-- let me pick

00:05:58.280 --> 00:06:00.190
another color-- minus
all of this stuff.

00:06:00.190 --> 00:06:03.250
So we have 2b dot a.

00:06:03.250 --> 00:06:06.040
So let me write it this way.

00:06:06.040 --> 00:06:08.790
I'm going to just leave
the minus b out front.

00:06:08.790 --> 00:06:14.280
So let me write a minus, and
then I'll put the b over here.

00:06:14.280 --> 00:06:19.384
And so you have 2b
dot a-- that color.

00:06:19.384 --> 00:06:20.800
b dot a is the
same thing as a dot

00:06:20.800 --> 00:06:26.720
b, so I'll just write 2a dot
b, and then minus a dot a.

00:06:30.270 --> 00:06:32.950
So remember, we're just working
on this green part right here.

00:06:32.950 --> 00:06:34.845
But since we already
have all of these dot

00:06:34.845 --> 00:06:37.540
products of a with
themselves and a dot b

00:06:37.540 --> 00:06:41.130
and b with themselves,
maybe let's just

00:06:41.130 --> 00:06:44.590
see what happens if we
actually try to evaluate these,

00:06:44.590 --> 00:06:46.690
and maybe this will simplify
some of our algebra.

00:06:46.690 --> 00:06:47.360
But you wouldn't have to.

00:06:47.360 --> 00:06:49.485
You could just keep going
with the a's and the b's.

00:06:49.485 --> 00:06:51.910
But maybe these evaluate to
relatively straightforward

00:06:51.910 --> 00:06:54.660
numbers since we actually
know what our a's and b's are.

00:06:54.660 --> 00:06:57.380
So let's try that out.

00:06:57.380 --> 00:07:01.790
So what is a dot a
going to be equal to?

00:07:01.790 --> 00:07:04.180
Well, you're just
really squaring

00:07:04.180 --> 00:07:06.370
each of the components
of a and adding them up.

00:07:06.370 --> 00:07:08.840
So the first component,
the i component,

00:07:08.840 --> 00:07:11.940
is 1 over the square root of 5.

00:07:11.940 --> 00:07:16.360
So 1 over the square root of
5 squared is going to be 1/5.

00:07:16.360 --> 00:07:20.290
And then negative 2 over
the square root of 5 squared

00:07:20.290 --> 00:07:23.750
is going to be plus 4/5.

00:07:23.750 --> 00:07:25.820
So our intuition was correct.

00:07:25.820 --> 00:07:27.610
When you take the dot
product of a and a,

00:07:27.610 --> 00:07:29.060
it's a very simple number.

00:07:29.060 --> 00:07:30.670
It is 1.

00:07:30.670 --> 00:07:32.700
Let's do the same thing with b.

00:07:32.700 --> 00:07:37.430
If I take b dot
b, what do I get?

00:07:37.430 --> 00:07:42.870
So 2 over the square root
of 14 squared is 4/14.

00:07:42.870 --> 00:07:48.110
And then 1 over the square root
of 14 squared is plus 1/14.

00:07:48.110 --> 00:07:50.990
And then 3 over the
square root of 14 squared

00:07:50.990 --> 00:07:55.280
is 9/14, plus 9/14.

00:07:55.280 --> 00:07:58.130
And this, once
again, adds up to 1.

00:07:58.130 --> 00:07:59.710
This, once again, adds up to 1.

00:07:59.710 --> 00:08:02.400
So it looks like things
are simplifying a good bit.

00:08:02.400 --> 00:08:04.190
So let's just try a dot b.

00:08:07.030 --> 00:08:10.160
So that's going to be 1 over
the square root of 5 times 2

00:08:10.160 --> 00:08:13.270
over the square root of 14.

00:08:13.270 --> 00:08:18.190
So it's going to be 2 over
the square root of 70.

00:08:18.190 --> 00:08:20.070
I just multiplied the x terms.

00:08:20.070 --> 00:08:22.280
And then we're going
to have negative 2

00:08:22.280 --> 00:08:25.730
over the square root of 5 times
1 over the square root of 14.

00:08:25.730 --> 00:08:29.910
So that's negative 2 over
the square root of 70.

00:08:29.910 --> 00:08:31.890
And then plus 0.

00:08:31.890 --> 00:08:33.539
This has no k
component, so it's going

00:08:33.539 --> 00:08:37.320
to be 0 times this, whatever
the k component is here,

00:08:37.320 --> 00:08:39.320
but that's still going to be 0.

00:08:39.320 --> 00:08:42.909
So this just evaluates all to 0.

00:08:42.909 --> 00:08:44.810
So that simplifies
things a good bit.

00:08:44.810 --> 00:08:50.540
This green expression right
here simplifies to-- this

00:08:50.540 --> 00:08:54.920
is 2 times 1, which is
just 2, minus 0, right?

00:08:54.920 --> 00:08:58.680
a dot b we just
figured out is 0.

00:08:58.680 --> 00:09:05.880
So this is just 2
times a minus-- now

00:09:05.880 --> 00:09:07.972
what is this going
to be equal to?

00:09:07.972 --> 00:09:11.490
a dot b we just figured
out. a dot b is 0.

00:09:11.490 --> 00:09:13.950
And then a dot a we
figured out is 1.

00:09:13.950 --> 00:09:18.240
So it's minus negative, so
this becomes a positive b, so

00:09:18.240 --> 00:09:20.550
plus b.

00:09:20.550 --> 00:09:26.950
So all of this mess over
here simplified to 2a

00:09:26.950 --> 00:09:30.490
plus b, which is the
exact same thing we

00:09:30.490 --> 00:09:32.140
have over here-- 2a plus b.

00:09:32.140 --> 00:09:34.162
So let's actually-- that
we're dotting it with.

00:09:34.162 --> 00:09:35.370
So let me write it over here.

00:09:35.370 --> 00:09:37.900
So this whole thing
is this over here.

00:09:37.900 --> 00:09:42.130
And then we have to take the dot
product with 2a plus b again.

00:09:42.130 --> 00:09:48.410
So we're going to take the dot
product with 2a plus b again.

00:09:48.410 --> 00:09:51.520
It doesn't matter. a dot b
is the same thing as b dot a.

00:09:51.520 --> 00:09:53.980
And they're the same number,
so it matters even less.

00:09:53.980 --> 00:09:55.580
And here, the
distributive property

00:09:55.580 --> 00:09:59.142
applies to the
vector dot product.

00:09:59.142 --> 00:10:00.350
Let me make sure we remember.

00:10:00.350 --> 00:10:03.555
This is a dot product
that we're taking.

00:10:03.555 --> 00:10:06.470
This is a dot-- let me
put it in a darker color.

00:10:06.470 --> 00:10:08.590
This is a dot product.

00:10:08.590 --> 00:10:11.150
So this whole thing
was 2a plus b.

00:10:11.150 --> 00:10:14.000
I'm dotting it with
another 2a plus b.

00:10:14.000 --> 00:10:16.650
And now this, you just use the
distributive property here.

00:10:16.650 --> 00:10:18.705
So you have 2a dot 2a.

00:10:18.705 --> 00:10:23.380
That's going to be
4 times a dot a.

00:10:26.670 --> 00:10:30.220
And then 2a dotted
with b is going

00:10:30.220 --> 00:10:34.730
to be plus 2 times a dot b.

00:10:34.730 --> 00:10:41.775
And then b dot 2a is going to
be plus 2 b dot a or a dot b.

00:10:41.775 --> 00:10:44.670
I want to just switch
colors actually.

00:10:44.670 --> 00:10:46.975
b-- I'll do it in the
same color as the b.

00:10:46.975 --> 00:10:52.280
So b dot this is
going to be 2 a dot b.

00:10:52.280 --> 00:10:57.789
And then b dot b,
so plus b dot b.

00:10:57.789 --> 00:10:59.330
So what is this
going to evaluate to?

00:10:59.330 --> 00:11:01.940
a dot a is 1, so
it's going to be 4.

00:11:01.940 --> 00:11:04.700
a dot b is 0.

00:11:04.700 --> 00:11:06.600
And a dot b is 0.

00:11:06.600 --> 00:11:08.170
And then this is 1.

00:11:08.170 --> 00:11:10.390
So we have 4 plus 1.

00:11:10.390 --> 00:11:13.850
4 plus 1 is equal to 5.

00:11:13.850 --> 00:11:17.330
So this whole mess, if
you're able to remember

00:11:17.330 --> 00:11:21.100
the triple product
expansion and kind of see

00:11:21.100 --> 00:11:23.080
the simplification
over here, it's

00:11:23.080 --> 00:11:24.830
actually a fairly
straightforward problem.

00:11:24.830 --> 00:11:27.910
This whole mess over here is 5.

00:11:27.910 --> 00:11:30.180
It evaluates to 5.


Vector triple product expansion (very optional)

WEBVTT
Kind: captions
Language: en

00:00:00.376 --> 00:00:01.750
What I want to do
with this video

00:00:01.750 --> 00:00:04.250
is cover something called the
triple product expansion--

00:00:04.250 --> 00:00:06.620
or Lagrange's
formula, sometimes.

00:00:06.620 --> 00:00:09.090
And it's really just
a simplification

00:00:09.090 --> 00:00:11.530
of the cross product
of three vectors,

00:00:11.530 --> 00:00:17.400
so if I take the cross product
of a, and then b cross c.

00:00:17.400 --> 00:00:19.820
And what we're
going to do is, we

00:00:19.820 --> 00:00:23.130
can express this really
as sum and differences

00:00:23.130 --> 00:00:24.266
of dot products.

00:00:24.266 --> 00:00:25.640
Well, not just
dot products-- dot

00:00:25.640 --> 00:00:27.099
products scaling
different vectors.

00:00:27.099 --> 00:00:28.431
You're going to see what I mean.

00:00:28.431 --> 00:00:30.540
But it simplifies this
expression a good bit,

00:00:30.540 --> 00:00:32.830
because cross products
are hard to take.

00:00:32.830 --> 00:00:34.230
They're computationally
intensive

00:00:34.230 --> 00:00:36.190
and, at least in my
mind, they're confusing.

00:00:36.190 --> 00:00:37.565
Now this isn't
something you have

00:00:37.565 --> 00:00:39.740
to know if you're going to
be dealing with vectors,

00:00:39.740 --> 00:00:41.089
but it's useful to know.

00:00:41.089 --> 00:00:42.880
My motivation for
actually doing this video

00:00:42.880 --> 00:00:47.230
is I saw some problems for the
Indian Institute of Technology

00:00:47.230 --> 00:00:51.340
entrance exam that seems to
expect that you know Lagrange's

00:00:51.340 --> 00:00:53.400
formula, or the triple
product expansion.

00:00:53.400 --> 00:00:55.730
So let's see how we
can simplify this.

00:00:55.730 --> 00:00:57.230
So to do that,
let's start taking

00:00:57.230 --> 00:01:01.480
the cross product of b and c.

00:01:01.480 --> 00:01:04.510
And in all of these
situations, I'm

00:01:04.510 --> 00:01:08.020
just going to assume--
let's say I have vector a.

00:01:08.020 --> 00:01:10.860
That's going to be a,
the x component of vector

00:01:10.860 --> 00:01:19.020
a times the unit of vector i
plus the y component of vector

00:01:19.020 --> 00:01:24.130
a times the unit vector j
plus the z component of vector

00:01:24.130 --> 00:01:25.580
a times unit vector k.

00:01:25.580 --> 00:01:27.830
And I could do the same
things for b and c.

00:01:27.830 --> 00:01:31.610
So if I say b sub y,
I'm talking about what's

00:01:31.610 --> 00:01:35.010
scaling the j component
in the b vector.

00:01:35.010 --> 00:01:38.370
So let's first take this
cross product over here.

00:01:38.370 --> 00:01:40.590
And if you've seen me
take cross products,

00:01:40.590 --> 00:01:43.370
you know that I like to do
these little determinants.

00:01:43.370 --> 00:01:44.620
Let me just take it over here.

00:01:44.620 --> 00:01:51.710
So b cross c is going to be
equal to the determinant.

00:01:51.710 --> 00:01:55.512
And I put an i, j, k up here.

00:01:55.512 --> 00:01:57.720
This is actually the definition
of the cross product,

00:01:57.720 --> 00:02:00.689
so no proof necessary to
show you why this is true.

00:02:00.689 --> 00:02:02.730
This is just one way to
remember the dot product,

00:02:02.730 --> 00:02:04.396
if you remember how
to take determinants

00:02:04.396 --> 00:02:06.190
of three-by-threes.

00:02:06.190 --> 00:02:11.310
And we'll put b's x
term, b's y coefficient,

00:02:11.310 --> 00:02:14.870
and b's z component.

00:02:14.870 --> 00:02:19.880
And then you do the same
thing for the c, cx, cy, cz.

00:02:19.880 --> 00:02:22.070
And then this is going
to be equal to-- so

00:02:22.070 --> 00:02:24.620
first you have the i component.

00:02:24.620 --> 00:02:28.940
So it's going to be the
i component times b.

00:02:28.940 --> 00:02:32.230
So you ignore this
column and this row.

00:02:32.230 --> 00:02:39.426
So bycz minus bzcy.

00:02:39.426 --> 00:02:40.800
So I'm just ignoring
all of this.

00:02:40.800 --> 00:02:41.750
And I'm looking
at this two-by-two

00:02:41.750 --> 00:02:42.943
over here, minus bzcy.

00:02:48.020 --> 00:02:51.420
And then we want to
subtract the j component.

00:02:51.420 --> 00:02:54.450
Remember, we alternate signs
when we take our determinant.

00:02:54.450 --> 00:02:56.180
Subtract that.

00:02:56.180 --> 00:02:59.050
And then we take out
that column and that row,

00:02:59.050 --> 00:03:05.090
so it's going to be bxcz--
this is a little monotonous,

00:03:05.090 --> 00:03:07.290
but hopefully, it'll have
an interesting result--

00:03:07.290 --> 00:03:09.333
bxcz minus bzcx.

00:03:16.630 --> 00:03:19.440
And then finally,
plus the k component.

00:03:19.440 --> 00:03:26.886
OK, we're going to have
bx times cy minus bycx.

00:03:33.860 --> 00:03:38.270
We just did the dot product, and
now we want to take the-- oh,

00:03:38.270 --> 00:03:39.950
sorry, we just did
the cross product.

00:03:39.950 --> 00:03:41.324
I don't want to
get you confused.

00:03:41.324 --> 00:03:43.990
We just took the cross
product of b and c.

00:03:43.990 --> 00:03:46.805
And now we need take the
cross product of that with a,

00:03:46.805 --> 00:03:50.022
or the cross product of a with
this thing right over here.

00:03:50.022 --> 00:03:50.730
So let's do that.

00:03:50.730 --> 00:03:52.430
Instead of rewriting
the vector, let

00:03:52.430 --> 00:03:54.850
me just set up
another matrix here.

00:03:54.850 --> 00:03:59.280
So let me write
my i j k up here.

00:03:59.280 --> 00:04:01.010
And then let me
write a's components.

00:04:01.010 --> 00:04:06.470
So we have a sub x,
a sub y, a sub z.

00:04:06.470 --> 00:04:09.040
And then let's clean
this up a little bit.

00:04:09.040 --> 00:04:09.810
Let's ignore this.

00:04:09.810 --> 00:04:13.090
We're just looking at-- no,
I want to do that in black.

00:04:13.090 --> 00:04:18.510
Let's do this in black, so
that we can kind of erase that.

00:04:18.510 --> 00:04:20.840
Now this is a
minus j times that.

00:04:20.840 --> 00:04:22.340
So what I'm going
to do is I'm going

00:04:22.340 --> 00:04:23.756
to get rid of the
minus and the j,

00:04:23.756 --> 00:04:28.210
but I am going to rewrite
this with the signs swapped.

00:04:28.210 --> 00:04:40.770
So if you swap the signs,
it's actually bzcx minus bxcz.

00:04:40.770 --> 00:04:43.260
So let me delete
everything else.

00:04:43.260 --> 00:04:46.139
So I just took the negative
and I multiplied it by this.

00:04:46.139 --> 00:04:48.180
I hope I'm not making any
careless mistakes here,

00:04:48.180 --> 00:04:51.240
so let me just check and make
my brush size little bit bigger,

00:04:51.240 --> 00:04:54.150
so I can erase that a
little more efficiently.

00:04:54.150 --> 00:04:54.700
There you go.

00:04:54.700 --> 00:04:57.710
And then we also want to get
rid of that right over there.

00:04:57.710 --> 00:05:01.480
Now let me get my brush size
back down to normal size.

00:05:01.480 --> 00:05:02.090
All right.

00:05:02.090 --> 00:05:05.630
So now let's just take
this cross product.

00:05:05.630 --> 00:05:10.359
So once again, set it
up as a determinant.

00:05:10.359 --> 00:05:11.900
And what I'm only
going to focus on--

00:05:11.900 --> 00:05:16.170
because it'll take the video,
or it'll take me forever

00:05:16.170 --> 00:05:18.970
if I were to do the i,
j, and k components--

00:05:18.970 --> 00:05:21.170
I'm just going to focus
on the i component,

00:05:21.170 --> 00:05:25.180
just on the x component
of this cross product.

00:05:25.180 --> 00:05:26.890
And then we can
see that we'll get

00:05:26.890 --> 00:05:28.740
the same result for
the j and the k.

00:05:28.740 --> 00:05:30.240
And then we can see
what, hopefully,

00:05:30.240 --> 00:05:32.490
this simplifies down to.

00:05:32.490 --> 00:05:36.240
So if we just focus on
the i component here,

00:05:36.240 --> 00:05:41.060
this is going to be
i times-- and we just

00:05:41.060 --> 00:05:43.360
look at this two-by-two
matrix right over here.

00:05:43.360 --> 00:05:45.890
We ignore i's column, i's row.

00:05:45.890 --> 00:05:49.780
And we have ay
times all of this.

00:05:49.780 --> 00:05:51.340
So let me just multiply it out.

00:05:51.340 --> 00:06:03.480
So it's ay times bxcy, minus
ay times by, times bycx.

00:06:07.260 --> 00:06:10.240
And then we're going
to want to subtract.

00:06:10.240 --> 00:06:13.134
We're going to have
minus az times this.

00:06:13.134 --> 00:06:14.050
So let's just do that.

00:06:14.050 --> 00:06:18.010
So it's minus, or
negative, azbzcx.

00:06:22.110 --> 00:06:23.880
And then we have a
negative az times this,

00:06:23.880 --> 00:06:26.425
so it's plus azbxcz.

00:06:29.472 --> 00:06:30.930
And now what I'm
going to do-- this

00:06:30.930 --> 00:06:33.150
is a little bit of a
trick for this proof

00:06:33.150 --> 00:06:37.030
right here, just so that we
get the results that I want.

00:06:37.030 --> 00:06:39.890
I'm just going to add and
subtract the exact same thing.

00:06:39.890 --> 00:06:43.501
So I'm going to add an axbxcx.

00:06:46.500 --> 00:06:50.160
And then I'm going to subtract
an axbxcx, minus axbxcx.

00:06:57.330 --> 00:06:59.820
So clearly, I have not
changed this expression.

00:06:59.820 --> 00:07:02.770
I've just added and
subtracted the same thing.

00:07:02.770 --> 00:07:04.250
And let's see what
we can simplify.

00:07:04.250 --> 00:07:07.270
Remember, this is
just the x component

00:07:07.270 --> 00:07:08.580
of our triple product.

00:07:08.580 --> 00:07:09.930
Just the x component.

00:07:09.930 --> 00:07:12.830
But to do this,
let me factor out.

00:07:12.830 --> 00:07:15.970
I'm going to factor out a bx.

00:07:15.970 --> 00:07:20.210
So let me do this,
let me get the bx.

00:07:20.210 --> 00:07:23.170
So if I were to
factor it out-- I'm

00:07:23.170 --> 00:07:26.070
going to factor it out of
this term that has a bx.

00:07:26.070 --> 00:07:28.060
I'm going to factor
it out of this term.

00:07:28.060 --> 00:07:30.860
And then I'm going to
factor it out of this term.

00:07:30.860 --> 00:07:35.030
So if I take the bx out,
I'm going to have an aycy.

00:07:37.650 --> 00:07:40.320
Actually, let me write it
a little bit differently.

00:07:40.320 --> 00:07:42.600
Let me factor it out
of this one first.

00:07:42.600 --> 00:07:46.440
So then it's going
to have an axcx.

00:07:46.440 --> 00:07:48.410
a sub x, c sub x.

00:07:48.410 --> 00:07:50.100
So I used this one up.

00:07:50.100 --> 00:07:53.030
And then I'll do this one now.

00:07:53.030 --> 00:07:58.700
Plus, if I factor the
bx out, I get ay cy.

00:07:58.700 --> 00:07:59.815
I've used that one now.

00:07:59.815 --> 00:08:00.815
And now I have this one.

00:08:00.815 --> 00:08:02.660
I'm going to factor the bx out.

00:08:02.660 --> 00:08:07.920
So I'm left with a plus az, cz.

00:08:07.920 --> 00:08:09.260
So that's all of those.

00:08:09.260 --> 00:08:10.790
So I've factored that out.

00:08:10.790 --> 00:08:14.530
And now, from these
right over here,

00:08:14.530 --> 00:08:16.180
let me factor out a negative cx.

00:08:20.610 --> 00:08:23.380
And so, if I do that-- let me go
to this term right over here--

00:08:23.380 --> 00:08:26.400
I'm going to have an axbx
when I factor it out.

00:08:26.400 --> 00:08:30.200
So an axbx, cross that out.

00:08:30.200 --> 00:08:32.590
And then, over here, I'm
going to have an ayby.

00:08:32.590 --> 00:08:34.370
Remember, I'm factoring
out a negative cx,

00:08:34.370 --> 00:08:39.460
so I'm going to have
a plus ay, sub by.

00:08:39.460 --> 00:08:44.530
And then, finally, I'm going
to have a plus az, az bz.

00:08:47.130 --> 00:08:49.310
And what is this?

00:08:49.310 --> 00:08:51.740
Well, this right
here, in green, this

00:08:51.740 --> 00:08:56.290
is the exact same thing as
the dot products of a and c.

00:08:56.290 --> 00:09:01.010
This is the dot product
of the vectors a and c.

00:09:01.010 --> 00:09:05.520
It's the dot product of
this vector and that vector.

00:09:05.520 --> 00:09:10.050
So that's the dot
of a and c times

00:09:10.050 --> 00:09:19.840
the x component
of b minus-- I'll

00:09:19.840 --> 00:09:22.660
do this in the same--
minus-- once again,

00:09:22.660 --> 00:09:29.790
this is the dot product of
a and b now, minus a dot b

00:09:29.790 --> 00:09:32.990
times the x component of c.

00:09:32.990 --> 00:09:34.490
And we can't forget,
all of this was

00:09:34.490 --> 00:09:36.450
multiplied by the unit vector i.

00:09:36.450 --> 00:09:40.140
We're looking at the x
component, or the i component

00:09:40.140 --> 00:09:42.550
of that whole triple product.

00:09:42.550 --> 00:09:46.070
So that's going
to be all of this.

00:09:46.070 --> 00:09:51.570
All of this is times
the unit vector i.

00:09:51.570 --> 00:09:53.510
Now, if we do this exact
same thing-- and I'm

00:09:53.510 --> 00:09:56.560
not going to do it, because
it's computationally intensive.

00:09:56.560 --> 00:10:00.700
But I think it won't be a
huge leap of faith for you.

00:10:00.700 --> 00:10:02.210
This is for the x component.

00:10:02.210 --> 00:10:04.830
If I were to do the exact same
thing for the y component,

00:10:04.830 --> 00:10:07.600
for the j component--
so it'll be plus--

00:10:07.600 --> 00:10:09.840
if I do the same thing
for the j component,

00:10:09.840 --> 00:10:11.840
we can really just
pattern match.

00:10:11.840 --> 00:10:14.500
We have bx, cx, that's
for the x component.

00:10:14.500 --> 00:10:19.440
We'll have by and c y
for the j component.

00:10:19.440 --> 00:10:21.540
And then this is not
component-specific,

00:10:21.540 --> 00:10:29.860
so it will be a dot c over here,
and minus a dot b over here.

00:10:29.860 --> 00:10:31.770
You can verify any of
these for yourself,

00:10:31.770 --> 00:10:33.240
if you don't believe me.

00:10:33.240 --> 00:10:35.600
But it's the exact same
process we just did.

00:10:35.600 --> 00:10:39.050
And then, finally, for the z
component, or the k component--

00:10:39.050 --> 00:10:41.930
let me put parentheses
over here-- same idea.

00:10:41.930 --> 00:10:47.240
You're going to have bz, cz.

00:10:47.240 --> 00:10:51.920
And then you're going to
have a dot b over there.

00:10:51.920 --> 00:10:57.920
And then you're going to
have a dot c over here.

00:10:57.920 --> 00:11:01.790
Now what does this become?

00:11:01.790 --> 00:11:04.500
How can we simplify this?

00:11:04.500 --> 00:11:07.780
Well, this right over here,
we can expand this out.

00:11:07.780 --> 00:11:10.720
We can factor out
an a dot c from all

00:11:10.720 --> 00:11:12.020
of these terms over here.

00:11:12.020 --> 00:11:14.340
And remember, this is going
to be multiplied times i.

00:11:14.340 --> 00:11:16.710
Actually, let me not
skip too many steps,

00:11:16.710 --> 00:11:20.340
just because I want you
to believe what I'm doing.

00:11:20.340 --> 00:11:24.724
So if we expand the i here--
instead of rewriting it,

00:11:24.724 --> 00:11:25.890
let me just do it like this.

00:11:25.890 --> 00:11:27.770
It's a little bit
messier, but let me just--

00:11:27.770 --> 00:11:31.320
so I could write this i
there and that i there.

00:11:31.320 --> 00:11:34.820
I'm kind of just distributing
that x unit vector,

00:11:34.820 --> 00:11:36.560
or the i unit vector.

00:11:36.560 --> 00:11:39.270
And let me do the
same thing for j.

00:11:39.270 --> 00:11:41.060
So I could put the j there.

00:11:41.060 --> 00:11:43.690
And I could put the
j right over there.

00:11:43.690 --> 00:11:46.460
And then I could do the
same thing for the k,

00:11:46.460 --> 00:11:49.890
put the k there, and
then put the k there.

00:11:49.890 --> 00:11:51.390
And now what are these?

00:11:51.390 --> 00:11:58.840
Well, this part right
over here is exactly

00:11:58.840 --> 00:12:05.910
the same thing as
a dot c times--

00:12:05.910 --> 00:12:10.400
and I'll write it
out here-- bx times i

00:12:10.400 --> 00:12:21.650
plus by times j,
plus bz times k.

00:12:21.650 --> 00:12:24.390
And then, from that,
we're going to subtract

00:12:24.390 --> 00:12:25.900
all of this, a dot b.

00:12:25.900 --> 00:12:32.617
We're going to subtract a dot
b times the exact same thing.

00:12:32.617 --> 00:12:34.200
And you're going to
notice, this right

00:12:34.200 --> 00:12:36.720
here is the same
thing as vector b.

00:12:36.720 --> 00:12:37.880
That is vector b.

00:12:37.880 --> 00:12:40.770
When you do it over here,
you're going to get vector c.

00:12:40.770 --> 00:12:42.520
So I'll just write it over here.

00:12:42.520 --> 00:12:44.870
You're just going
to get vector c.

00:12:44.870 --> 00:12:48.600
So just like that, we
have a simplification

00:12:48.600 --> 00:12:51.110
for our triple product.

00:12:51.110 --> 00:12:53.150
I know it took us a
long time to get here,

00:12:53.150 --> 00:12:54.590
but this is a simplification.

00:12:54.590 --> 00:12:57.150
It might not look like one,
but computationally it is.

00:12:57.150 --> 00:12:58.470
It's easier to do.

00:12:58.470 --> 00:13:05.760
If I have-- I'll try to
color-code it-- a cross b

00:13:05.760 --> 00:13:09.820
cross-- let me do it in
all different colors-- c,

00:13:09.820 --> 00:13:13.690
we just saw that this is going
to be equivalent to-- and

00:13:13.690 --> 00:13:16.430
one way to think about it is,
it's going to be, you take

00:13:16.430 --> 00:13:20.190
the first vector times the dot
product of-- the first vector

00:13:20.190 --> 00:13:22.210
in this second dot
product, the one

00:13:22.210 --> 00:13:23.716
that we have our
parentheses around,

00:13:23.716 --> 00:13:25.340
the one we would have
to do first-- you

00:13:25.340 --> 00:13:27.370
take your first vector there.

00:13:27.370 --> 00:13:29.790
So it's vector b.

00:13:29.790 --> 00:13:32.510
And you multiply that times the
dot product of the other two

00:13:32.510 --> 00:13:36.035
vectors, so a dot c.

00:13:38.730 --> 00:13:46.850
And from that, you subtract
the second vector multiplied

00:13:46.850 --> 00:13:53.900
by the dot product of the
other two vectors, of a dot b.

00:13:53.900 --> 00:13:57.030
And we're done.

00:13:57.030 --> 00:13:59.260
This is our triple
product expansion.

00:13:59.260 --> 00:14:02.130
Now, once again,
this isn't something

00:14:02.130 --> 00:14:04.042
that you really have to know.

00:14:04.042 --> 00:14:05.750
You could always,
obviously, multiply it.

00:14:05.750 --> 00:14:09.660
You could actually
do it by hand.

00:14:09.660 --> 00:14:10.930
You don't have to know this.

00:14:10.930 --> 00:14:14.080
But if you have
really hairy vectors,

00:14:14.080 --> 00:14:16.055
or if this was some type
of math competition,

00:14:16.055 --> 00:14:18.425
and sometimes it simplifies
real fast when you reduce it

00:14:18.425 --> 00:14:20.870
to dot products, this is
a useful thing to know,

00:14:20.870 --> 00:14:24.001
Lagrange's formula, or the
triple product expansion.


IIT JEE trigonometric maximum

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:04.950
The maximum value of the
expression 1 over sine squared

00:00:04.950 --> 00:00:07.400
of theta plus 3
sine theta cosine

00:00:07.400 --> 00:00:11.710
theta plus 5 cosine
squared of theta is?

00:00:11.710 --> 00:00:13.920
So let's rewrite this.

00:00:13.920 --> 00:00:18.610
So this is 1 over, so I have
a sine squared of theta.

00:00:22.230 --> 00:00:23.970
And then, at least
in my brain, whenever

00:00:23.970 --> 00:00:25.595
I see a sine squared
of theta, I always

00:00:25.595 --> 00:00:27.520
look for a cosine
squared of theta.

00:00:27.520 --> 00:00:30.240
Because I know that when I take
the sum of them it equals 1.

00:00:30.240 --> 00:00:32.870
And I don't have just one
cosine squared theta here,

00:00:32.870 --> 00:00:35.070
I have five cosine
squared theta.

00:00:35.070 --> 00:00:36.800
So let me just take one of them.

00:00:36.800 --> 00:00:40.250
So I have a plus
cosine squared theta.

00:00:40.250 --> 00:00:43.010
And since I took one of them,
I only have four of these left.

00:00:43.010 --> 00:00:47.510
So plus 4 cosine squared theta.

00:00:47.510 --> 00:00:48.810
And then I have this stuff.

00:00:48.810 --> 00:00:54.480
Plus 3 sine of theta
cosine of theta.

00:00:54.480 --> 00:00:56.200
So what this first
step allowed me to do

00:00:56.200 --> 00:00:59.230
is just turn these two
characters right over here,

00:00:59.230 --> 00:01:01.220
sine squared theta
plus cosine squared

00:01:01.220 --> 00:01:04.500
theta, that is equal to 1.

00:01:04.500 --> 00:01:11.100
So we've simplified it to 1
over 1 plus-- now let's think

00:01:11.100 --> 00:01:15.340
about how we can write
cosine squared of theta.

00:01:15.340 --> 00:01:17.070
I'll write our identities here.

00:01:17.070 --> 00:01:19.370
Cosine squared of theta.

00:01:19.370 --> 00:01:22.390
And we've prove these in
the trigonometry playlist.

00:01:22.390 --> 00:01:31.530
This is equal to 1 plus cosine
of 2 theta all of that over 2.

00:01:31.530 --> 00:01:34.170
And my goal here is, I really
just want to get everything--

00:01:34.170 --> 00:01:35.836
well, I really just
want to simplify it.

00:01:35.836 --> 00:01:37.080
And maybe we'll do calculus.

00:01:37.080 --> 00:01:39.706
We'll use a little calculus
to find the minimum value

00:01:39.706 --> 00:01:41.330
of the denominator,
which would give us

00:01:41.330 --> 00:01:44.390
the maximum value
of the numerator.

00:01:44.390 --> 00:01:47.420
So let's see, cosine squared
of theta is equal to this.

00:01:47.420 --> 00:01:51.670
So 4 times this is just going
to be-- so 4 times this.

00:01:51.670 --> 00:01:53.470
The 4 divided by 2 is 2.

00:01:53.470 --> 00:01:56.040
So it's going to be 2
times this numerator.

00:01:56.040 --> 00:02:01.590
So it's going to be 2
plus 2 cosine of 2 theta.

00:02:01.590 --> 00:02:03.850
That's this term right here.

00:02:03.850 --> 00:02:06.730
And then this term
right over here,

00:02:06.730 --> 00:02:08.190
we could use the trig identity.

00:02:08.190 --> 00:02:13.070
That sine of two theta
is equal to 2 sine

00:02:13.070 --> 00:02:16.010
of theta cosine of theta.

00:02:16.010 --> 00:02:21.140
Or you divide both sides by
2, you get 1/2 sine of 2 theta

00:02:21.140 --> 00:02:26.530
is equal to sine of
theta cosine of theta.

00:02:26.530 --> 00:02:28.850
So this right over here,
this part right over here

00:02:28.850 --> 00:02:31.220
is going to be 1/2
sine of 2 theta.

00:02:31.220 --> 00:02:33.020
But we're multiplying it by 3.

00:02:33.020 --> 00:02:36.970
So it's going to be plus
3/2 sine of 2 theta.

00:02:41.470 --> 00:02:45.200
And let's see, this part
over here clearly simplifies.

00:02:45.200 --> 00:02:45.997
This is 3.

00:02:45.997 --> 00:02:47.080
So let me just rewrite it.

00:02:47.080 --> 00:02:53.940
This is 1 over 3 plus
2 cosine of 2 theta

00:02:53.940 --> 00:02:59.340
plus 3/2 sine of 2 theta.

00:02:59.340 --> 00:03:01.589
And we're really just
looking for the minimum.

00:03:01.589 --> 00:03:04.130
We're looking for the minimum
value of the denominator, which

00:03:04.130 --> 00:03:06.671
would give us the same as the
maximum value of the numerator.

00:03:06.671 --> 00:03:09.190
It would just be 1 over
this minimum value.

00:03:09.190 --> 00:03:12.510
So let's see how low we can
get, assuming we're above 0.

00:03:12.510 --> 00:03:16.130
Let's see how low we can get
for this denominator right here.

00:03:16.130 --> 00:03:19.140
We're just going to look
for its minimum value.

00:03:19.140 --> 00:03:21.350
So one thing we can do,
just to simplify things,

00:03:21.350 --> 00:03:24.430
the minimum value of this
is going to be the same.

00:03:24.430 --> 00:03:28.180
The min of this thing-- I
don't want to write it there

00:03:28.180 --> 00:03:30.650
because it kind of
confuses the problem.

00:03:30.650 --> 00:03:39.270
The minimum value of 3 plus 2
cosine 2 theta plus 3/2 sine

00:03:39.270 --> 00:03:42.880
of 2 theta is going to be the
same thing as the minimum value

00:03:42.880 --> 00:03:45.670
of 3 plus-- I'm just going
to do the substitution

00:03:45.670 --> 00:03:46.930
that 2 theta is equal to x.

00:03:46.930 --> 00:03:48.638
That just simplifies
things a little bit.

00:03:48.638 --> 00:03:49.870
You don't have to do that.

00:03:49.870 --> 00:03:58.370
But just to, so 3 plus 2
cosine of x plus 3/2 sine of x.

00:03:58.370 --> 00:04:00.027
So this is a pretty
simple expression.

00:04:00.027 --> 00:04:02.110
Let's see how we can figure
out its minimum value.

00:04:02.110 --> 00:04:04.800
And my temptation is to take
the derivative, find out

00:04:04.800 --> 00:04:07.990
where the derivative
is equal to 0,

00:04:07.990 --> 00:04:11.730
and then that will either be
a maximum or a minimum point.

00:04:11.730 --> 00:04:13.090
So let's take the derivative.

00:04:13.090 --> 00:04:14.570
The derivative of
this expression

00:04:14.570 --> 00:04:16.690
right over here
with respect to x.

00:04:16.690 --> 00:04:19.019
Well, derivative of 3
with respect to x is 0.

00:04:19.019 --> 00:04:24.990
Derivative of cosine of x
is negative 2 sine of x.

00:04:24.990 --> 00:04:27.160
Derivative of 3/2
sine of x is going

00:04:27.160 --> 00:04:30.610
to be plus 3/2 cosine of x.

00:04:30.610 --> 00:04:32.447
And then, that is
going to be equal to 0.

00:04:32.447 --> 00:04:34.780
We want to find where the
slope is 0 because it's either

00:04:34.780 --> 00:04:37.220
going to be a maximum
or minimum point.

00:04:37.220 --> 00:04:39.840
And let's see, we
can add, we could

00:04:39.840 --> 00:04:41.530
add 2 sine of x to both sides.

00:04:41.530 --> 00:04:47.660
So we get 3/2 cosine of x
is equal to 2 sine of x.

00:04:47.660 --> 00:04:54.140
And then, we can divide both
sides of this equation by,

00:04:54.140 --> 00:04:55.430
let's divide it by 2 first.

00:04:55.430 --> 00:04:56.929
I don't want to
skip too many steps.

00:04:56.929 --> 00:05:02.200
So 3/4 cosine of x is
equal to sine of x.

00:05:02.200 --> 00:05:04.830
Then we could divide both
sides by cosine of x.

00:05:04.830 --> 00:05:08.280
So we get 3 over
4 is equal to sine

00:05:08.280 --> 00:05:13.440
of x over cosine of x, which is
the same thing as the tangent

00:05:13.440 --> 00:05:14.260
of x.

00:05:14.260 --> 00:05:18.850
So an x value that gives us--
or the tangent of an x value

00:05:18.850 --> 00:05:20.750
that gives us 3/4
is going to give us

00:05:20.750 --> 00:05:23.020
either a maximum
or a minimum point.

00:05:23.020 --> 00:05:24.490
So let's think about this.

00:05:24.490 --> 00:05:25.990
Let's think about
this a little bit.

00:05:25.990 --> 00:05:28.120
Let me draw my unit circle.

00:05:28.120 --> 00:05:31.320
So let's think about
two x values that

00:05:31.320 --> 00:05:33.880
will give us the tangent of 3/4.

00:05:33.880 --> 00:05:36.930
So let me draw my unit circle.

00:05:36.930 --> 00:05:40.680
That's a unit circle.

00:05:40.680 --> 00:05:41.613
Let me draw a unit.

00:05:44.330 --> 00:05:45.820
This is always the hardest part.

00:05:45.820 --> 00:05:47.850
So let me draw this.

00:05:47.850 --> 00:05:51.000
All right, there
is my unit circle.

00:05:51.000 --> 00:05:54.606
So how can I get a triangle,
or, well, yeah, let's think

00:05:54.606 --> 00:05:55.230
about that way.

00:05:55.230 --> 00:05:58.950
How can I get a triangle where
an angle is the tangent of 3/4.

00:05:58.950 --> 00:06:01.681
Remember, tangent is
opposite over adjacent.

00:06:01.681 --> 00:06:02.180
Right?

00:06:02.180 --> 00:06:04.290
Tangent is opposite
over adjacent.

00:06:04.290 --> 00:06:09.960
So if this is my triangle
right over here, if this is x,

00:06:09.960 --> 00:06:13.270
opposite over adjacent
is equal to 3/4.

00:06:13.270 --> 00:06:16.520
So opposite could be 3
and adjacent could be 4.

00:06:16.520 --> 00:06:19.472
And we hopefully
immediately recognize this.

00:06:19.472 --> 00:06:23.120
This is a 3, 4, 5 triangle
because it's a right triangle.

00:06:23.120 --> 00:06:27.050
3 squared plus 4 squared
is 25 which is 5 squared.

00:06:27.050 --> 00:06:29.770
So this is a 3, 4, 5 triangle.

00:06:29.770 --> 00:06:32.490
Now, there's two tangent values.

00:06:32.490 --> 00:06:33.730
So x could be like this.

00:06:33.730 --> 00:06:37.610
So this obviously isn't a
unit hypotenuse over here.

00:06:37.610 --> 00:06:40.030
But we could divide everything
by 5, and it would be.

00:06:40.030 --> 00:06:42.460
So we could have this situation.

00:06:42.460 --> 00:06:44.800
We could have this
situation over here.

00:06:44.800 --> 00:06:45.870
Where this is x.

00:06:45.870 --> 00:06:48.220
If this is the unit circle,
the hypotenuse is 1.

00:06:48.220 --> 00:06:52.880
This is 3/5 and this is 4/5.

00:06:52.880 --> 00:06:57.280
This would, tangent of x here,
tangent of x would give us 3/4.

00:06:57.280 --> 00:07:00.250
But is this going to give us a
maximum value or minimum value?

00:07:00.250 --> 00:07:04.800
Well over here, both
cosine of x and sine of x

00:07:04.800 --> 00:07:06.040
are going to be positive.

00:07:06.040 --> 00:07:08.670
So both of these are going
to be positive values.

00:07:08.670 --> 00:07:12.870
So it's going to, probably,
maximize this expression

00:07:12.870 --> 00:07:13.790
over here.

00:07:13.790 --> 00:07:17.460
Now the other x that gives us
the same tangent, and remember,

00:07:17.460 --> 00:07:21.330
the tangent is really just the
slope of the radius in the unit

00:07:21.330 --> 00:07:24.350
circle, would be this angle.

00:07:24.350 --> 00:07:27.750
This has the same tangent value.

00:07:27.750 --> 00:07:32.690
So this x over here.

00:07:32.690 --> 00:07:37.210
In this case, the tangent
is still going to be 3/4.

00:07:37.210 --> 00:07:40.500
But over here, the sine
and cosine are negative.

00:07:40.500 --> 00:07:45.930
So over here, the
x-coordinate, or the cosine,

00:07:45.930 --> 00:07:48.360
is going to be negative 4/5.

00:07:48.360 --> 00:07:51.340
And the sine value,
or the y value,

00:07:51.340 --> 00:07:53.390
is going to be negative 3/5.

00:07:53.390 --> 00:07:58.250
And this will give
us a minimum point,

00:07:58.250 --> 00:08:02.480
because here both the sine
and the cosine are negative.

00:08:02.480 --> 00:08:06.460
So let's use this
x right over here.

00:08:06.460 --> 00:08:09.000
And notice, we don't even have
to figure out what the x is.

00:08:09.000 --> 00:08:12.950
Because we know that if the
tangent over here is 4/5,

00:08:12.950 --> 00:08:16.290
either both the sine
is going to be 3/5,

00:08:16.290 --> 00:08:18.262
and the cosine is
going to be 4/5,

00:08:18.262 --> 00:08:19.720
which will give us
a maximum point.

00:08:19.720 --> 00:08:22.620
Or the tangent could be 3/4.

00:08:22.620 --> 00:08:24.700
And then the sine
will be negative 3/5,

00:08:24.700 --> 00:08:26.672
and the cosine will
be negative 4/5.

00:08:26.672 --> 00:08:27.880
So let's use these over here.

00:08:27.880 --> 00:08:30.670
So the minimum is
going to be equal to 3

00:08:30.670 --> 00:08:33.758
plus 2 times cosine of x.

00:08:33.758 --> 00:08:35.049
We're using this one over here.

00:08:35.049 --> 00:08:36.370
So 2 times cosine of x.

00:08:36.370 --> 00:08:38.785
Cosine of x here
is negative 4/5.

00:08:42.220 --> 00:08:47.130
And then plus 3/2
times the sine of x.

00:08:47.130 --> 00:08:48.900
Sine of x here is negative 3/5.

00:08:51.970 --> 00:08:54.690
And what is this
going to be equal to?

00:08:54.690 --> 00:09:01.530
This is going to be equal to 3
plus, this is negative 8/5, 3,

00:09:01.530 --> 00:09:09.540
maybe I should write 3
minus 8/5 minus 9/10.

00:09:09.540 --> 00:09:11.900
And so this is going
to be equal to-- we

00:09:11.900 --> 00:09:18.390
could put everything over 10
--30 over 10 minus 16 over 10,

00:09:18.390 --> 00:09:19.720
that's 8/5.

00:09:19.720 --> 00:09:21.630
Minus 9/10.

00:09:21.630 --> 00:09:23.180
And this gives us what?

00:09:23.180 --> 00:09:25.970
This gives us 5/10.

00:09:25.970 --> 00:09:29.080
5/10 or 1/2.

00:09:29.080 --> 00:09:33.060
So the minimum value for
our denominator-- everything

00:09:33.060 --> 00:09:35.750
we've been dealing with so
far has been our denominator--

00:09:35.750 --> 00:09:39.090
the minimum value of all
of this business over here,

00:09:39.090 --> 00:09:41.980
the minimum value is 1/2.

00:09:41.980 --> 00:09:45.470
So the maximum value, that
this whole expression takes,

00:09:45.470 --> 00:09:47.170
is when the minimum
value is 1/2.

00:09:47.170 --> 00:09:52.410
So we get 1 over 1/2
half which is equal to 2.

00:09:52.410 --> 00:09:54.250
And we're done.