Extreme derivative word problem (advanced)

WEBVTT
Kind: captions
Language: en

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I just got sent this
problem, and it's a

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pretty meaty problem.

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A lot harder than what
you'd normally find

00:00:04.910 --> 00:00:05.580
in most textbooks.

00:00:05.580 --> 00:00:08.470
So I thought it would help
us all to work it out.

00:00:08.470 --> 00:00:10.190
And it's one of those problems
that when you first read it,

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your eyes kind of glaze over,
but when you understand what

00:00:12.630 --> 00:00:15.340
they're talking about, it's
reasonably interesting.

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So they say, the curve in the
figure above is the parabola

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y is equal to x squared.

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So this curve right there is
y is equal to x squared.

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Let us define a normal line as
a line whose first quadrant

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intersection with the parabola
is perpendicular

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to the parabola.

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So this is the first
quadrant, right here.

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And they're saying that a
normal line is something, when

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the first quadrant intersection
with the parabola is

00:00:40.570 --> 00:00:41.720
normal to the parabola.

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So if I were to draw a tangent
line right there, this line is

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normal to that tangent line.

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That's all that's saying.

00:00:49.070 --> 00:00:52.470
So this is a normal
line, right there.

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Normal line.

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Fair enough.

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5 normal lines are
shown in the figure.

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1, 2, 3, 4, 5.

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Good enough.

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And these all look
perpendicular, or normal to

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the parabola in the first
quadrant intersection,

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so that makes sense.

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For a while, the x-coordinate
of the second quadrant

00:01:10.420 --> 00:01:14.140
intersection of the normal line
of the parabola gets smaller,

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as the x-coordinate of the
first quadrant intersection

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gets smaller.

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So let's see what happens as
the x-quadrant of the first

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intersection gets smaller.

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So this is where I left
off in that dense text.

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So if I start at this point
right here, my x-coordinate

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right there would look
something like this.

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Let me go down.

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my x-coordinate is
right around there.

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And then as I move to a smaller
x-coordinate to, say, this one

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right here, what happened
to the normal line?

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Or even more important, what
happened to the intersection

00:01:42.380 --> 00:01:44.590
of the normal line in
the second quadrant?

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This is the second
quadrant, right here.

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So when I had a larger
x-value here, my normal

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line intersected here,
in the second quadrant.

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Then when I brought my x-value
in, when I lowered my x-value,

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my x-value here, because this
is the next point, right here,

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my x-value at the intersection
here, went-- actually,

00:02:04.960 --> 00:02:05.910
their wording is bad.

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They're saying that
the second quadrant

00:02:07.570 --> 00:02:09.510
intersection gets smaller.

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But actually, it's not
really getting smaller.

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It's getting less negative.

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I guess smaller could be just
absolute value or magnitude,

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but it's just getting
less negative.

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It's moving there, but
it's actually becoming

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a larger number, right?

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It's becoming less negative,
but a larger number.

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But if we think in absolute
value, I guess it's

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getting smaller, right?

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As we went from that point to
that point, as we moved the x

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in for the intersection of the
first quadrant, the second

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quadrant intersection also
moved in a bit, from

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that line to that line.

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Fair enough.

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But eventually, a normal line
second quadrant intersection

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gets as small as it can get.

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So if we keep lowering our
x-value in the first quadrant,

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so we keep on pulling in the
first quadrant, as we

00:02:53.210 --> 00:02:54.480
get to this point.

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And then this point intersects
the second quadrant,

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right there.

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And then, if you go even
smaller x-values in the first

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quadrant then your normal line
starts intersecting in the

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second quadrant, further and
further negative numbers.

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So you can kind of view this
as the highest value, or the

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smallest absolute value, at
which the normal line can

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intersect in the
second quadrant

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Let me make that clear.

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Up here, you were intersecting
when you had a large x in the

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first quadrant, you had a large
negative x in the second

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quadrant intersection.

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And then as you lowered your
x-value, here, you had a

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smaller negative value.

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Up until you got to this point,
right here, you got this, which

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you can view as the smallest
negative value could get, and

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then when you pulled in your x
even more, these normal lines

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started to push out again,
out in the second quadrant.

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That's, I think, what
they're talking about.

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The extreme normal line
is shown as a thick

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line in the figure.

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Right.

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This is the extreme normal
line, right there.

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So this is the extreme
one, that deep, bold one.

00:04:01.500 --> 00:04:04.040
Extreme normal line.

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After this point, when you pull
in your x-values even more, the

00:04:08.600 --> 00:04:10.680
intersection in your second
quadrant starts to

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push out some.

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And you can think of the
extreme case, if you draw the

00:04:14.770 --> 00:04:18.360
normal line down here, your
intersection with the second

00:04:18.360 --> 00:04:20.260
quadrant is going to be way out
here someplace, although it

00:04:20.260 --> 00:04:22.430
seems like it's kind of
asymptoting a little bit.

00:04:22.430 --> 00:04:22.990
But I don't know.

00:04:22.990 --> 00:04:25.060
Let's read the rest
of the problem.

00:04:25.060 --> 00:04:27.385
Once the normal line passes
the extreme normal line, the

00:04:27.385 --> 00:04:30.130
x-coordinates of their second
quadrant intersections what the

00:04:30.130 --> 00:04:32.300
parabola start to increase.

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And they're really, when they
say they start to increase,

00:04:34.520 --> 00:04:36.100
they're actually just
becoming more negative.

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That wording is bad.

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I should change this to
more, more negative.

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Or they're becoming
larger negative numbers.

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Because once you get below
this, then all of a sudden

00:04:46.450 --> 00:04:49.460
the x-intersections start
to push out more in

00:04:49.460 --> 00:04:50.390
the second quadrant.

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Fair enough.

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The figures show 2
pairs of normal lines.

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Fair enough.

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The 2 normal lines of a pair
have the same second quadrant

00:04:58.740 --> 00:05:01.730
intersection with the parabola,
but 1 is above the extreme

00:05:01.730 --> 00:05:03.590
normal line, in the first
quadrant, the other

00:05:03.590 --> 00:05:05.510
is below it.

00:05:05.510 --> 00:05:07.180
Right, fair enough.

00:05:07.180 --> 00:05:10.280
For example, this guy right
here, this is when we

00:05:10.280 --> 00:05:11.360
had a large x-value.

00:05:11.360 --> 00:05:13.760
He intersects with the
second quadrant there.

00:05:13.760 --> 00:05:15.940
Then if you lower and lower the
x-value, if you lower it

00:05:15.940 --> 00:05:18.930
enough, you pass the extreme
normal line, and then you get

00:05:18.930 --> 00:05:22.890
to this point, and then this
point, he intersects, or

00:05:22.890 --> 00:05:24.010
actually, you go to this point.

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So if you pull in your x-value
enough, you once again

00:05:27.160 --> 00:05:29.650
intersect at that same point
in the second quadrant.

00:05:29.650 --> 00:05:32.510
So hopefully I'm making some
sense to you, as I try to make

00:05:32.510 --> 00:05:34.410
some sense of this problem.

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OK.

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Now what do they want to know?

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And I think I only have time
for the first part of this.

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Maybe I'll do the second
part in the another video.

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Find the equation of the
extreme normal line.

00:05:45.850 --> 00:05:48.790
Well, that seems very daunting
at first, but I think our

00:05:48.790 --> 00:05:51.790
toolkit of derivatives, and
what we know about equations

00:05:51.790 --> 00:05:54.710
of a line, should be
able to get us there.

00:05:54.710 --> 00:05:58.470
So what's the slope of
the tangent line at any

00:05:58.470 --> 00:05:59.490
point on this curve?

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Well, we just take the
derivative of y equals

00:06:01.540 --> 00:06:04.910
x squared, and y prime
is just equal to 2x.

00:06:04.910 --> 00:06:14.040
This is the slope of the
tangent at any point x.

00:06:14.040 --> 00:06:18.290
So if I want to know the slope
of the tangent at x0, at some

00:06:18.290 --> 00:06:22.250
particular x, I would just
say, well, let me just say,

00:06:22.250 --> 00:06:25.010
slope, it would be 2 x0.

00:06:25.010 --> 00:06:29.490
Or let me just say, f of
x0 is equal to 2 x0.

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This is the slope at
any particular x0 of

00:06:32.760 --> 00:06:34.310
the tangent line.

00:06:34.310 --> 00:06:38.550
Now, the normal line slope
is perpendicular to this.

00:06:38.550 --> 00:06:41.720
So the perpendicular line, and
I won't review it here, but the

00:06:41.720 --> 00:06:44.730
perpendicular line has a
negative inverse slope.

00:06:44.730 --> 00:06:54.460
So the slope of normal line at
x0 will be the negative inverse

00:06:54.460 --> 00:06:57.590
of this, because this is the
slope of the tangent line x0.

00:06:57.590 --> 00:07:03.090
So it'll be equal to
minus 1 over 2 x0.

00:07:03.090 --> 00:07:04.510
Fair enough.

00:07:04.510 --> 00:07:10.100
Now, what is the equation of
the normal line at x0 let's say

00:07:10.100 --> 00:07:12.570
that this is my x0 in question.

00:07:12.570 --> 00:07:16.500
What is the equation of
the normal line there?

00:07:16.500 --> 00:07:18.960
Well, we can just use
the point-slope form

00:07:18.960 --> 00:07:20.330
of our equation.

00:07:20.330 --> 00:07:23.570
So this point right here
will be on the normal line.

00:07:23.570 --> 00:07:27.230
And that's the
point x0 squared.

00:07:27.230 --> 00:07:31.030
Because this the graph of
y equals x0, x squared.

00:07:31.030 --> 00:07:35.720
So this normal line will
also have this point.

00:07:35.720 --> 00:07:38.970
So we could say that the
equation of the normal line,

00:07:38.970 --> 00:07:48.300
let me write it down, would
be equal to, this is just a

00:07:48.300 --> 00:07:50.480
point-slope definition
of a line.

00:07:50.480 --> 00:07:55.480
You say, y minus the y-point,
which is just x0 squared,

00:07:55.480 --> 00:07:59.810
that's that right there, is
equal to the slope of the

00:07:59.810 --> 00:08:08.865
normal line minus 1 over
2 x0 times x minus the

00:08:08.865 --> 00:08:10.050
x-point that we're at.

00:08:10.050 --> 00:08:12.680
Minus x, minus x0.

00:08:12.680 --> 00:08:15.780
This is the equation
of the normal line.

00:08:15.780 --> 00:08:16.260
So let's see.

00:08:16.260 --> 00:08:19.620
And what we care about is when
x0 is greater than 0, right?

00:08:19.620 --> 00:08:22.740
We care about the normal line
when we're in the first

00:08:22.740 --> 00:08:25.410
quadrant, we're in all of
these values right there.

00:08:25.410 --> 00:08:28.710
So that's my equation
of the normal line.

00:08:28.710 --> 00:08:32.150
And let's solve it
explicitly in terms of x.

00:08:32.150 --> 00:08:33.800
So y is a function of x.

00:08:33.800 --> 00:08:38.780
Well, if I add x0 squared
to both sides, I get y is

00:08:38.780 --> 00:08:42.760
equal to, actually, let
me multiply this guy out.

00:08:42.760 --> 00:08:52.340
I get minus 1/2 x0 times x, and
then I have plus, plus, because

00:08:52.340 --> 00:08:55.880
I have a minus times
a minus, plus 1/2.

00:08:55.880 --> 00:08:59.110
The x0 and the over the
x0, they cancel out.

00:08:59.110 --> 00:09:01.340
And then I have to add
this x0 to both sides.

00:09:01.340 --> 00:09:05.560
So all I did so far, this is
just this part right there.

00:09:05.560 --> 00:09:07.040
That's this right there.

00:09:07.040 --> 00:09:10.260
And then I have to add this to
both sides of the equation, so

00:09:10.260 --> 00:09:14.130
then I have plus x0 squared.

00:09:14.130 --> 00:09:18.300
So this is the equation of the
normal line, in mx plus b form.

00:09:18.300 --> 00:09:22.860
This is its slope, this is
the m, and then this is its

00:09:22.860 --> 00:09:24.740
y-intercept right here.

00:09:24.740 --> 00:09:27.280
That's kind of the b.

00:09:27.280 --> 00:09:29.200
Now, what do we care about?

00:09:29.200 --> 00:09:32.600
We care about where
this thing intersects.

00:09:32.600 --> 00:09:35.990
We care about where it
intersects the parabola.

00:09:35.990 --> 00:09:38.130
And the parabola, that's pretty
straightforward, that's just

00:09:38.130 --> 00:09:41.590
y is equal to x squared.

00:09:41.590 --> 00:09:43.960
So to figure out where they
intersect, we just have

00:09:43.960 --> 00:09:46.660
to set the 2 y's to be
equal to each other.

00:09:46.660 --> 00:09:50.110
So they intersect, the x-values
where they intersect, x

00:09:50.110 --> 00:09:52.660
squared, this y would have
to be equal to that y.

00:09:52.660 --> 00:09:55.940
Or we could just substitute
this in for that y.

00:09:55.940 --> 00:10:02.870
So you get x squared is equal
to minus 1 over 2 x0 times x,

00:10:02.870 --> 00:10:10.180
plus 1/2 plus x0 squared.

00:10:10.180 --> 00:10:11.020
Fair enough.

00:10:11.020 --> 00:10:15.350
And let's put this in a
quadratic equation, or try to

00:10:15.350 --> 00:10:17.670
solve this, so we can apply
the quadratic equation.

00:10:17.670 --> 00:10:20.410
So let's put all of this
stuff on the lefthand side.

00:10:20.410 --> 00:10:29.420
So you get x squared plus
1 over 2 x0 times x minus

00:10:29.420 --> 00:10:35.680
all of this, 1/2 plus x0
squared is equal to 0.

00:10:35.680 --> 00:10:38.100
All I did is, I took all of
this stuff and I put it on the

00:10:38.100 --> 00:10:39.770
lefthand side of the equation.

00:10:39.770 --> 00:10:42.340
Now, this is just a standard
quadratic equation, so we can

00:10:42.340 --> 00:10:47.680
figure out now where the
x-values that satisfy this

00:10:47.680 --> 00:10:52.420
quadratic equation will tell
us where our normal line

00:10:52.420 --> 00:10:54.890
and our parabola intersect.

00:10:54.890 --> 00:10:57.850
So let's just apply the
quadratic equation here.

00:10:57.850 --> 00:11:00.690
So the potential x-values,
where they intersect, x is

00:11:00.690 --> 00:11:02.950
equal to minus b, I'm
just applying the

00:11:02.950 --> 00:11:04.400
quadratic equation.

00:11:04.400 --> 00:11:10.950
So minus b is minus 1 over
2 x0, plus or minus the

00:11:10.950 --> 00:11:14.060
square root of b squared.

00:11:14.060 --> 00:11:15.200
So that's that squared.

00:11:15.200 --> 00:11:22.520
So it's one over four
x0 squared minus 4ac.

00:11:22.520 --> 00:11:26.570
So minus 4 times 1 times
this minus thing.

00:11:26.570 --> 00:11:29.320
So I'm going to have a minus
times a minus is a plus, so

00:11:29.320 --> 00:11:31.370
it's just 4 times this,
because there was one there.

00:11:31.370 --> 00:11:35.720
So plus 4 times
this, right here.

00:11:35.720 --> 00:11:42.940
4 times this is just
2 plus 4 x0 squared.

00:11:42.940 --> 00:11:46.840
All I did is, this
is 4ac right here.

00:11:46.840 --> 00:11:48.190
Well, minus 4ac.

00:11:48.190 --> 00:11:50.295
The minus and the minus
canceled out, so

00:11:50.295 --> 00:11:50.980
you got a plus.

00:11:50.980 --> 00:11:51.820
There's a 1.

00:11:51.820 --> 00:12:00.010
So 4 times c is just
2 plus 4x squared.

00:12:00.010 --> 00:12:03.410
I just multiply this by two,
and of course all of this

00:12:03.410 --> 00:12:07.290
should be over 2 times
a. a is just 2 there.

00:12:07.290 --> 00:12:09.000
So let's see if I
can simplify this.

00:12:09.000 --> 00:12:09.890
Remember what we're doing.

00:12:09.890 --> 00:12:14.740
We're just figuring out where
the normal line and the

00:12:14.740 --> 00:12:16.810
parabola intersect.

00:12:16.810 --> 00:12:18.540
Now, what do we get here.

00:12:18.540 --> 00:12:21.350
This looks like a little
hairy beast here.

00:12:21.350 --> 00:12:24.980
Let me see if I can simplify
this a little bit.

00:12:24.980 --> 00:12:31.310
So let us factor out--
let me rewrite this.

00:12:34.650 --> 00:12:40.290
I can just divide everything by
1/2, so this is minus 1 over 4

00:12:40.290 --> 00:12:47.110
x0, I just divided this by 2,
plus or minus 1/2, that's just

00:12:47.110 --> 00:12:51.720
this 1/2 right there, times the
square root, let me see what

00:12:51.720 --> 00:12:53.560
I can simplify out of here.

00:12:53.560 --> 00:13:03.300
So if I factor out a 4 over
x0 squared, then what does

00:13:03.300 --> 00:13:05.630
my expression become?

00:13:05.630 --> 00:13:12.000
This term right here will
become an x to the fourth, x0

00:13:12.000 --> 00:13:17.140
to the fourth, plus, now,
what does this term become?

00:13:17.140 --> 00:13:25.240
This term becomes
a 1/2 x0 squared.

00:13:25.240 --> 00:13:28.110
And just to verify this,
multiply 4 times 1/2, you

00:13:28.110 --> 00:13:31.110
get 2, and then the x0
squares cancel out.

00:13:31.110 --> 00:13:34.410
So write this term times that,
will equal 2, and then you have

00:13:34.410 --> 00:13:38.280
plus-- now we factored a four
out of this and the x0

00:13:38.280 --> 00:13:40.762
squared, so plus 1/16.

00:13:40.762 --> 00:13:45.060
Let me scroll over
a little bit.

00:13:45.060 --> 00:13:46.840
And you can verify
that this works out.

00:13:46.840 --> 00:13:50.280
If you were to multiply this
out, you should get this

00:13:50.280 --> 00:13:51.340
business right here.

00:13:54.200 --> 00:13:56.680
I see the home stretch here,
because this should actually

00:13:56.680 --> 00:13:58.890
factor out quite neatly.

00:13:58.890 --> 00:14:01.090
So what does this equal?

00:14:01.090 --> 00:14:04.120
So the intersection of
our normal line and our

00:14:04.120 --> 00:14:06.650
parabola is equal to this.

00:14:06.650 --> 00:14:12.510
Minus 1 over 4 x0 plus or
minus 1/2 times the square

00:14:12.510 --> 00:14:13.290
root of this business.

00:14:13.290 --> 00:14:16.350
And the square root, this
thing right here is

00:14:16.350 --> 00:14:19.160
4 over x0 squared.

00:14:19.160 --> 00:14:19.880
Now what's this?

00:14:19.880 --> 00:14:23.680
This is actually, lucky
for us, a perfect square.

00:14:23.680 --> 00:14:25.850
And I won't go into details,
because then the video will get

00:14:25.850 --> 00:14:28.090
too long, but I think you can
recognize that this is

00:14:28.090 --> 00:14:33.580
x0 squared, plus 1/4.

00:14:33.580 --> 00:14:36.410
If you don't believe me,
square this thing right here.

00:14:36.410 --> 00:14:38.480
You'll get this
expression right there.

00:14:38.480 --> 00:14:40.370
And luckily enough, this is
a perfect square, so we

00:14:40.370 --> 00:14:42.650
can actually take the
square root of it.

00:14:42.650 --> 00:14:46.110
And so we get, the point
at which they intersect,

00:14:46.110 --> 00:14:48.730
our normal line and our
parabola, and this is

00:14:48.730 --> 00:14:50.970
quite a hairy problem.

00:14:50.970 --> 00:14:54.960
The points where they intersect
is minus 1 over 4 x0, plus

00:14:54.960 --> 00:14:56.990
or minus 1/2 times the
square root of this.

00:14:56.990 --> 00:14:59.270
The square root of this is the
square root of this, which is

00:14:59.270 --> 00:15:05.810
just 2 over x0 times the square
root of this, which is

00:15:05.810 --> 00:15:09.590
x0 squared plus 1/4.

00:15:09.590 --> 00:15:17.860
And if I were to rewrite all of
this, I'd get minus 1 over 4 x0

00:15:17.860 --> 00:15:21.090
plus, let's see, this 1/2 and
this 2 cancel out, right?

00:15:21.090 --> 00:15:22.810
So these cancel out.

00:15:22.810 --> 00:15:26.650
So plus or minus, now I
just have a one over

00:15:26.650 --> 00:15:27.960
x0 times x0 squared.

00:15:27.960 --> 00:15:35.360
So I have 1 over x0-- oh sorry,
let me, we have to be very

00:15:35.360 --> 00:15:39.475
careful there-- x0 squared
divided by x0 is just x0, let

00:15:39.475 --> 00:15:42.400
me do that in a yellow color so
you know what I'm dealing with.

00:15:42.400 --> 00:15:46.940
This term multiplied by this
term is just x0, and then

00:15:46.940 --> 00:15:51.560
you have a plus 1/4 x0.

00:15:51.560 --> 00:15:53.630
And this is all a
parentheses here.

00:15:53.630 --> 00:15:57.280
So these are the two points at
which the normal curve and

00:15:57.280 --> 00:15:58.610
our parabola intersect.

00:15:58.610 --> 00:15:59.810
Let me just be very clear.

00:15:59.810 --> 00:16:04.230
Those 2 points are, for if
this is my x0 that we're

00:16:04.230 --> 00:16:05.380
dealing with, right there.

00:16:05.380 --> 00:16:07.405
It's this point and this point.

00:16:07.405 --> 00:16:10.070
And we have a plus or minus
here, so this is going to be

00:16:10.070 --> 00:16:12.920
the plus version, and this is
going to be the minus version.

00:16:12.920 --> 00:16:15.680
In fact, the plus version
should simplify into x0.

00:16:15.680 --> 00:16:17.670
Let's see if that's the case.

00:16:17.670 --> 00:16:22.360
Let's see if the plus version
actually simplifies to x0.

00:16:22.360 --> 00:16:23.830
So these are our two points.

00:16:23.830 --> 00:16:26.600
If I take the plus version,
that should be our first

00:16:26.600 --> 00:16:27.770
quadrant intersection.

00:16:27.770 --> 00:16:35.980
So x is equal to minus 1/4
x0 plus x0 plus 1/4 x0.

00:16:35.980 --> 00:16:38.320
And, good enough, it does
actually cancel out.

00:16:38.320 --> 00:16:39.530
That cancels out.

00:16:39.530 --> 00:16:42.140
So x0 is one of the points
of intersection, which

00:16:42.140 --> 00:16:43.260
makes complete sense.

00:16:43.260 --> 00:16:45.940
Because that's how we even
defined the problem.

00:16:45.940 --> 00:16:48.540
But, so this is the first
quadrant intersection.

00:16:48.540 --> 00:16:50.770
So that's the first
quadrant intersection.

00:16:50.770 --> 00:16:53.430
The second quadrant
intersection will be

00:16:53.430 --> 00:16:55.690
where we take the minus
sign right there.

00:16:55.690 --> 00:16:59.890
So x, I'll just call it in the
second quadrant intersection,

00:16:59.890 --> 00:17:06.240
it'd be equal to minus 1/4 x0
minus this stuff over here,

00:17:06.240 --> 00:17:08.020
minus the stuff there.

00:17:08.020 --> 00:17:19.310
So minus x0 minus
1 over 4 mine x0.

00:17:19.310 --> 00:17:20.850
Now what do we have?

00:17:20.850 --> 00:17:21.270
So let's see.

00:17:21.270 --> 00:17:24.150
We have a minus 1 over 4
x0, minus 1 over 4 x0.

00:17:24.150 --> 00:17:33.020
So this is equal to minus x0,
minus x0, minus 1 over 2 x0.

00:17:33.020 --> 00:17:36.760
So if I take minus 1/4 minus
1/4, I get minus 1/2.

00:17:36.760 --> 00:17:39.780
And so my second quadrant
intersection, all this work

00:17:39.780 --> 00:17:41.900
I did got me this result.

00:17:41.900 --> 00:17:43.990
My second quadrant
intersection, I hope I

00:17:43.990 --> 00:17:46.090
don't run out of space.

00:17:46.090 --> 00:17:49.730
My second quadrant
intersection, of the normal

00:17:49.730 --> 00:17:59.730
line and the parabola, is
minus x0 minus 1 over 2 x0.

00:17:59.730 --> 00:18:02.970
Now this by itself is a pretty
neat result we just got, but

00:18:02.970 --> 00:18:05.880
we're unfortunately not
done with the problem.

00:18:05.880 --> 00:18:10.550
Because the problem wants us to
find that point, the maximum

00:18:10.550 --> 00:18:11.620
point of intersection.

00:18:11.620 --> 00:18:13.420
They call this the
extreme normal line.

00:18:13.420 --> 00:18:16.930
The extreme normal line is
when our second quadrant

00:18:16.930 --> 00:18:19.560
intersection essentially
achieves a maximum point.

00:18:19.560 --> 00:18:21.750
I know they call it the
smallest point, but it's the

00:18:21.750 --> 00:18:24.870
smallest negative value, so
it's really a maximum point.

00:18:24.870 --> 00:18:27.440
So how do we figure out
that maximum point?

00:18:27.440 --> 00:18:31.940
Well, we have our second
quadrant intersection as

00:18:31.940 --> 00:18:35.460
a function of our
first quadrant x.

00:18:35.460 --> 00:18:38.950
I could rewrite this as, my
second quadrant intersection as

00:18:38.950 --> 00:18:45.210
a function of x0 is equal to
minus x minus 1 over 2 x0.

00:18:45.210 --> 00:18:48.270
So this is going to reach a
minimum or a maximum point when

00:18:48.270 --> 00:18:50.790
its derivative is equal to 0.

00:18:50.790 --> 00:18:52.910
This is a very unconventional
notation, and that's

00:18:52.910 --> 00:18:54.530
probably the hardest
thing about this problem.

00:18:54.530 --> 00:18:58.390
But let's take this derivative
with respect to x0.

00:18:58.390 --> 00:19:01.470
So my second quadrant
intersection, the derivative of

00:19:01.470 --> 00:19:05.910
that with respect to x0, is
equal to, this is pretty

00:19:05.910 --> 00:19:06.580
straightforward.

00:19:06.580 --> 00:19:12.860
It's equal to minus 1, and then
I have a minus 1/2 times, this

00:19:12.860 --> 00:19:14.800
is the same thing as
x to the minus 1.

00:19:14.800 --> 00:19:20.630
So it's minus 1 times x0
to the minus 2, right?

00:19:20.630 --> 00:19:23.320
I could have rewritten
this as minus 1/2 times

00:19:23.320 --> 00:19:24.850
x0 to the minus 1.

00:19:24.850 --> 00:19:26.560
So you just put its
exponent out front and

00:19:26.560 --> 00:19:28.090
decrement it by 1.

00:19:28.090 --> 00:19:31.870
And so this is the derivative
with respect to my first

00:19:31.870 --> 00:19:33.770
quadrant intersection.

00:19:33.770 --> 00:19:35.760
So let me simplify this.

00:19:35.760 --> 00:19:41.820
So x, my second quadrant
intersection, the derivative of

00:19:41.820 --> 00:19:45.670
it with respect to my first
quadrant intersection, is equal

00:19:45.670 --> 00:19:52.110
to minus 1, the minus 1/2 and
the minus 1 become a positive

00:19:52.110 --> 00:19:58.190
when you multiply them, and so
plus 1/2 over x0 squared.

00:19:58.190 --> 00:20:01.280
Now, this'll reach a maximum
or minimum when it equals 0.

00:20:01.280 --> 00:20:06.600
So let's set that equal to
0, and then solve this

00:20:06.600 --> 00:20:08.410
problem right there.

00:20:08.410 --> 00:20:10.000
Well, we add one to both sides.

00:20:10.000 --> 00:20:16.180
We get 1 over 2 x0 squared is
equal to 1, or you could just

00:20:16.180 --> 00:20:20.510
say that that means that 2 x0
squared must be equal to

00:20:20.510 --> 00:20:23.750
1, if we just invert both
sides of this equation.

00:20:23.750 --> 00:20:32.740
Or we could say that x0 squared
is equal to 1/2, or if we take

00:20:32.740 --> 00:20:35.780
the square roots of both sides
of that equation, we get x0 is

00:20:35.780 --> 00:20:40.030
equal to 1 over the
square root of 2.

00:20:40.030 --> 00:20:42.470
So we're really, really,
really close now.

00:20:42.470 --> 00:20:46.000
We've just figured out the
x0 value that gives us

00:20:46.000 --> 00:20:47.530
our extreme normal line.

00:20:47.530 --> 00:20:48.910
This value right here.

00:20:48.910 --> 00:20:51.080
Let me do it in a
nice deeper color.

00:20:51.080 --> 00:20:53.640
This value right here, that
gives us the extreme normal

00:20:53.640 --> 00:20:57.410
line, that over there is
x0 is equal to 1 over

00:20:57.410 --> 00:20:58.930
the square root of 2.

00:20:58.930 --> 00:21:01.390
Now, they want us to figure
out the equation of the

00:21:01.390 --> 00:21:02.910
extreme normal line.

00:21:02.910 --> 00:21:05.710
Well, the equation of the
extreme normal line we already

00:21:05.710 --> 00:21:06.670
figured out right here.

00:21:06.670 --> 00:21:07.910
It's this.

00:21:07.910 --> 00:21:11.900
The equation of the normal line
is that thing, right there.

00:21:11.900 --> 00:21:15.400
So if we want the equation of
the normal line at this extreme

00:21:15.400 --> 00:21:18.120
point, right here, the one that
creates the extreme normal

00:21:18.120 --> 00:21:21.140
line, I just substitute 1 over
the square root of 2 in for x0.

00:21:21.140 --> 00:21:22.400
So what do I get?

00:21:22.400 --> 00:21:26.800
I get, and this is the home
stretch, and this is quite

00:21:26.800 --> 00:21:28.440
a beast of a problem.

00:21:28.440 --> 00:21:31.670
y minus x0 squared.

00:21:31.670 --> 00:21:34.290
x0 squared is 1/2, right?

00:21:34.290 --> 00:21:36.430
1 over the square root
of 2 squared is 1/2.

00:21:36.430 --> 00:21:42.510
Is equal to minus 1 over 2 x0.

00:21:42.510 --> 00:21:43.660
So let's be careful here.

00:21:43.660 --> 00:21:48.560
So minus 1/2 times 1 over x0.

00:21:48.560 --> 00:21:52.670
One over x0 is the square
root of 2, right?

00:21:52.670 --> 00:21:57.530
All of that times x minus x0.

00:21:57.530 --> 00:22:02.080
So that's 1 over the square
root of 2. x0 is one

00:22:02.080 --> 00:22:02.870
over square root of 2.

00:22:02.870 --> 00:22:04.850
So let's simplify
this a little bit.

00:22:04.850 --> 00:22:07.570
So the equation of our normal
line, assuming I haven't made

00:22:07.570 --> 00:22:13.400
any careless mistakes, is
equal to, so y minus 1/2

00:22:13.400 --> 00:22:15.060
is equal to, let's see.

00:22:15.060 --> 00:22:21.410
If we multiply this minus
square root of 2 over 2x, and

00:22:21.410 --> 00:22:25.310
then if I multiply these
square root of 2 over

00:22:25.310 --> 00:22:26.690
this, it becomes one.

00:22:26.690 --> 00:22:29.120
And then I have a minus
and a minus, so that

00:22:29.120 --> 00:22:31.290
I have a plus 1/2.

00:22:31.290 --> 00:22:32.230
I think that's right.

00:22:32.230 --> 00:22:37.590
Yeah, plus 1/2, this
times this times that

00:22:37.590 --> 00:22:39.170
is equal to plus 1/2.

00:22:39.170 --> 00:22:41.190
And then, we're at
the home stretch.

00:22:41.190 --> 00:22:44.650
So we just add 1/2 to both
sides of this equation, and we

00:22:44.650 --> 00:22:51.150
get our extreme normal line
equation, which is y is

00:22:51.150 --> 00:22:54.670
equal to minus square
root of 2 over 2x.

00:22:54.670 --> 00:23:00.100
If you add 1/2 to both sides of
this equation, you get plus 1.

00:23:00.100 --> 00:23:01.300
And there you go.

00:23:01.300 --> 00:23:04.830
That's the equation of that
line there, assuming I haven't

00:23:04.830 --> 00:23:06.020
made any careless mistakes.

00:23:06.020 --> 00:23:09.320
But even if I have, I think you
get the idea of hopefully how

00:23:09.320 --> 00:23:12.710
to do this problem, which
is quite a beastly one.


Trig Implicit Differentiation Example

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:04.050
I've been asked implicitly
differentiate the equation

00:00:04.050 --> 00:00:10.390
tangent of x over y is
equal to x plus y.

00:00:10.390 --> 00:00:14.150
And I've done several implicit
differentiation videos, but

00:00:14.150 --> 00:00:17.440
this tends to be one of the
biggest sources of pain for

00:00:17.440 --> 00:00:18.720
first year calculus students.

00:00:18.720 --> 00:00:21.040
So I thought I would give
at least another example.

00:00:21.040 --> 00:00:22.860
It never hurts to see
as many as possible.

00:00:22.860 --> 00:00:24.290
So let's do this one.

00:00:24.290 --> 00:00:26.680
So to implicitly differentiate
this, we just apply the

00:00:26.680 --> 00:00:29.363
derivative with respect to x
operator to both sides

00:00:29.363 --> 00:00:29.970
of the equation.

00:00:29.970 --> 00:00:33.290
The derivative with this
respect to x -- the derivative

00:00:33.290 --> 00:00:35.420
of the left side with respect
to x is the same as the

00:00:35.420 --> 00:00:40.580
derivative of the right
side with respect to x.

00:00:40.580 --> 00:00:42.790
The right side's going to be
very straightforward, but

00:00:42.790 --> 00:00:44.770
the left side is a
little bit tricky.

00:00:44.770 --> 00:00:47.380
So let's do that on
the side over here.

00:00:47.380 --> 00:00:52.020
Let me write the left hand side
a little bit differently.

00:00:52.020 --> 00:00:52.990
I'm going to do it in
a different color.

00:00:52.990 --> 00:01:00.410
Let me say that a is equal
to the tangent of b.

00:01:00.410 --> 00:01:09.380
And let me say that b
is equal to x over y.

00:01:09.380 --> 00:01:11.620
Then a is clearly
the same thing.

00:01:11.620 --> 00:01:14.860
I mean if I just substituted
b back in here, a, this

00:01:14.860 --> 00:01:18.090
whole thing I could
re-write as just a.

00:01:18.090 --> 00:01:20.930
So if we're taking the
derivative of a with respect

00:01:20.930 --> 00:01:23.740
to x, that's what we
want to do right here.

00:01:23.740 --> 00:01:26.570
Let me just take the derivative
of both sides of this.

00:01:26.570 --> 00:01:36.500
This would be the derivative of
a with respect to x is equal to

00:01:36.500 --> 00:01:38.610
the derivative of x
with respect to x.

00:01:38.610 --> 00:01:41.210
Well that's pretty
straightforward, that's just 1.

00:01:41.210 --> 00:01:44.390
Plus the derivative of
y with respect to x.

00:01:44.390 --> 00:01:45.430
So let me write it like this.

00:01:45.430 --> 00:01:48.820
I'll write the derivative
operator, the derivative

00:01:48.820 --> 00:01:53.770
oh y with respect to x.

00:01:53.770 --> 00:01:54.350
That's all we did.

00:01:54.350 --> 00:01:56.520
We just applied the derivative
operator to y, and we don't

00:01:56.520 --> 00:01:58.650
know what this thing is,
we're going to solve for it.

00:01:58.650 --> 00:02:01.180
But obviously, I can't just
leave this here, the derivative

00:02:01.180 --> 00:02:02.360
of a with respect to x.

00:02:02.360 --> 00:02:04.610
We just solved for a, and
a is just this thing

00:02:04.610 --> 00:02:05.930
right here, right?

00:02:05.930 --> 00:02:09.450
a is tangent of b, and
b is just y over x.

00:02:09.450 --> 00:02:11.730
The reason why I wrote it this
way is because I wanted to show

00:02:11.730 --> 00:02:14.870
you that when you take the
derivative of this, it just

00:02:14.870 --> 00:02:16.500
comes out of the chain rule.

00:02:16.500 --> 00:02:18.840
It's not some type of new
voodoo magic that you

00:02:18.840 --> 00:02:20.090
haven't learned yet.

00:02:20.090 --> 00:02:22.200
So the derivative -- let
me just write down the

00:02:22.200 --> 00:02:23.990
chain rule right here.

00:02:23.990 --> 00:02:30.930
The derivative of a with
respect to x is equal to the

00:02:30.930 --> 00:02:35.280
derivative of a with respect
to b times the derivative

00:02:35.280 --> 00:02:37.580
of b with respect to x.

00:02:37.580 --> 00:02:39.720
That's just the chain rule and
it's very easy to remember,

00:02:39.720 --> 00:02:43.040
because the db's cancel out and
you're just left with the

00:02:43.040 --> 00:02:45.800
derivative of a with respect to
x, if you just treated these

00:02:45.800 --> 00:02:47.470
like regular fractions.

00:02:47.470 --> 00:02:50.275
So what's the derivative
of a with respect to b?

00:02:55.020 --> 00:03:01.570
Well, that's just 1 over
cosine squared of b.

00:03:01.570 --> 00:03:03.570
And if you don't have that
memorized, it's actually not

00:03:03.570 --> 00:03:07.400
too hard to prove to yourself
if you just write this as sine

00:03:07.400 --> 00:03:10.670
of b over cosine of b, but this
tends to be one of the trig

00:03:10.670 --> 00:03:12.130
derivatives that most
people memorize.

00:03:12.130 --> 00:03:14.230
I think I've already made a
video where I proved this.

00:03:14.230 --> 00:03:16.840
And some books still write this
as secant squared of b, but we

00:03:16.840 --> 00:03:19.070
know that secant squared is the
same thing as 1 over

00:03:19.070 --> 00:03:20.340
cosine squared.

00:03:20.340 --> 00:03:25.320
I like to keep it in kind of
the fundamental trig functions,

00:03:25.320 --> 00:03:27.360
or trig ratios as opposed to
things like secant

00:03:27.360 --> 00:03:28.490
and cosecant.

00:03:28.490 --> 00:03:31.090
Then what's the derivative
of b with respect to x?

00:03:37.030 --> 00:03:38.260
So this is pretty interesting.

00:03:38.260 --> 00:03:39.710
Let me re-write b, actually.

00:03:39.710 --> 00:03:45.730
Let me write b is equal to
x times y to the minus 1.

00:03:45.730 --> 00:03:48.520
So the derivative of b with
respect to x, we could do a

00:03:48.520 --> 00:03:50.470
little bit of chain
rule right here.

00:03:50.470 --> 00:03:53.680
We could say -- let me write
this -- the derivative of b

00:03:53.680 --> 00:03:57.530
with respect to x is equal to
the derivative of x times

00:03:57.530 --> 00:03:58.790
y to the negative 1.

00:03:58.790 --> 00:04:01.300
So the derivative of x is 1.

00:04:01.300 --> 00:04:07.360
times y to the negative 1 plus
the derivative of y -- so

00:04:07.360 --> 00:04:08.030
let me just write this.

00:04:08.030 --> 00:04:12.320
Plus the derivative with
respect to x of y to

00:04:12.320 --> 00:04:17.930
the minus 1 times the
first term, times x.

00:04:17.930 --> 00:04:20.470
So this thing right here, and
clearly I haven't completely

00:04:20.470 --> 00:04:21.190
simplified it yet.

00:04:21.190 --> 00:04:22.890
I still have to figure out
what this thing is here.

00:04:22.890 --> 00:04:25.010
But I just simply applied
the product rule here.

00:04:25.010 --> 00:04:27.990
Your derivative of the first
term, derivative of x is 1

00:04:27.990 --> 00:04:30.380
times the second term plus
the derivative of the second

00:04:30.380 --> 00:04:31.310
term times the first term.

00:04:31.310 --> 00:04:32.700
That's all I did there.

00:04:32.700 --> 00:04:35.170
So the derivative of b
with respect to x is just

00:04:35.170 --> 00:04:36.560
this thing right there.

00:04:36.560 --> 00:04:42.290
So it equals -- let me do it in
the yellow -- so it's times --

00:04:42.290 --> 00:04:43.520
oh, I'll do it in the blue
since I already wrote it.

00:04:43.520 --> 00:04:47.290
This is the blue, derivative of
b with respect to x is y to the

00:04:47.290 --> 00:04:52.580
minus 1, or 1 over y plus the
derivative with respect to

00:04:52.580 --> 00:04:59.590
x of 1 over y times x.

00:04:59.590 --> 00:05:01.180
So let me write that down here.

00:05:01.180 --> 00:05:04.330
So we just figured out, or
we're almost done figuring

00:05:04.330 --> 00:05:07.400
out, what the derivative of a
with respect to x is, and we

00:05:07.400 --> 00:05:08.450
could throw that in there.

00:05:08.450 --> 00:05:09.230
But we're not done.

00:05:09.230 --> 00:05:12.280
What's the derivative of 1
over y with respect to x?

00:05:12.280 --> 00:05:14.990
Well, do the chain rule again.

00:05:17.520 --> 00:05:18.830
And I want to be very
clear with this.

00:05:18.830 --> 00:05:21.570
I know this might seem a little
bit cumbersome what I'm doing

00:05:21.570 --> 00:05:24.020
here, but I think it might
make a little bit of sense.

00:05:24.020 --> 00:05:28.390
Let me just set c is
equal to 1 over y.

00:05:28.390 --> 00:05:32.550
So the derivative of c with
respect to x, just from the

00:05:32.550 --> 00:05:35.580
chain rule, is equal to the
derivative of c with respect

00:05:35.580 --> 00:05:40.090
to y times the derivative
of y with respect to x.

00:05:40.090 --> 00:05:43.140
What's the derivative of
c with respect to y?

00:05:43.140 --> 00:05:44.930
Well this is the same thing
as -- I could re-write

00:05:44.930 --> 00:05:46.350
this as y to the minus 1.

00:05:46.350 --> 00:05:51.160
So it's minus y to
the minus 2 power.

00:05:51.160 --> 00:05:52.910
That's what this thing is.

00:05:52.910 --> 00:05:55.740
This thing is that right there.

00:05:55.740 --> 00:05:57.220
And I don't know what the
derivative of y with

00:05:57.220 --> 00:05:58.020
respect to x is.

00:05:58.020 --> 00:05:59.690
That's what we're
trying to solve for.

00:05:59.690 --> 00:06:02.390
So it's that times
the derivative of y

00:06:02.390 --> 00:06:03.540
with respect to x.

00:06:03.540 --> 00:06:05.340
That just comes out
of the chain rule.

00:06:05.340 --> 00:06:11.400
So this thing right here, this
is the derivative of this thing

00:06:11.400 --> 00:06:13.830
with respect to x, which is the
same thing as derivative

00:06:13.830 --> 00:06:15.770
of c with respect to x.

00:06:15.770 --> 00:06:19.210
So I can write this little
piece right here, I can

00:06:19.210 --> 00:06:25.240
re-write this little piece as
minus y to the minus 2 dy

00:06:25.240 --> 00:06:28.910
dx, and then, of course,
that there is times x.

00:06:28.910 --> 00:06:33.910
And then we had the plus 1 over
y, and all of that was times

00:06:33.910 --> 00:06:38.050
the 1 over cosine squared of b.

00:06:38.050 --> 00:06:40.660
So now we've simplified
this a good bit.

00:06:40.660 --> 00:06:42.840
I hope going into the chain
rule didn't confuse you,

00:06:42.840 --> 00:06:45.020
because I really want to hit
the point home that all of

00:06:45.020 --> 00:06:48.320
these implicit differentiation
problems, these dy dx's just

00:06:48.320 --> 00:06:50.570
don't, it's not some rule
that you should memorize.

00:06:50.570 --> 00:06:52.890
They come out naturally
from the chain rule.

00:06:52.890 --> 00:06:56.930
So we solved da dx,
that is equal to this

00:06:56.930 --> 00:06:59.230
expression right here.

00:06:59.230 --> 00:07:07.130
Let me write it, it's equal to
1 over cosine squared of b.

00:07:07.130 --> 00:07:07.880
Well what's b?

00:07:07.880 --> 00:07:10.640
I wrote it's cos x over y.

00:07:10.640 --> 00:07:16.920
Cosine squared of x over y
times all of this stuff over

00:07:16.920 --> 00:07:19.840
here, times all of this mess.

00:07:19.840 --> 00:07:25.670
1 over y plus, or maybe I
should say minus, minus -- if I

00:07:25.670 --> 00:07:32.486
just simplify this, this is x
over y squared times dy dx.

00:07:36.660 --> 00:07:39.000
Then that is equal to
the right hand side.

00:07:39.000 --> 00:07:48.490
It is equal to 1 plus dy dx.

00:07:48.490 --> 00:07:51.420
And now all we have to
do is solve for dy dx.

00:07:51.420 --> 00:07:53.990
So let me just review
how we got here.

00:07:53.990 --> 00:07:56.300
I went through the chain rule
at every step of the way, but

00:07:56.300 --> 00:07:58.170
once you get the hang of it,
you can literally just go

00:07:58.170 --> 00:07:59.360
straight down this way.

00:07:59.360 --> 00:08:01.380
The way you think about
it is -- the right hand

00:08:01.380 --> 00:08:02.033
side I think you get it.

00:08:02.033 --> 00:08:04.380
The derivative of x is 1, the
derivative of y with respect

00:08:04.380 --> 00:08:06.560
to x, well that's just dy dx.

00:08:06.560 --> 00:08:09.010
But the left hand side, you
take the derivative of

00:08:09.010 --> 00:08:11.630
the whole thing with
respect to x over y.

00:08:11.630 --> 00:08:14.100
So that's just the derivative
of tangent is 1 over

00:08:14.100 --> 00:08:15.020
cosine squared.

00:08:15.020 --> 00:08:18.620
So it's 1 over cosine squared
of x over y, and you multiply

00:08:18.620 --> 00:08:23.530
that times the derivative of
x over y with respect to x.

00:08:23.530 --> 00:08:26.770
And the derivative of x over
y with respect to x is the

00:08:26.770 --> 00:08:28.970
derivative of-- and it gets
complicated, that's why it's

00:08:28.970 --> 00:08:31.590
good to do it on the side here
-- but it's the derivative of

00:08:31.590 --> 00:08:34.150
x, which is 1 times 1 over y.

00:08:34.150 --> 00:08:39.680
Which is that term plus the
derivative of 1 over Y with

00:08:39.680 --> 00:08:44.200
respect to X, which is minus
1 over y squared dy dx, from

00:08:44.200 --> 00:08:46.620
the chain rule, times dx.

00:08:46.620 --> 00:08:48.140
That's why it was good to do
over to the side so we don't

00:08:48.140 --> 00:08:49.360
make a careless mistake.

00:08:49.360 --> 00:08:51.410
But once you get used to it you
could actually do that in your

00:08:51.410 --> 00:08:53.980
head, and of course, that
equals the right hand side.

00:08:53.980 --> 00:08:56.520
So from here on out it's
just pure algebra.

00:08:56.520 --> 00:08:59.140
Just to solve for our dy dx.

00:08:59.140 --> 00:09:01.490
So a good place to start is to
multiply both sides of this

00:09:01.490 --> 00:09:04.910
equation times cosine
squared of x over y.

00:09:04.910 --> 00:09:07.420
So obviously, that'll
turn to 1 on this side.

00:09:07.420 --> 00:09:14.970
And the left hand side will be
1 over y minus x over y squared

00:09:14.970 --> 00:09:23.690
dy dx is equal to -- I have to
multiply both side of the

00:09:23.690 --> 00:09:26.730
equation times this denominator
right here -- is equal to

00:09:26.730 --> 00:09:32.530
cosine squared of x over y plus
cosine squared of

00:09:32.530 --> 00:09:35.190
x over y dy dx.

00:09:39.420 --> 00:09:40.190
Now what can we do.

00:09:40.190 --> 00:09:44.210
We can subtract this cosine
squared of x over y from both

00:09:44.210 --> 00:09:52.110
sides of the equation, and we
get 1 over y minus cosine

00:09:52.110 --> 00:09:53.710
squared of x over y.

00:09:53.710 --> 00:09:55.780
All I did is I subtracted
this from both sides of the

00:09:55.780 --> 00:09:57.590
equation, so essentially I
moved it over to the

00:09:57.590 --> 00:09:59.040
left hand side.

00:09:59.040 --> 00:10:01.040
What I'm trying to do is I'm
going to try to separate the

00:10:01.040 --> 00:10:04.810
non dy dx terms from
the dy dx terms.

00:10:04.810 --> 00:10:06.750
So I want to bring this
dy dx term over to

00:10:06.750 --> 00:10:07.950
the right hand side.

00:10:07.950 --> 00:10:11.550
So let me add x over y
squared dy dx to both sides.

00:10:11.550 --> 00:10:17.260
So then that is equal to x over
y -- let me write that in the

00:10:17.260 --> 00:10:21.070
color that I originally wrote
it in, a slightly

00:10:21.070 --> 00:10:21.470
different color.

00:10:21.470 --> 00:10:27.110
So that is x over y squared --
I'll write the dy dx in orange.

00:10:27.110 --> 00:10:34.120
dy dx, and then you have this
term, plus cosine squared

00:10:34.120 --> 00:10:36.880
of x over y dy dx.

00:10:40.950 --> 00:10:43.000
I think we're in
the home stretch.

00:10:43.000 --> 00:10:46.410
Let's factor this dy dx out
from the right hand side.

00:10:46.410 --> 00:10:56.770
So this is equal to dy dx
times x over y squared plus

00:10:56.770 --> 00:11:01.220
cosine squared of x over y.

00:11:01.220 --> 00:11:04.180
And that is equal to this thing
over here, it's equal to

00:11:04.180 --> 00:11:09.250
1 over y minus cosine
squared of x over y.

00:11:09.250 --> 00:11:12.240
Now to solve for dy dx, we just
have to divide both sides of

00:11:12.240 --> 00:11:15.450
this equation by this
expression right here.

00:11:15.450 --> 00:11:16.900
And then what do we get?

00:11:16.900 --> 00:11:21.970
We get, if we just divide both
sides by that, we get 1 over y

00:11:21.970 --> 00:11:27.210
minus cosine squared of x over
y divided by this whole

00:11:27.210 --> 00:11:28.720
business right there.

00:11:28.720 --> 00:11:36.190
x over y squared plus cosine
squared of x over y is

00:11:36.190 --> 00:11:42.150
equal to our dy dx.

00:11:42.150 --> 00:11:43.370
And then we're done.

00:11:43.370 --> 00:11:46.460
We just applied the chain rule
multiple times and we were able

00:11:46.460 --> 00:11:50.600
to implicitly differentiate
tangent of y over x is

00:11:50.600 --> 00:11:51.600
equal to x plus y.

00:11:51.600 --> 00:11:55.980
The hard part really is getting
to this step right here.

00:11:55.980 --> 00:11:59.470
After this step it's literally
just pure algebra just to solve

00:11:59.470 --> 00:12:04.730
for the dy dx's, and then you
get that answer right there.

00:12:04.730 --> 00:12:07.380
Anyway, hopefully you
found that useful.


Dot and cross product comparison/intuition

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:04.610
We've known for several videos
now that the dot product of

00:00:04.610 --> 00:00:11.420
two nonzero vectors, a and b,
is equal to the length of

00:00:11.420 --> 00:00:17.540
vector a times the length of
vector b times the cosine of

00:00:17.540 --> 00:00:20.820
the angle between them.

00:00:20.820 --> 00:00:22.620
Let me draw a and b just
to make it clear.

00:00:22.620 --> 00:00:27.900
If that's my vector a and
that's my vector b right

00:00:27.900 --> 00:00:31.600
there, the angle between
them is this angle.

00:00:31.600 --> 00:00:33.430
And we defined it in this way.

00:00:33.430 --> 00:00:35.390
And actually, if you ever want
to solve-- if you have two

00:00:35.390 --> 00:00:37.260
vectors and you want to solve
for that angle, and I've never

00:00:37.260 --> 00:00:38.470
done this before explicitly.

00:00:38.470 --> 00:00:41.040
And I thought, well, I might
as well do it right now.

00:00:41.040 --> 00:00:42.760
You could just solve
for your theta.

00:00:42.760 --> 00:00:51.460
So it would be a dot b divided
by the lengths of your two

00:00:51.460 --> 00:00:54.300
vectors multiplied by each
other is equal to

00:00:54.300 --> 00:00:55.800
the cosine of theta.

00:00:55.800 --> 00:00:58.840
And then to solve for theta
you would have to take the

00:00:58.840 --> 00:01:01.880
inverse cosine of both sides,
or the arc cosine of both

00:01:01.880 --> 00:01:08.840
sides, and you would get theta
is equal to arc cosine of a

00:01:08.840 --> 00:01:15.450
dot b over the magnitudes or
the lengths of the products

00:01:15.450 --> 00:01:18.180
of, or the lengths of each of
those vectors, multiplied by

00:01:18.180 --> 00:01:19.280
each other.

00:01:19.280 --> 00:01:22.170
So if I give you two arbitrary
vectors-- and the neat thing

00:01:22.170 --> 00:01:24.210
about it is, this might be
pretty straightforward.

00:01:24.210 --> 00:01:26.440
If I just drew something in two
dimensions right here, you

00:01:26.440 --> 00:01:28.500
could just take your protractor
out and measure

00:01:28.500 --> 00:01:29.320
this angle.

00:01:29.320 --> 00:01:32.440
But if a and b each have a
hundred components, it becomes

00:01:32.440 --> 00:01:35.320
hard to visualize the
idea of an angle

00:01:35.320 --> 00:01:36.890
between the two vectors.

00:01:36.890 --> 00:01:39.110
But you don't need to visualize
them anymore because

00:01:39.110 --> 00:01:41.840
you just have to calculate
this thing right here.

00:01:41.840 --> 00:01:46.130
You just have to calculate
this value right there.

00:01:46.130 --> 00:01:49.760
And then go to your calculator
and then type in arc cosine,

00:01:49.760 --> 00:01:53.140
or the inverse cosine that are
the equivalent functions, and

00:01:53.140 --> 00:01:54.440
it'll give you an angle.

00:01:54.440 --> 00:01:57.520
And that, by definition, is the
angle between those two

00:01:57.520 --> 00:01:59.120
vectors, which is a
very neat concept.

00:01:59.120 --> 00:02:00.970
And then you can start
addressing issues of

00:02:00.970 --> 00:02:03.500
perpendicularity and
whatever else.

00:02:03.500 --> 00:02:04.620
This was a bit of a tangent.

00:02:04.620 --> 00:02:08.300
But the other outcome that I
painstakingly proved to you in

00:02:08.300 --> 00:02:14.430
the previous video was that
the length of the cross

00:02:14.430 --> 00:02:18.270
product of two vectors is equal
to-- it's a very similar

00:02:18.270 --> 00:02:18.810
expression.

00:02:18.810 --> 00:02:23.680
It's equal to the product of the
two vectors' lengths, so

00:02:23.680 --> 00:02:27.280
the length of a times the length
of b times the sine of

00:02:27.280 --> 00:02:28.930
the angle between them.

00:02:28.930 --> 00:02:31.290
Times the sign of the
angle between them.

00:02:31.290 --> 00:02:33.120
So it's the same angle.

00:02:33.120 --> 00:02:36.820
So what I want to do is take
these two ideas and this was a

00:02:36.820 --> 00:02:39.950
bit of a diversion there just
to kind of show you how to

00:02:39.950 --> 00:02:41.900
solve for theta because I
realized I've never done that

00:02:41.900 --> 00:02:42.510
for you before.

00:02:42.510 --> 00:02:45.260
But what I want to do is I want
to take this expression

00:02:45.260 --> 00:02:50.800
up here and this expression up
here and see if we can develop

00:02:50.800 --> 00:02:54.590
an intuition, at least in R3
because right now we've only

00:02:54.590 --> 00:02:56.700
defined our cross product.

00:02:56.700 --> 00:03:00.580
Or the cross product of two
vectors is only defined in R3.

00:03:00.580 --> 00:03:03.780
Let's take these two ideas in
R3 and see if we can develop

00:03:03.780 --> 00:03:04.340
an intuition.

00:03:04.340 --> 00:03:07.940
And I've done a very similar
video in the physics playlist

00:03:07.940 --> 00:03:10.920
where I compare the dot product
to the cross product.

00:03:10.920 --> 00:03:14.400
Now, if I'm talking about--
let me redraw my vectors.

00:03:14.400 --> 00:03:19.820
So the length of a--
so let me draw a.

00:03:19.820 --> 00:03:22.420
b, I want to do it
bigger than that.

00:03:22.420 --> 00:03:25.090
So let me do it like that.

00:03:25.090 --> 00:03:28.540
So that is my vector b.

00:03:28.540 --> 00:03:29.910
So this is b.

00:03:29.910 --> 00:03:31.070
That is a.

00:03:31.070 --> 00:03:34.200
What is the length of a times
the length of b times the

00:03:34.200 --> 00:03:35.850
cosine of the angle?

00:03:35.850 --> 00:03:38.080
So let me do that right there.

00:03:38.080 --> 00:03:40.000
So this is the angle.

00:03:40.000 --> 00:03:42.970
So the length of a if I were to
draw these vectors is this

00:03:42.970 --> 00:03:43.760
length right here.

00:03:43.760 --> 00:03:45.390
This is the length of a.

00:03:45.390 --> 00:03:47.230
It's this distance right
here, the way

00:03:47.230 --> 00:03:48.330
I've drawn this vector.

00:03:48.330 --> 00:03:51.680
So this is, literally, the
length of vector a.

00:03:51.680 --> 00:03:54.920
And I'm doing it in R3 or maybe
a version of it that I

00:03:54.920 --> 00:03:58.160
can fit onto my little
blackboard right here.

00:03:58.160 --> 00:04:01.150
So it'll just be the length
of this line right there.

00:04:01.150 --> 00:04:03.080
And then the length of
b is the length of

00:04:03.080 --> 00:04:06.180
that line right there.

00:04:06.180 --> 00:04:08.910
So that is the length of b.

00:04:11.530 --> 00:04:13.650
Let me rewrite this
thing up here.

00:04:13.650 --> 00:04:19.790
Let me write it as b, the length
of b times the length--

00:04:19.790 --> 00:04:20.570
and I want to be careful.

00:04:20.570 --> 00:04:22.710
I don't want to do the dot there
because you'll think

00:04:22.710 --> 00:04:23.640
it's a dot product.

00:04:23.640 --> 00:04:26.560
Times a cosine of theta.

00:04:26.560 --> 00:04:28.630
All I did is I rearranged
this thing here.

00:04:28.630 --> 00:04:31.340
It's the same thing
as a dot b.

00:04:31.340 --> 00:04:35.460
Well what is a times the
cosine of theta?

00:04:35.460 --> 00:04:37.640
Let's get out our basic
trigonometry

00:04:37.640 --> 00:04:39.370
tools-- SOH CAH TOA.

00:04:45.160 --> 00:04:52.270
Cosine of theta is equal to
adjacent over hypotenuse.

00:04:52.270 --> 00:04:55.680
So if I drop, if I create a
right triangle here, and let

00:04:55.680 --> 00:04:58.620
me introduce some new colors
just to ease the monotony.

00:04:58.620 --> 00:05:01.970
If I drop a right triangle
right here and I create a

00:05:01.970 --> 00:05:05.490
right triangle right there, and
this is theta, than what

00:05:05.490 --> 00:05:08.650
is the cosine of theta?

00:05:08.650 --> 00:05:10.230
It's equal to this.

00:05:10.230 --> 00:05:11.490
Let me do it another color.

00:05:11.490 --> 00:05:15.130
It's equal to this,
the adjacent side.

00:05:15.130 --> 00:05:16.820
It's equal to this little
magenta thing.

00:05:16.820 --> 00:05:18.770
Not all of b, just this
part that goes

00:05:18.770 --> 00:05:20.290
up to my right angle.

00:05:20.290 --> 00:05:21.570
That's my adjacent.

00:05:21.570 --> 00:05:23.950
I want to do it a little
bit bigger.

00:05:23.950 --> 00:05:29.020
It's equal to the adjacent
side over the hypotenuse.

00:05:29.020 --> 00:05:30.250
So let me write this down.

00:05:30.250 --> 00:05:35.850
So cosine of theta is equal to
this little adjacent side.

00:05:35.850 --> 00:05:37.020
I'm just going to write
it like that.

00:05:37.020 --> 00:05:44.880
Is equal to this adjacent side
over the hypotenuse.

00:05:44.880 --> 00:05:46.640
But what is the hypotenuse?

00:05:46.640 --> 00:05:48.970
It is the length of vector a.

00:05:48.970 --> 00:05:49.620
It's this.

00:05:49.620 --> 00:05:51.210
That's my hypotenuse
right there.

00:05:51.210 --> 00:05:55.430
So my hypotenuse is the
length of vector a.

00:05:55.430 --> 00:05:58.040
And so if I multiply both sides
by the length of vector

00:05:58.040 --> 00:06:06.660
a I get the length of vector a
times the cosine of theta is

00:06:06.660 --> 00:06:09.520
equal to the adjacent side.

00:06:09.520 --> 00:06:10.770
I'll do that in magenta.

00:06:13.300 --> 00:06:18.910
So this expression right here,
which was just a dot b can be

00:06:18.910 --> 00:06:22.790
rewritten as-- I just told you
that the length of vector a

00:06:22.790 --> 00:06:25.210
times cosine of theta is equal
to this little magenta

00:06:25.210 --> 00:06:25.970
adjacent side.

00:06:25.970 --> 00:06:29.500
So this is equal to
the adjacent side.

00:06:29.500 --> 00:06:38.420
So you can view a dot b as being
equal to the length of

00:06:38.420 --> 00:06:44.350
vector b-- that length-- times
that adjacent side.

00:06:44.350 --> 00:06:45.910
And you're saying, Sal, what
does that do for me?

00:06:45.910 --> 00:06:48.360
Well what it tells you
is you're multiplying

00:06:48.360 --> 00:06:52.840
essentially, the length of
vector b times the amount of

00:06:52.840 --> 00:06:56.830
vector a that's going in the
same direction as vector b.

00:06:56.830 --> 00:07:00.190
You can kind of view this as
the shadow of vector a.

00:07:00.190 --> 00:07:02.420
And I'll talk about projections
in the future.

00:07:02.420 --> 00:07:04.930
And I'll more formally define
them, but if the word

00:07:04.930 --> 00:07:07.140
projection helps you, just
think of that word.

00:07:07.140 --> 00:07:10.530
If you have a light that shines
down from above, this

00:07:10.530 --> 00:07:15.190
adjacent side is kind of like
the shadow of a onto vector b.

00:07:15.190 --> 00:07:18.630
And you can imagine, if these
two vectors-- if our two

00:07:18.630 --> 00:07:25.180
vectors looked more like this,
if they were really going in

00:07:25.180 --> 00:07:26.110
the same direction.

00:07:26.110 --> 00:07:30.760
Let's say that's vector a and
that's vector b, then the

00:07:30.760 --> 00:07:33.730
adjacent side that I care about
is going to-- they're

00:07:33.730 --> 00:07:35.070
going to have a lot
more in common.

00:07:35.070 --> 00:07:38.950
The part of a that is going in
the same direction of b will

00:07:38.950 --> 00:07:39.840
be a lot larger.

00:07:39.840 --> 00:07:45.840
So this will have a larger
dot product.

00:07:45.840 --> 00:07:47.920
Because the dot product is
essentially saying, how much

00:07:47.920 --> 00:07:50.010
of those vectors are going
in the same direction?

00:07:50.010 --> 00:07:52.950
But it's just a number, so it
will just be this adjacent

00:07:52.950 --> 00:07:56.310
side times the length of b.

00:07:56.310 --> 00:07:59.730
And what if I had vectors that
are pretty perpendicular to

00:07:59.730 --> 00:08:00.850
each other?

00:08:00.850 --> 00:08:02.930
So what if I had two vectors
that were like this?

00:08:02.930 --> 00:08:08.390
What if my vector a looked
like that and my vector b

00:08:08.390 --> 00:08:10.370
looked like that?

00:08:10.370 --> 00:08:14.940
Well now the adjacent, the way
I define it here, if I had to

00:08:14.940 --> 00:08:17.060
make a right triangle
like that, the

00:08:17.060 --> 00:08:18.880
adjacent side's very small.

00:08:18.880 --> 00:08:21.440
So you're dot product, even
though a is still a reasonably

00:08:21.440 --> 00:08:25.740
large vector, is now much
smaller because a and b have

00:08:25.740 --> 00:08:28.170
very little commonality
in the same direction.

00:08:28.170 --> 00:08:30.880
And you can do it
the other way.

00:08:30.880 --> 00:08:33.070
You could draw this down like
that and you could do the

00:08:33.070 --> 00:08:34.760
adjacent the other way, but it
doesn't matter because these

00:08:34.760 --> 00:08:37.159
a's and b's are arbitrary.

00:08:37.159 --> 00:08:44.460
So the take away is the fact
that a dot b is equal to the

00:08:44.460 --> 00:08:49.480
lengths of each of those times
the cosine of theta.

00:08:49.480 --> 00:08:55.590
To me it says that the dot
product tells me how much are

00:08:55.590 --> 00:08:57.100
my vectors moving together?

00:08:57.100 --> 00:08:59.040
Or the product of the
part of the vectors

00:08:59.040 --> 00:09:01.660
that are moving together.

00:09:01.660 --> 00:09:10.320
Product of the lengths of the
vectors that are moving

00:09:10.320 --> 00:09:11.946
together or in the
same direction.

00:09:15.130 --> 00:09:22.210
You could view this adjacent
side here as the part of a

00:09:22.210 --> 00:09:25.050
that's going in the
direction of b.

00:09:25.050 --> 00:09:27.080
That's the part of a that's
going in the direction of b.

00:09:27.080 --> 00:09:30.180
So you're multiplying
that times b itself.

00:09:30.180 --> 00:09:31.620
So that's what the
dot product is.

00:09:31.620 --> 00:09:34.340
How much are two things going
in the same direction.

00:09:34.340 --> 00:09:38.000
And notice, when two things are
orthogonal or when they're

00:09:38.000 --> 00:09:41.990
perpendicular-- when a dot b is
equal to 0, we say they're

00:09:41.990 --> 00:09:42.910
perpendicular.

00:09:42.910 --> 00:09:45.110
And that makes complete sense
based on this kind of

00:09:45.110 --> 00:09:47.930
intuition of what the dot
product is doing.

00:09:47.930 --> 00:09:49.210
Because that means that
they're perfectly

00:09:49.210 --> 00:09:51.040
perpendicular.

00:09:51.040 --> 00:09:53.620
So that's b and that's a.

00:09:53.620 --> 00:09:57.700
And so the adjacent part of a,
if I had to draw a right

00:09:57.700 --> 00:09:59.390
trianlge, it would come
straight down.

00:09:59.390 --> 00:10:01.440
And if I were to say the
projection of a and I

00:10:01.440 --> 00:10:02.070
haven't draw that.

00:10:02.070 --> 00:10:05.400
Or if I put a light shining down
from above and I'd say

00:10:05.400 --> 00:10:07.670
what's the shadow of a onto b?

00:10:07.670 --> 00:10:08.440
You'd get nothing.

00:10:08.440 --> 00:10:09.120
You'd get 0.

00:10:09.120 --> 00:10:12.010
This arrow has no width, even
though I've drawn it to have

00:10:12.010 --> 00:10:12.530
some width.

00:10:12.530 --> 00:10:13.820
It has no width.

00:10:13.820 --> 00:10:15.890
So you would have
a 0 down here.

00:10:15.890 --> 00:10:17.930
The part of a that goes in
the same direction as b.

00:10:17.930 --> 00:10:20.270
No part of this vector
goes in the same

00:10:20.270 --> 00:10:21.550
direction as this vector.

00:10:21.550 --> 00:10:25.560
So you're going to have this 0
kind of adjacent side times b,

00:10:25.560 --> 00:10:27.030
so you're going to get
something that's 0.

00:10:27.030 --> 00:10:29.150
So hopefully that makes
a little sense.

00:10:29.150 --> 00:10:31.870
Now let's think about
the cross product.

00:10:31.870 --> 00:10:37.520
The cross product tells us well,
the length of a cross b,

00:10:37.520 --> 00:10:42.510
I painstakingly showed, you is
equal to the length of a times

00:10:42.510 --> 00:10:47.750
the length of b times the sin
of the angle between them.

00:10:47.750 --> 00:10:50.500
So let me do the same example.

00:10:50.500 --> 00:10:53.370
Let me draw my two vectors.

00:10:53.370 --> 00:10:59.300
That's my vector a and
this is my vector b.

00:10:59.300 --> 00:11:01.480
And now sin-- SOH CAH TOA.

00:11:01.480 --> 00:11:03.400
So sin of theta, let
me write that.

00:11:03.400 --> 00:11:09.340
Sin of theta-- SOH CAH TOA-- is
equal to opposite over the

00:11:09.340 --> 00:11:10.020
hypotenuse.

00:11:10.020 --> 00:11:12.870
So if I were to draw a little
right triangle here, so if I

00:11:12.870 --> 00:11:17.900
were to draw a perpendicular
right there, this is theta.

00:11:17.900 --> 00:11:21.300
What is the sin of theta equal
to in this context?

00:11:21.300 --> 00:11:24.750
The sin of theta is
equal to what?

00:11:24.750 --> 00:11:28.370
It's equal to this
side over here.

00:11:28.370 --> 00:11:31.640
Let me call that just
the opposite.

00:11:31.640 --> 00:11:34.840
It's equal to the opposite
side over the hypotenuse.

00:11:34.840 --> 00:11:36.540
So the hypotenuse is
the length of this

00:11:36.540 --> 00:11:38.760
vector a right there.

00:11:38.760 --> 00:11:41.760
It's the length of
this vector a.

00:11:41.760 --> 00:11:45.440
So the hypotenuse is the length
over my vector a.

00:11:45.440 --> 00:11:48.500
So if I multiply both sides of
this by my length of vector a,

00:11:48.500 --> 00:11:52.710
I get the length of vector a
times the sin of theta is

00:11:52.710 --> 00:11:55.570
equal to the opposite side.

00:11:55.570 --> 00:11:59.290
So if we rearrange this a little
bit, I can rewrite this

00:11:59.290 --> 00:12:01.300
as equal to-- I'm just
going to swap them.

00:12:01.300 --> 00:12:02.800
I have to do the dot
product as well.

00:12:02.800 --> 00:12:06.820
This is equal to b, the length
of vector b, times the length

00:12:06.820 --> 00:12:09.920
of vector a sin of theta.

00:12:09.920 --> 00:12:12.720
Well this thing is just the
opposite side as I've defined

00:12:12.720 --> 00:12:14.300
it right here.

00:12:14.300 --> 00:12:18.305
So this right here is just
the opposite side,

00:12:18.305 --> 00:12:20.680
this side right there.

00:12:20.680 --> 00:12:23.460
So when we're taking the cross
product, we're essentially

00:12:23.460 --> 00:12:28.540
multiplying the length of vector
b times the part of a

00:12:28.540 --> 00:12:31.040
that's going perpendicular
to b.

00:12:31.040 --> 00:12:33.680
This opposite side is the
part of a that's going

00:12:33.680 --> 00:12:35.110
perpendicular to b.

00:12:35.110 --> 00:12:37.930
So they're kind of
opposite ideas.

00:12:37.930 --> 00:12:40.640
The dot product, you're
multiplying the part of a

00:12:40.640 --> 00:12:44.570
that's going in the same
direction as b with b.

00:12:44.570 --> 00:12:46.400
While when you're taking the
cross product, you're

00:12:46.400 --> 00:12:48.760
multiplying the part of
a that's going in the

00:12:48.760 --> 00:12:54.720
perpendicular direction to
b with the length of b.

00:12:54.720 --> 00:12:56.540
It's a measure, especially when
you take the length of

00:12:56.540 --> 00:12:58.790
this, it's a measure of
how perpendicular

00:12:58.790 --> 00:13:00.000
these two guys are.

00:13:00.000 --> 00:13:03.590
And this is, it's a measure of
how much do they move in the

00:13:03.590 --> 00:13:05.040
same direction?

00:13:05.040 --> 00:13:07.490
And let's just look at
a couple of examples.

00:13:07.490 --> 00:13:08.810
So if you take two
right triangles.

00:13:11.750 --> 00:13:16.020
So if that's a and that's-- or
if you take two vectors that

00:13:16.020 --> 00:13:22.840
are perpendicular to each other,
the length of a cross b

00:13:22.840 --> 00:13:26.570
is going to be equal to-- if we
just use this formula right

00:13:26.570 --> 00:13:30.060
there-- the length of a
times the length of b.

00:13:30.060 --> 00:13:32.000
And what's the sin
of 90 degrees?

00:13:32.000 --> 00:13:33.250
It's 1.

00:13:33.250 --> 00:13:36.910
So in this case you kind of have
maximized the length of

00:13:36.910 --> 00:13:37.970
your cross product.

00:13:37.970 --> 00:13:39.660
This is as high as it can go.

00:13:39.660 --> 00:13:42.690
Because sin of theta, it's
a maximum value.

00:13:42.690 --> 00:13:46.940
Sin of theta is always less
than or equal to 1.

00:13:46.940 --> 00:13:48.980
So this is as good as you're
ever going to get.

00:13:48.980 --> 00:13:52.480
This is the highest possible
value when you have perfectly

00:13:52.480 --> 00:13:54.630
perpendicular vectors.

00:13:54.630 --> 00:13:57.870
Now, when is-- actually, just to
kind of go back to make the

00:13:57.870 --> 00:13:58.780
same point here.

00:13:58.780 --> 00:14:01.830
When do you get the maximum
value for your cosine of-- for

00:14:01.830 --> 00:14:03.760
your dot product?

00:14:03.760 --> 00:14:08.230
Well, it's when your two
vectors are collinear.

00:14:08.230 --> 00:14:11.500
If my vector a looks like that
and my vector b is essentially

00:14:11.500 --> 00:14:14.250
another vector that's going
in the same direction,

00:14:14.250 --> 00:14:15.410
then theta is 0.

00:14:15.410 --> 00:14:16.970
There's no angle between them.

00:14:16.970 --> 00:14:23.080
And then you have a dot b is
equal to the magnitude or the

00:14:23.080 --> 00:14:26.440
length of vector a times the
length of vector b times the

00:14:26.440 --> 00:14:27.870
cosine of the angle
between them.

00:14:27.870 --> 00:14:29.190
The cosine of the angle between
them, the cosine of

00:14:29.190 --> 00:14:30.680
that angle is 0.

00:14:30.680 --> 00:14:33.180
Or the angle is 0, so the
cosine of that is 1.

00:14:33.180 --> 00:14:38.020
So when you have two vectors
that go exactly in the same

00:14:38.020 --> 00:14:41.230
direction or they're collinear,
you kind of

00:14:41.230 --> 00:14:43.700
maximize your dot product.

00:14:43.700 --> 00:14:46.510
You maximize your cross product
when they're perfectly

00:14:46.510 --> 00:14:48.080
perpendicular to each other.

00:14:48.080 --> 00:14:50.960
And just to make the analogy
clear, when they're

00:14:50.960 --> 00:14:53.550
perpendicular to each other
you've minimized-- or at least

00:14:53.550 --> 00:14:54.640
the magnitude of your
dot product.

00:14:54.640 --> 00:14:57.080
You can get negative dot
products, but the absolute

00:14:57.080 --> 00:15:00.360
size of your dot product, the
absolute value of your dot

00:15:00.360 --> 00:15:02.270
product is minimized
when they're

00:15:02.270 --> 00:15:03.710
perpendicular to each other.

00:15:03.710 --> 00:15:10.090
Similarly, if you were to take
two vectors that are collinear

00:15:10.090 --> 00:15:12.350
and they're moving in the same
direction, so if that's vector

00:15:12.350 --> 00:15:16.280
a, and then I have vector b that
just is another vector

00:15:16.280 --> 00:15:19.480
that I want to draw them
on top of each other.

00:15:19.480 --> 00:15:20.560
But I think you get the idea.

00:15:20.560 --> 00:15:24.540
Let's say vector
b is like that.

00:15:24.540 --> 00:15:25.860
Then theta is 0.

00:15:25.860 --> 00:15:27.110
You can't even see it.

00:15:27.110 --> 00:15:28.190
It's been squeezed out.

00:15:28.190 --> 00:15:31.170
I've just brought these two
things on top of each other.

00:15:31.170 --> 00:15:35.670
And then the cross product in
this situation, a cross b is

00:15:35.670 --> 00:15:39.525
equal to-- well, the length of
both of these things times the

00:15:39.525 --> 00:15:40.070
sin of theta.

00:15:40.070 --> 00:15:41.620
Sin of 0 is 0.

00:15:41.620 --> 00:15:43.600
So it's just 0.

00:15:43.600 --> 00:15:48.300
So two collinear vectors, the
magnitude of their cross

00:15:48.300 --> 00:15:49.810
product is 0.

00:15:49.810 --> 00:15:56.110
But the magnitude of their dot
product, the a dot b, is going

00:15:56.110 --> 00:15:57.070
to be maximized.

00:15:57.070 --> 00:15:59.330
It's going to be as high
as you can get.

00:15:59.330 --> 00:16:02.540
It's going to be the length of
a times the length of b.

00:16:02.540 --> 00:16:04.450
Now the opposite scenario
is right here.

00:16:04.450 --> 00:16:07.210
When they're perpendicular to
each other, the cross product

00:16:07.210 --> 00:16:09.750
is maximized because it's
measuring on how much of the

00:16:09.750 --> 00:16:14.930
vectors-- how much of the
perpendicular part of a is--

00:16:14.930 --> 00:16:17.430
multiplying that times
the length of b.

00:16:17.430 --> 00:16:20.490
And then when you have two
orthogonal vectors, your dot

00:16:20.490 --> 00:16:22.330
product is minimized,
or the absolute

00:16:22.330 --> 00:16:23.400
value of your dot product.

00:16:23.400 --> 00:16:26.840
So a dot b in this case,
is equal to 0.

00:16:26.840 --> 00:16:29.420
Anyway, I wanted to make all
of this clear because

00:16:29.420 --> 00:16:31.300
sometimes you kind of get into
the formulas and the

00:16:31.300 --> 00:16:34.720
definitions and you lose the
intuition about what are all

00:16:34.720 --> 00:16:36.940
of these ideas really for?

00:16:36.940 --> 00:16:39.130
And actually, before I move on,
let me just make another

00:16:39.130 --> 00:16:42.210
kind of idea about what the
cross product can be

00:16:42.210 --> 00:16:43.520
interpreted as.

00:16:43.520 --> 00:16:48.020
Because a cross product tends
to give people more trouble.

00:16:48.020 --> 00:16:52.220
That's my a and that's my b.

00:16:52.220 --> 00:16:55.820
What if I wanted to figure
out the area of this

00:16:55.820 --> 00:16:58.110
parallelogram?

00:16:58.110 --> 00:17:00.410
If I were to shift a and have
that there and if I were to

00:17:00.410 --> 00:17:03.420
shift b and draw a line parallel
to b, and if I wanted

00:17:03.420 --> 00:17:08.089
to figure out the area of this
parallelogram right there, how

00:17:08.089 --> 00:17:11.000
would I do it just using
regular geometry?

00:17:11.000 --> 00:17:15.369
Well I would drop a
perpendicular right there.

00:17:18.670 --> 00:17:25.020
This is perpendicular and this
length is h for height.

00:17:25.020 --> 00:17:30.020
Then the area of this, the area
of the parallelogram is

00:17:30.020 --> 00:17:34.140
just equal to the length of my
base, which is just the length

00:17:34.140 --> 00:17:36.650
of vector b times my height.

00:17:39.640 --> 00:17:41.670
But what is my height?

00:17:41.670 --> 00:17:45.330
Let me just draw a little
theta there.

00:17:45.330 --> 00:17:47.430
Let me do a green theta,
it's more visible.

00:17:47.430 --> 00:17:48.530
So theta.

00:17:48.530 --> 00:17:53.150
So we know already that the sin
of this theta is equal to

00:17:53.150 --> 00:17:55.460
the opposite over
the hypotenuse.

00:17:55.460 --> 00:17:59.740
So it's equal to the height
over the hypotenuse.

00:17:59.740 --> 00:18:02.350
The hypotenuse is just the
length of vector a.

00:18:02.350 --> 00:18:04.520
So it's just the length
of vector a.

00:18:04.520 --> 00:18:08.590
Or we could just solve for
height and we'd get the height

00:18:08.590 --> 00:18:11.040
is equal to the length
of vector a

00:18:11.040 --> 00:18:13.000
times the sin of theta.

00:18:13.000 --> 00:18:14.480
So I can rewrite this here.

00:18:14.480 --> 00:18:19.110
I can replace it with that and
I get the area of this

00:18:19.110 --> 00:18:23.210
parallelogram is equal to the
length of vector b times the

00:18:23.210 --> 00:18:27.110
length of vector a sin theta.

00:18:27.110 --> 00:18:30.660
Well this is just the length
of the cross product of the

00:18:30.660 --> 00:18:35.670
two vectors, a cross b.

00:18:35.670 --> 00:18:36.670
This is the same thing.

00:18:36.670 --> 00:18:39.200
I mean you can rearrange
the a and the b.

00:18:39.200 --> 00:18:42.780
So we now have another way of
thinking about what the cross

00:18:42.780 --> 00:18:43.670
product is.

00:18:43.670 --> 00:18:47.130
The cross product of two
vectors, or at least the

00:18:47.130 --> 00:18:51.220
magnitude or the length of the
cross product of two vectors--

00:18:51.220 --> 00:18:52.320
obviously, the cross product
you're going

00:18:52.320 --> 00:18:53.580
to get a third vector.

00:18:53.580 --> 00:18:58.190
But the length of that third
vector is equal to the area of

00:18:58.190 --> 00:19:02.490
the parallelogram that's defined
or that's kind of--

00:19:02.490 --> 00:19:05.220
that you can create from
those two vectors.

00:19:05.220 --> 00:19:08.080
Anyway, hopefully you found this
a little bit intuitive

00:19:08.080 --> 00:19:10.750
and it'll give you a little bit
more of kind of a sense of

00:19:10.750 --> 00:19:13.910
what the dot product and cross
product are all about.