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So -- So, yesterday we learned
about the questions of planes
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and how to think of 3x3 linear
systems in terms of
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intersections of planes and how
to think about them
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geometrically.
And, that in particular led us
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to see which cases actually we
don't have a unique solution to
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the system,
but maybe we have no solutions
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or infinitely many solutions
because maybe the line at
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intersection of two of the
planes happens to be parallel to
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the other plane.
So, today, we'll start by
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looking at the equations of
lines.
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And, so in a way it seems like
something which we've already
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seen last time because we have
seen that we can think of a line
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as the intersection of two
planes.
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And, we know what equations of
planes look like.
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So, we could describe a line by
two equations telling us about
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the two planes that intersect on
the line.
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But that's not the most
convenient way to think about
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the line usually,
though, because when you have
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these two questions,
have you solve them?
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Well, OK, you can,
but it takes a bit of effort.
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So, instead,
there is another representation
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of a line.
So, if you have a line in
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space, well, you can imagine may
be that you have a point on it.
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And, that point is moving in
time.
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And, the line is the trajectory
of a point as time varies.
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So, think of a line as the
trajectory of a moving point.
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And, so when we think of the
trajectory of the moving point,
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that's called a parametric
equation.
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OK, so we are going to learn
about parametric equations of
00:03:01.000 --> 00:03:07.000
lines.
So, let's say for example that
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we are looking at the line.
So, to specify a line in space,
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I can do that by giving you two
points on the line or by giving
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you a point and a vector
parallel to the line.
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For example,
so let's say I give you two
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points on the line:
(-1,2,2), and the other point
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will be (1,3,-1).
So, OK, it's pretty good
00:03:40.000 --> 00:03:43.000
because we have two points in
that line.
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Now, how do we find all the
other points?
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Well, the other points in
between these guys and also on
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either side.
Let's imagine that we have a
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point that's moving on the line,
and at time zero,
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it's here at Q0.
And, in a unit time,
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I'm not telling you what the
unit is.
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It could be a second.
It could be an hour.
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It could be a year.
At t=1, it's going to be at Q1.
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And, it moves at a constant
speed.
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So, maybe at time one half,
it's going to be here.
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Times two, it would be over
there.
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And, in fact,
that point didn't start here.
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Maybe it's always been moving
on that line.
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At time minus two,
it was down there.
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So, let's say Q(t) is a moving
point, and at t=0 it's at Q0.
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And, let's say that it moves.
Well, we couldn't make it move
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in any way we want.
But, probably the easiest to
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find, so our role is going to
find formulas for a position of
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this moving point in terms of t.
And, we'll use that to say,
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well,
any point on the line is of
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this form where you have to plug
in the current value of t
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depending on when it's hit by
the moving point.
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So, perhaps it's easiest to do
it if we make it move at a
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constant speed on the line,
and that speed is chosen so
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that at time one,
it's at Q1.
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So, the question we want to
answer is, what is the position
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at time t, so,
the point Q(t)?
00:06:03.000 --> 00:06:08.000
Well, to answer that we have an
easy observation,
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which is that the vector from
Q0 to Q of t is proportional to
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the vector from Q0 to Q1.
And, what's the proportionality
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factor here?
Yeah, it's exactly t.
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At time one,
Q0 Q is exactly the same.
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Maybe I should draw another
picture again.
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I have Q0.
I have Q1, and after time t,
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I'm here at Q of t where this
vector from Q0 Q(t) is actually
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going to be t times the vector
Q0 Q1.
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So, when t increases,
it gets longer and longer.
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So, does everybody see this now?
Is that OK?
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Any questions about that?
Yes?
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OK, so I will try to avoid
using blue.
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Thanks for, that's fine.
So, OK, I will not use blue
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anymore.
OK, well, first let me just
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make everything white just for
now.
00:07:46.000 --> 00:07:49.000
This is the vector from Q0 to
Q(t).
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This is the point Q(t).
OK, is it kind of visible now?
00:07:56.000 --> 00:08:03.000
OK, thanks for pointing it out.
I will switch to brighter
00:08:03.000 --> 00:08:09.000
colors.
So, OK, so apart from that,
00:08:09.000 --> 00:08:13.000
I claim now we can find the
position of its moving point
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because,
well, this vector,
00:08:15.000 --> 00:08:19.000
Q0Q1 we can find from the
coordinates of Q0 and Q1.
00:08:19.000 --> 00:08:26.000
So, we just subtract the
coordinates of Q0 from those of
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Q1 will get that vector Q0 Q1 is
00:08:32.000 --> 00:08:36.000
OK,
so, if I look at it,
00:08:36.000 --> 00:08:44.000
well, so let's call x(t),
y(t), and z(t) the coordinates
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of the point that's moving on
the line.
00:08:50.000 --> 00:09:00.000
Then we get x of t minus,
well, actually plus one equals
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t times two.
I'm writing the components of
00:09:07.000 --> 00:09:13.000
Q0Q(t).
And here, I'm writing t times
00:09:13.000 --> 00:09:19.000
Q0Q1.
y(t) minus two equals t,
00:09:19.000 --> 00:09:28.000
and z(t) minus two equals -3t.
So, in other terms,
00:09:28.000 --> 00:09:34.000
the more familiar way that we
used to write these equations,
00:09:34.000 --> 00:09:42.000
let me do it that way instead,
minus one plus 2t,
00:09:42.000 --> 00:09:53.000
y(t) = 2 t, z(t) = 2 - 3t.
And, if you prefer,
00:09:53.000 --> 00:10:02.000
I can just say Q(t) is Q0 plus
t times vector Q0Q1.
00:10:02.000 --> 00:10:07.000
OK, so that's our first
parametric equation of a line in
00:10:07.000 --> 00:10:10.000
this class.
And, I hope you see it's not
00:10:10.000 --> 00:10:13.000
extremely hard.
In fact, parametric equations
00:10:13.000 --> 00:10:17.000
of lines always look like that.
x, y, and z are functions of t
00:10:17.000 --> 00:10:22.000
but are of the form a constant
plus a constant times t.
00:10:22.000 --> 00:10:26.000
The coefficients of t tell us
about a vector along the line.
00:10:26.000 --> 00:10:33.000
Here, we have a vector,
Q0Q1, which is .
00:10:33.000 --> 00:10:37.000
And, the constant terms tell us
about where we are at t=0.
00:10:37.000 --> 00:10:41.000
If I plug t=0 these guys go
away, I get minus 1,2,
00:10:41.000 --> 00:10:46.000
2.
That's my starting point.
00:10:46.000 --> 00:10:59.000
OK, so, any questions about
that?
00:10:59.000 --> 00:11:05.000
No?
OK, so let's see,
00:11:05.000 --> 00:11:12.000
now, what we can do with these
parametric equations.
00:11:12.000 --> 00:11:26.000
So, one application is to think
about the relative position of a
00:11:26.000 --> 00:11:36.000
line and a plane with respect to
each other.
00:11:36.000 --> 00:11:44.000
So, let's say that we take
still the same line up there,
00:11:44.000 --> 00:11:53.000
and let's consider the plane
with the equation x 2y 4z = 7.
00:11:53.000 --> 00:11:55.000
OK, so I'm giving you this
plane.
00:11:55.000 --> 00:11:58.000
And, the questions that we are
going to ask ourselves are,
00:11:58.000 --> 00:12:00.000
well, does the line intersect
the plane?
00:12:00.000 --> 00:12:02.000
And, where does it intersect
the plane?
00:12:22.000 --> 00:12:28.000
So, let's start with the first
primary question that maybe we
00:12:28.000 --> 00:12:32.000
should try to understand.
We have these points.
00:12:32.000 --> 00:12:35.000
We have this plane,
and we have these points,
00:12:35.000 --> 00:12:38.000
Q0 and Q1.
I'm going to draw them in
00:12:38.000 --> 00:12:42.000
completely random places.
Well, are Q0 and Q1 on the same
00:12:42.000 --> 00:12:47.000
side of a plane or on different
sides, on opposite sides of the
00:12:47.000 --> 00:12:50.000
planes?
Could it be that maybe one of
00:12:50.000 --> 00:12:59.000
the points is in the plane?
So, I think I'm going to let
00:12:59.000 --> 00:13:05.000
you vote on that.
So, is that readable?
00:13:05.000 --> 00:13:08.000
Is it too small?
OK, so anyway,
00:13:08.000 --> 00:13:12.000
the question says,
relative to the plane,
00:13:12.000 --> 00:13:16.000
x 2y 4z = 7.
This point, Q0 and Q1,
00:13:16.000 --> 00:13:22.000
are they on the same side,
on opposite sides,
00:13:22.000 --> 00:13:29.000
is one of them on the plane,
or we can't decide?
00:13:29.000 --> 00:13:41.000
OK, that should be better.
So, I see relatively few
00:13:41.000 --> 00:13:46.000
answers.
OK, it looks like also a lot of
00:13:46.000 --> 00:13:51.000
you have forgotten the cards
and, so I see people raising two
00:13:51.000 --> 00:13:55.000
fingers, I see people raising
three fingers.
00:13:55.000 --> 00:13:57.000
And, I see people raising four
fingers.
00:13:57.000 --> 00:14:01.000
I don't see anyone answering
number one.
00:14:01.000 --> 00:14:03.000
So, the general idea seems to
be that either they are on
00:14:03.000 --> 00:14:07.000
opposite sides.
Maybe one of them is on the
00:14:07.000 --> 00:14:10.000
plane.
Well, let's try to see.
00:14:10.000 --> 00:14:14.000
Is one of them on the plane?
Well, let's check.
00:14:14.000 --> 00:14:20.000
OK, so let's look at the point,
sorry.
00:14:20.000 --> 00:14:25.000
I have one blackboard to use
here.
00:14:25.000 --> 00:14:31.000
So, I take the point Q0,
which is at (-1,2,2).
00:14:31.000 --> 00:14:37.000
Well, if I plug that into the
plane equation,
00:14:37.000 --> 00:14:44.000
so, x 2y 4z will equal minus
one plus two times two plus four
00:14:44.000 --> 00:14:48.000
times two.
That's, well,
00:14:48.000 --> 00:14:52.000
four plus eight,
12 minus one,
00:14:52.000 --> 00:14:54.000
11.
That, I think,
00:14:54.000 --> 00:15:01.000
is bigger than seven.
OK, so Q0 is not in the plane.
00:15:01.000 --> 00:15:07.000
Let's try again with Q1.
(1,3, - 1) well,
00:15:07.000 --> 00:15:15.000
if we plug that into x 2y 4z,
we'll have one plus two times
00:15:15.000 --> 00:15:20.000
three makes seven.
But, we add four times negative
00:15:20.000 --> 00:15:23.000
one.
We add up with three less than
00:15:23.000 --> 00:15:25.000
seven.
Well, that one is not in the
00:15:25.000 --> 00:15:27.000
plane, either.
So, I don't think,
00:15:27.000 --> 00:15:32.000
actually, that the answer
should be number three.
00:15:32.000 --> 00:15:37.000
So, let's get rid of answer
number three.
00:15:37.000 --> 00:15:42.000
OK, let's see,
in light of this,
00:15:42.000 --> 00:15:50.000
are you willing to reconsider
your answer?
00:15:50.000 --> 00:15:53.000
OK, so I think now everyone
seems to be interested in
00:15:53.000 --> 00:15:57.000
answering number two.
And, I would agree with that
00:15:57.000 --> 00:16:00.000
answer.
So, let's think about it.
00:16:00.000 --> 00:16:02.000
These points are not in the
plane, but they are not in the
00:16:02.000 --> 00:16:05.000
plane in different ways.
One of them somehow overshoots;
00:16:05.000 --> 00:16:08.000
we get 11.
The other one we only get 3.
00:16:08.000 --> 00:16:12.000
That's less than seven.
If you think about how a plan
00:16:12.000 --> 00:16:15.000
splits space into two half
spaces on either side,
00:16:15.000 --> 00:16:22.000
well, one of them is going to
be the point where x 2y 4z is
00:16:22.000 --> 00:16:27.000
less than seven.
And, the other one will be,
00:16:27.000 --> 00:16:32.000
so, that's somehow this side.
And, that's where Q1 is.
00:16:32.000 --> 00:16:43.000
And, the other side is where x
2y 4z is actually bigger than
00:16:43.000 --> 00:16:47.000
seven.
And, to go from one to the
00:16:47.000 --> 00:16:53.000
other, well, x 2y 4z needs to go
through the value seven.
00:16:53.000 --> 00:16:57.000
If you're moving along any path
from Q0 to Q1,
00:16:57.000 --> 00:17:02.000
this thing will change
continuously from 11 to 3.
00:17:02.000 --> 00:17:05.000
At some time,
it has to go through 7.
00:17:05.000 --> 00:17:09.000
Does that make sense?
So, to go from Q0 to Q1 we need
00:17:09.000 --> 00:17:12.000
to cross P at some place.
So, they're on opposite sides.
00:17:31.000 --> 00:17:37.000
OK, now that doesn't quite
finish answering the question
00:17:37.000 --> 00:17:43.000
that we had, which was,
where does the line intersect
00:17:43.000 --> 00:17:46.000
the plane?
But, why can't we do the same
00:17:46.000 --> 00:17:48.000
thing?
Now, we know not only the
00:17:48.000 --> 00:17:51.000
points Q0 and Q1,
we know actually any point on
00:17:51.000 --> 00:17:55.000
the line because we have a
parametric equation up there
00:17:55.000 --> 00:17:57.000
telling us,
where is the point that's
00:17:57.000 --> 00:18:04.000
moving on the line at time t?
So, what about the moving
00:18:04.000 --> 00:18:08.000
point, Q(t)?
Well, let's plug its
00:18:08.000 --> 00:18:10.000
coordinates into the plane
equation.
00:18:10.000 --> 00:18:24.000
So, we'll take x(t) 2y(t) 4z(t).
OK, that's equal to,
00:18:24.000 --> 00:18:34.000
well, (-1 2t) 2( 2 t) 4( 2 -
3t).
00:18:34.000 --> 00:18:41.000
So, if you simplify this a bit,
you get 2t 2t -12t.
00:18:41.000 --> 00:18:46.000
That should be -8t.
And, the constant term is minus
00:18:46.000 --> 00:18:54.000
one plus four plus eight is 11.
OK, and we have to compare that
00:18:54.000 --> 00:18:57.000
with seven.
OK, the question is,
00:18:57.000 --> 00:19:07.000
is this ever equal to seven?
Well, so, Q(t) is in the plane
00:19:07.000 --> 00:19:16.000
exactly when -8t 11 equals
seven.
00:19:16.000 --> 00:19:20.000
And, that' the same.
If you manipulate this,
00:19:20.000 --> 00:19:27.000
you will get t equals one half.
In fact, that's not very
00:19:27.000 --> 00:19:30.000
surprising.
If you look at these values,
00:19:30.000 --> 00:19:32.000
11 and three,
you see that seven is actually
00:19:32.000 --> 00:19:35.000
right in between.
It's the average of these two
00:19:35.000 --> 00:19:39.000
numbers.
So, it would make sense that
00:19:39.000 --> 00:19:44.000
it's halfway in between Q0 and
Q1, but we will get seven.
00:19:44.000 --> 00:19:50.000
OK, and that at that time,
Q at time one half,
00:19:50.000 --> 00:19:59.000
well, let's plug the values.
So, minus one plus 2t will be
00:19:59.000 --> 00:20:04.000
zero.
Two plus t will be two and a
00:20:04.000 --> 00:20:11.000
half of five halves,
and two minus three halves will
00:20:11.000 --> 00:20:15.000
be one half, OK?
So, this is where the line
00:20:15.000 --> 00:20:16.000
intersects the plane.
00:20:43.000 --> 00:20:47.000
So, you see that's actually a
pretty easy way of finding where
00:20:47.000 --> 00:20:49.000
a line on the plane intersects
each other.
00:20:49.000 --> 00:20:52.000
If we can find a parametric
equation of a line and an
00:20:52.000 --> 00:20:55.000
equation of a plane,
but we basically just plug one
00:20:55.000 --> 00:20:59.000
into the other,
and see at what time the moving
00:20:59.000 --> 00:21:04.000
point hits the plane so that we
know where this.
00:21:04.000 --> 00:21:23.000
OK, other questions about this?
Yes?
00:21:23.000 --> 00:21:30.000
Sorry, can you say that?
Yes, so what if we don't get a
00:21:30.000 --> 00:21:32.000
solution?
What happens?
00:21:32.000 --> 00:21:36.000
So, indeed our line could have
been parallel to the plane or
00:21:36.000 --> 00:21:38.000
maybe even contained in the
plane.
00:21:38.000 --> 00:21:42.000
Well, if the line is parallel
to the plane then maybe what
00:21:42.000 --> 00:21:46.000
happens is that what we plug in
the positions of the moving
00:21:46.000 --> 00:21:48.000
point,
we actually get something that
00:21:48.000 --> 00:21:50.000
never equals seven because maybe
we get actually a constant.
00:21:50.000 --> 00:21:53.000
Say that we had gotten,
I don't know,
00:21:53.000 --> 00:21:56.000
13 all the time.
Well, when is 13 equal to seven?
00:21:56.000 --> 00:21:59.000
The answer is never.
OK, so that's what would tell
00:21:59.000 --> 00:22:02.000
you that the line is actually
parallel to the plane.
00:22:02.000 --> 00:22:06.000
You would not find a solution
to the equation that you get at
00:22:06.000 --> 00:22:13.000
the end.
Yes?
00:22:13.000 --> 00:22:16.000
So, if there's no solution at
all to the equation that you
00:22:16.000 --> 00:22:19.000
get, it means that at no time is
the traveling point going to be
00:22:19.000 --> 00:22:22.000
in the plane.
That means the line really does
00:22:22.000 --> 00:22:25.000
not have the plane ever.
So, it has to be parallel
00:22:25.000 --> 00:22:27.000
outside of it.
On the other hand,
00:22:27.000 --> 00:22:30.000
if a line is inside the plane,
then that means that no matter
00:22:30.000 --> 00:22:33.000
what time you choose,
you always get seven.
00:22:33.000 --> 00:22:37.000
OK, that's what would happen if
a line is in the plane.
00:22:37.000 --> 00:22:44.000
You always get seven.
So, maybe I should write this
00:22:44.000 --> 00:22:54.000
down.
So, if a line is in the plane
00:22:54.000 --> 00:23:10.000
then plugging x(t),
y(t), z(t) into the equation,
00:23:10.000 --> 00:23:18.000
we always get,
well, here in this case seven
00:23:18.000 --> 00:23:22.000
or whatever the value should be
for the plane,
00:23:22.000 --> 00:23:34.000
If the line is parallel to the
plane -- -- in fact,
00:23:34.000 --> 00:23:45.000
we, well, get,
let's see, another constant.
00:23:45.000 --> 00:23:49.000
So, in fact,
you know, when you plug in
00:23:49.000 --> 00:23:51.000
these things,
normally you get a quantity
00:23:51.000 --> 00:23:54.000
that's of a form,
something times t plus a
00:23:54.000 --> 00:23:57.000
constant because that's what you
plug into the equation of a
00:23:57.000 --> 00:23:59.000
plane.
And so, in general,
00:23:59.000 --> 00:24:01.000
you have an equation of the
form, something times t plus
00:24:01.000 --> 00:24:05.000
something equals something.
And, that usually has a single
00:24:05.000 --> 00:24:08.000
solution.
And, the special case is if
00:24:08.000 --> 00:24:11.000
this coefficient of t turns out
to be zero in the end,
00:24:11.000 --> 00:24:14.000
and that's actually going to
happen,
00:24:14.000 --> 00:24:20.000
exactly when the line is either
parallel or in the plane.
00:24:20.000 --> 00:24:24.000
In fact, if you think this
through carefully,
00:24:24.000 --> 00:24:26.000
the coefficient of t that you
get here,
00:24:26.000 --> 00:24:30.000
see, it's one times two plus
two times one plus four times
00:24:30.000 --> 00:24:33.000
minus three.
It's the dot product between
00:24:33.000 --> 00:24:37.000
the normal vector of a plane and
the vector along the line.
00:24:37.000 --> 00:24:41.000
So, see, this coefficient
becomes zero exactly when the
00:24:41.000 --> 00:24:44.000
line is perpendicular to the
normal vector.
00:24:44.000 --> 00:24:46.000
That means it's parallel to the
plane.
00:24:46.000 --> 00:24:51.000
So, everything makes sense.
OK, if you're confused about
00:24:51.000 --> 00:24:55.000
what I just said,
you can ignore it.
00:24:55.000 --> 00:25:03.000
OK, more questions? No?
OK, so if not,
00:25:03.000 --> 00:25:09.000
let's move on to linear
parametric equations.
00:25:09.000 --> 00:25:13.000
So, I hope you've seen here
that parametric equations are a
00:25:13.000 --> 00:25:18.000
great way to think about lines.
There are also a great way to
00:25:18.000 --> 00:25:22.000
think about actually any curve,
any trajectory that can be
00:25:22.000 --> 00:25:34.000
traced by a moving point.
So -- -- more generally,
00:25:34.000 --> 00:26:00.000
we can use parametric equations
-- -- for arbitrary motions --
00:26:00.000 --> 00:26:15.000
-- in the plane or in space.
So, let's look at an example.
00:26:15.000 --> 00:26:20.000
Let's take, so,
it's a famous curve called a
00:26:20.000 --> 00:26:23.000
cycloid.
A cycloid is something that you
00:26:23.000 --> 00:26:27.000
can actually see sometimes at
night when people are biking If
00:26:27.000 --> 00:26:31.000
you have something that reflects
light on the wheel.
00:26:31.000 --> 00:26:33.000
So, let me explain what's the
definition of a cycloid.
00:27:05.000 --> 00:27:07.000
So, I should say,
I've seen a lecture where,
00:27:07.000 --> 00:27:10.000
actually, the professor had a
volunteer on a unicycle to
00:27:10.000 --> 00:27:13.000
demonstrate how that works.
But, I didn't arrange for that,
00:27:13.000 --> 00:27:17.000
so instead I will explain it to
you using more conventional
00:27:17.000 --> 00:27:23.000
means.
So, let's say that we have a
00:27:23.000 --> 00:27:31.000
wheel that's rolling on a
horizontal ground.
00:27:31.000 --> 00:27:34.000
And, as it's rolling of course
it's going to turn.
00:27:34.000 --> 00:27:40.000
So, it's going to move forward
to a new position.
00:27:40.000 --> 00:27:45.000
And, now, let's mention that we
have a point that's been painted
00:27:45.000 --> 00:27:47.000
red on the circumference of the
wheel.
00:27:47.000 --> 00:27:51.000
And, initially,
that point is here.
00:27:51.000 --> 00:27:53.000
So, as the wheel stops
rotating, well,
00:27:53.000 --> 00:27:57.000
of course, it moves forward,
and so it turns on itself.
00:27:57.000 --> 00:28:02.000
So, that point starts falling
back behind the point of contact
00:28:02.000 --> 00:28:07.000
because the wheel is rotating at
the same time as it's moving
00:28:07.000 --> 00:28:12.000
forward.
And so, the cycloid is the
00:28:12.000 --> 00:28:21.000
trajectory of this moving point.
OK, so the cycloid is obtained
00:28:21.000 --> 00:28:27.000
by considering,
so we have a wheel,
00:28:27.000 --> 00:28:38.000
let's say, of radius a.
So, this height here is (a)
00:28:38.000 --> 00:28:47.000
rolling on the floor which is
the x axis.
00:28:47.000 --> 00:28:53.000
And, let's -- And,
we have a point,
00:28:53.000 --> 00:29:01.000
P, that's painted on the wheel.
Initially, it's at the origin.
00:29:01.000 --> 00:29:04.000
But, of course,
as time goes by,
00:29:04.000 --> 00:29:13.000
it moves on the wheel.
P is a point on the rim of the
00:29:13.000 --> 00:29:21.000
wheel, and it starts at the
origin.
00:29:21.000 --> 00:29:27.000
So, the question is,
what happens?
00:29:27.000 --> 00:29:32.000
In particular,
can we find the position of
00:29:32.000 --> 00:29:37.000
this point, x(t),
y(t), as a function of time?
00:29:37.000 --> 00:29:42.000
So, that's the reason why I
have this computer.
00:29:42.000 --> 00:29:48.000
So, I'm not sure it will be
very easy to visualize,
00:29:48.000 --> 00:29:54.000
but so we have a wheel,
well, I hope you can vaguely
00:29:54.000 --> 00:30:00.000
see that there's a circle that's
moving.
00:30:00.000 --> 00:30:05.000
The wheel is green here.
And, there's a radius that's
00:30:05.000 --> 00:30:09.000
been painted blue in it.
And, that radius rotates around
00:30:09.000 --> 00:30:12.000
the wheel as the wheel is moving
forward.
00:30:12.000 --> 00:30:23.000
So, now, let's try to paint,
actually, the trajectory of a
00:30:23.000 --> 00:30:26.000
point.
[LAUGHTER]
00:30:26.000 --> 00:30:30.000
OK, so that's what the cycloid
looks like.
00:30:30.000 --> 00:30:37.000
[APPLAUSE]
OK, so -- So the cycloid,
00:30:37.000 --> 00:30:47.000
well, I guess it doesn't quite
look like what I've drawn.
00:30:47.000 --> 00:30:52.000
It looks like it goes a bit
higher up, which will be the
00:30:52.000 --> 00:30:57.000
trajectory of this red point.
And, see, it hits the bottom
00:30:57.000 --> 00:31:01.000
once in a while.
It forms these arches because
00:31:01.000 --> 00:31:04.000
when the wheel has rotated by a
full turn,
00:31:04.000 --> 00:31:07.000
then you're basically back at
the same situation,
00:31:07.000 --> 00:31:09.000
except a bit further along the
route.
00:31:09.000 --> 00:31:13.000
So, if we do it once more,
you see the point now is at the
00:31:13.000 --> 00:31:18.000
top, and now it's at the bottom.
And then we start again.
00:31:18.000 --> 00:31:23.000
It's at the top,
and then again at the bottom.
00:31:23.000 --> 00:31:40.000
OK.
No.
00:31:40.000 --> 00:31:48.000
[LAUGHTER]
OK, so the question that we
00:31:48.000 --> 00:31:58.000
want to answer is what is the
position x(t),
00:31:58.000 --> 00:32:05.000
y(t), of the point P?
OK, so actually,
00:32:05.000 --> 00:32:07.000
I'm writing x(t),
y(t).
00:32:07.000 --> 00:32:10.000
That means that I have,
maybe I'm expressing the
00:32:10.000 --> 00:32:13.000
position in terms of time.
Let's see, is time going to be
00:32:13.000 --> 00:32:15.000
a good thing to do?
Well, suddenly,
00:32:15.000 --> 00:32:20.000
the position changes over time.
But doesn't actually matter how
00:32:20.000 --> 00:32:24.000
fast the wheel is rolling?
No, because I can just play the
00:32:24.000 --> 00:32:27.000
motion fast-forward.
The wheel will be going faster,
00:32:27.000 --> 00:32:29.000
but the trajectory is still the
same.
00:32:29.000 --> 00:32:32.000
So, in fact,
time is not the most relevant
00:32:32.000 --> 00:32:36.000
thing here.
What matters to us now is how
00:32:36.000 --> 00:32:39.000
far the wheel has gone.
So, we could use as a
00:32:39.000 --> 00:32:44.000
parameter, for example,
the distance by which the wheel
00:32:44.000 --> 00:32:46.000
has moved.
We can do even better because
00:32:46.000 --> 00:32:49.000
we see that, really,
the most complicated thing that
00:32:49.000 --> 00:32:50.000
happens here is really the
rotation.
00:32:50.000 --> 00:32:55.000
So, maybe we can actually use
the angle by which the wheel has
00:32:55.000 --> 00:32:57.000
turned to parameterize the
motion.
00:32:57.000 --> 00:33:02.000
So, there's various choices.
You can choose whichever one
00:33:02.000 --> 00:33:04.000
you prefer.
But, I think here,
00:33:04.000 --> 00:33:07.000
we will get the simplest answer
if we parameterize things by the
00:33:07.000 --> 00:33:10.000
angle.
So, in fact,
00:33:10.000 --> 00:33:23.000
instead of t I will be using
what's called theta as a
00:33:23.000 --> 00:33:36.000
function of the angle,
theta, by which the wheel has
00:33:36.000 --> 00:33:50.000
rotated.
So, how are we going to do that?
00:33:50.000 --> 00:33:57.000
Well, because we are going to
try to use our new knowledge,
00:33:57.000 --> 00:34:03.000
let's try to do it using
vectors in a smart way.
00:34:03.000 --> 00:34:07.000
So, let me draw a picture of
the wheel after things have
00:34:07.000 --> 00:34:12.000
rotated by a certain amount.
So, maybe my point,
00:34:12.000 --> 00:34:18.000
P, now, is here.
And, so the wheel has rotated
00:34:18.000 --> 00:34:21.000
by this angle here.
And, I want to find the
00:34:21.000 --> 00:34:23.000
position of my point,
P, OK?
00:34:23.000 --> 00:34:29.000
So, the position of this point,
P, is going to be the same as
00:34:29.000 --> 00:34:35.000
knowing the vector OP from the
origin to this moving point.
00:34:35.000 --> 00:34:39.000
So, I haven't really simplify
the problem yet because we don't
00:34:39.000 --> 00:34:43.000
really know about vector OP.
But, maybe we know about
00:34:43.000 --> 00:34:47.000
simpler vectors where some will
be OP.
00:34:47.000 --> 00:34:50.000
So, let's see,
let's give names to a few of
00:34:50.000 --> 00:34:52.000
our points.
For example,
00:34:52.000 --> 00:34:54.000
let's say that this will be
point A.
00:34:54.000 --> 00:34:58.000
A is the point where the wheel
is touching the road.
00:34:58.000 --> 00:35:02.000
And, B will be the center of
the wheel.
00:35:02.000 --> 00:35:07.000
Then, it looks like maybe I
have actually a chance of
00:35:07.000 --> 00:35:12.000
understanding vectors like maybe
OA doesn't look quite so scary,
00:35:12.000 --> 00:35:16.000
or AB doesn't look too bad.
BP doesn't look too bad.
00:35:16.000 --> 00:35:27.000
And, if I sum them together,
I will obtain OP.
00:35:27.000 --> 00:35:35.000
So, let's do that.
So, now we've greatly
00:35:35.000 --> 00:35:39.000
simplified the problem.
We had to find one vector that
00:35:39.000 --> 00:35:42.000
we didn't know.
Now we have to find three
00:35:42.000 --> 00:35:47.000
vectors which we don't know.
But, you will see each of them
00:35:47.000 --> 00:35:50.000
as fairly easy to think about.
So, let's see.
00:35:50.000 --> 00:35:56.000
Should we start with vector OA,
maybe?
00:35:56.000 --> 00:36:04.000
So, OA has two components.
One of them should be very easy.
00:36:04.000 --> 00:36:06.000
Well, the y component is just
going to be zero,
00:36:06.000 --> 00:36:10.000
OK?
It's directed along the x axis.
00:36:10.000 --> 00:36:15.000
What about the x component?
So, OA is the distance by which
00:36:15.000 --> 00:36:21.000
the wheel has traveled to get to
its current position.
00:36:21.000 --> 00:36:23.000
Yeah.
I hear a lot of people saying R
00:36:23.000 --> 00:36:25.000
theta.
Let me actually say a(theta)
00:36:25.000 --> 00:36:28.000
because I've called a the radius
of the wheel.
00:36:28.000 --> 00:36:33.000
So, this distance is a(theta).
Why is it a(theta)?
00:36:33.000 --> 00:36:36.000
Well, that's because the wheel,
well, there's an assumption
00:36:36.000 --> 00:36:38.000
which is that the wheel is
rolling on something normal like
00:36:38.000 --> 00:36:40.000
a road,
and not on, maybe,
00:36:40.000 --> 00:36:45.000
ice, or something like that.
S So, it's rolling without
00:36:45.000 --> 00:36:48.000
slipping.
So, that means that this
00:36:48.000 --> 00:36:53.000
distance on the road is actually
equal to the distance here on
00:36:53.000 --> 00:36:57.000
the circumference of the wheel.
This point, P,
00:36:57.000 --> 00:37:01.000
was there, and the amount by
which the things have moved can
00:37:01.000 --> 00:37:06.000
be measured either here or here.
These are the same distances.
00:37:06.000 --> 00:37:15.000
OK, so, that makes it a(theta),
and maybe I should justify by
00:37:15.000 --> 00:37:22.000
saying amount by which the wheel
has rolled,
00:37:22.000 --> 00:37:30.000
has moved, is equal to the,
so, the distance from O to A is
00:37:30.000 --> 00:37:37.000
equal to the arc length on the
circumference of the circle from
00:37:37.000 --> 00:37:40.000
A to P.
And, you know that if you have
00:37:40.000 --> 00:37:42.000
a sector corresponding to an
angle, theta,
00:37:42.000 --> 00:37:45.000
then its length is a times
theta, provided that,
00:37:45.000 --> 00:37:48.000
of course, you express the
angel in radians.
00:37:48.000 --> 00:37:58.000
That's the reason why we always
used radians in math.
00:37:58.000 --> 00:38:01.000
Now, let's think about vector
AB and vector BP.
00:38:30.000 --> 00:38:39.000
OK, so AB is pretty easy,
right, because it's pointing
00:38:39.000 --> 00:38:45.000
straight up, and its length is
a.
00:38:45.000 --> 00:38:55.000
So, it's just zero, a.
Now, the most serious one we've
00:38:55.000 --> 00:39:00.000
kept for the end.
What about vector BP?
00:39:00.000 --> 00:39:04.000
So, vector BP,
we know two things about it.
00:39:04.000 --> 00:39:17.000
We know actually its length,
so, the magnitude of BP -- --
00:39:17.000 --> 00:39:23.000
a.
And, we know it makes an angle,
00:39:23.000 --> 00:39:29.000
theta, with the vertical.
So, that should let us find its
00:39:29.000 --> 00:39:34.000
components.
Let's draw a closer picture.
00:39:34.000 --> 00:39:40.000
Now, in the picture I'm going
to center things at B.
00:39:40.000 --> 00:39:44.000
So, I have my point P.
Here I have theta.
00:39:44.000 --> 00:39:49.000
This length is A.
Well, what are the components
00:39:49.000 --> 00:39:57.000
of BP?
Well, the X component is going
00:39:57.000 --> 00:39:59.000
to be?
Almost.
00:39:59.000 --> 00:40:03.000
I hear people saying things
about a, but I agree with a.
00:40:03.000 --> 00:40:04.000
I hear some cosines.
I hear some sines.
00:40:04.000 --> 00:40:07.000
I think it's actually the sine.
Yes.
00:40:07.000 --> 00:40:10.000
It's a(sin(theta)),
except it's going to the left.
00:40:10.000 --> 00:40:18.000
So, actually it will have a
negative a(sin(theta)).
00:40:18.000 --> 00:40:23.000
And, the vertical component,
well, it will be a(cos(theta)),
00:40:23.000 --> 00:40:27.000
but also negative because we
are going downwards.
00:40:27.000 --> 00:40:46.000
So, it's negative a(cos(theta)).
So, now we can answer the
00:40:46.000 --> 00:40:52.000
initial question because vector
OP, well, we just add up OA,
00:40:52.000 --> 00:40:57.000
AB, and BP.
So, the X component will be
00:40:57.000 --> 00:41:09.000
a(theta) - a(sin(theta)).
And, a-a(cos(theta)).
00:41:09.000 --> 00:41:25.000
OK.
So, any questions about that?
00:41:25.000 --> 00:41:29.000
OK, so, what's the answer?
Because this thing here is the
00:41:29.000 --> 00:41:35.000
x coordinate as a function of
theta, and that one is the y
00:41:35.000 --> 00:41:39.000
coordinate as a function of
theta.
00:41:39.000 --> 00:41:44.000
So, now, just to show you that
we can do a lot of things when
00:41:44.000 --> 00:41:48.000
we have a parametric equation,
here is a small mystery.
00:41:48.000 --> 00:41:54.000
So, what happens exactly near
the bottom point?
00:41:54.000 --> 00:41:57.000
What does the curve look like?
The computer tells us,
00:41:57.000 --> 00:41:59.000
well, it looks like it has some
sort of pointy thing,
00:41:59.000 --> 00:42:02.000
but isn't that something of a
display?
00:42:02.000 --> 00:42:12.000
Is it actually what happens?
So, what do you think happens
00:42:12.000 --> 00:42:19.000
near the bottom point?
Remember, we had that picture.
00:42:19.000 --> 00:42:24.000
Let me show you once more,
where you have these
00:42:24.000 --> 00:42:28.000
corner-like things at the
bottom.
00:42:28.000 --> 00:42:31.000
Well, actually,
is it indeed a corner with some
00:42:31.000 --> 00:42:34.000
angle between the two
directions?
00:42:34.000 --> 00:42:38.000
Does it make an angle?
Or, is it actually a smooth
00:42:38.000 --> 00:42:42.000
curve without any corner,
but we don't see it because
00:42:42.000 --> 00:42:46.000
it's too small to be visible on
the computer screen?
00:42:46.000 --> 00:42:50.000
Does it actually make a loop?
Does it actually come down and
00:42:50.000 --> 00:42:55.000
then back up without going to
the left or to the right and
00:42:55.000 --> 00:43:01.000
without making an angle?
So, yeah, I see the majority
00:43:01.000 --> 00:43:05.000
votes for answers number two or
four.
00:43:05.000 --> 00:43:08.000
And, well, at this point,
we can't quite tell.
00:43:08.000 --> 00:43:10.000
So, let's try to figure it out
from these formulas.
00:43:10.000 --> 00:43:17.000
The way to answer that for sure
is to actually look at the
00:43:17.000 --> 00:43:23.000
formulas.
OK, so question that we are
00:43:23.000 --> 00:43:34.000
trying to answer now is what
happens near the bottom point?
00:43:52.000 --> 00:43:58.000
OK, so how do we answer that?
Well, we should probably try to
00:43:58.000 --> 00:44:03.000
find simpler formulas for these
things.
00:44:03.000 --> 00:44:06.000
Well, to simplify,
let's divide everything by a.
00:44:06.000 --> 00:44:08.000
Let's rescale everything by a.
If you want,
00:44:08.000 --> 00:44:12.000
let's say that we take the unit
of length to be the radius of
00:44:12.000 --> 00:44:15.000
our wheel.
So, instead of measuring things
00:44:15.000 --> 00:44:18.000
in feet or meters,
we'll just measure them in
00:44:18.000 --> 00:44:25.000
radius.
So, take the length unit to be
00:44:25.000 --> 00:44:32.000
equal to the radius.
So, that means we'll have a=1.
00:44:32.000 --> 00:44:35.000
Then, our formulas are slightly
simpler.
00:44:35.000 --> 00:44:45.000
We get x(theta) is theta -
sin(theta), and y equals 1 - cos
00:44:45.000 --> 00:44:49.000
(theta).
OK, so, if we want to
00:44:49.000 --> 00:44:52.000
understand what these things
look like, maybe we should try
00:44:52.000 --> 00:44:56.000
to take some approximation.
OK, so what about
00:44:56.000 --> 00:45:00.000
approximations?
Well, probably you know that if
00:45:00.000 --> 00:45:07.000
I take the sine of a very small
angle, it's close to the actual
00:45:07.000 --> 00:45:12.000
angle itself if theta is very
small.
00:45:12.000 --> 00:45:18.000
And, you know that the cosine
of an angle that's very small is
00:45:18.000 --> 00:45:21.000
close to one.
Well, that's pretty good.
00:45:21.000 --> 00:45:23.000
If we use that,
we will get theta minus theta,
00:45:23.000 --> 00:45:26.000
one minus one,
it looks like it's not precise
00:45:26.000 --> 00:45:29.000
enough.
We just get zero and zero.
00:45:29.000 --> 00:45:31.000
That's not telling us much
about what happens.
00:45:31.000 --> 00:45:39.000
OK, so we need actually better
approximations than that.
00:45:39.000 --> 00:45:50.000
So -- So, hopefully you have
seen in one variable calculus
00:45:50.000 --> 00:45:57.000
something called Taylor
expansion.
00:45:57.000 --> 00:46:14.000
That's [GROANS].
I see that -- OK,
00:46:14.000 --> 00:46:17.000
so if you have not seen Taylor
expansion,
00:46:17.000 --> 00:46:21.000
or somehow it was so traumatic
that you've blocked it out of
00:46:21.000 --> 00:46:24.000
your memory,
let me just remind you that
00:46:24.000 --> 00:46:27.000
Taylor expansion is a way to get
a better approximation than just
00:46:27.000 --> 00:46:32.000
looking at the function,
its derivative.
00:46:32.000 --> 00:46:42.000
So -- And, here's an example of
where it actually comes in handy
00:46:42.000 --> 00:46:52.000
in real life.
So, Taylor approximation says
00:46:52.000 --> 00:47:01.000
that if t is small,
then the value of the function,
00:47:01.000 --> 00:47:04.000
f(t), is approximately equal
to,
00:47:04.000 --> 00:47:07.000
well, our first guess,
of course, would be f(0).
00:47:07.000 --> 00:47:12.000
That's our first approximation.
If we want to be a bit more
00:47:12.000 --> 00:47:15.000
precise, we know that when we
change by t,
00:47:15.000 --> 00:47:17.000
well, t times the derivative
comes in,
00:47:17.000 --> 00:47:23.000
that's for linear approximation
to how the function changes.
00:47:23.000 --> 00:47:28.000
Now, if we want to be even more
precise, there's another term,
00:47:28.000 --> 00:47:32.000
which is t^2 over two times the
second derivative.
00:47:32.000 --> 00:47:37.000
And, if we want to be even more
precise, you will have t^3 over
00:47:37.000 --> 00:47:41.000
six times the third derivative
at zero.
00:47:41.000 --> 00:47:43.000
OK, and you can continue,
and so on.
00:47:43.000 --> 00:47:49.000
But, we won't need more.
So, if you use this here,
00:47:49.000 --> 00:47:53.000
it tells you that the sine of a
smaller angle,
00:47:53.000 --> 00:47:57.000
theta, well,
yeah, it looks like theta.
00:47:57.000 --> 00:48:01.000
But, if we want to be more
precise, then we should add
00:48:01.000 --> 00:48:06.000
minus theta cubed over six.
And, cosine of theta,
00:48:06.000 --> 00:48:12.000
well, it's not quite one.
It's close to one minus theta
00:48:12.000 --> 00:48:16.000
squared over two.
OK, so these are slightly
00:48:16.000 --> 00:48:21.000
better approximations of sine
and cosine for small angles.
00:48:21.000 --> 00:48:28.000
So, now, if we try to figure
out, again, what happens to our
00:48:28.000 --> 00:48:31.000
x of theta, well,
it would be,
00:48:31.000 --> 00:48:36.000
sorry, theta minus theta cubed
over six.
00:48:36.000 --> 00:48:44.000
That's theta cubed over six.
And y, on the other hand,
00:48:44.000 --> 00:48:53.000
is going to be one minus that.
That's about theta squared over
00:48:53.000 --> 00:48:57.000
two.
So, now, which one of them is
00:48:57.000 --> 00:49:01.000
bigger when theta is small?
Yeah, y is much larger.
00:49:01.000 --> 00:49:03.000
OK, if you take the cube of a
very small number,
00:49:03.000 --> 00:49:06.000
it becomes very,
very, very small.
00:49:06.000 --> 00:49:09.000
So, in fact,
we can look at that.
00:49:09.000 --> 00:49:15.000
So, x, an absolute value,
is much smaller than y.
00:49:15.000 --> 00:49:17.000
And, in fact,
what we can do is we can look
00:49:17.000 --> 00:49:21.000
at the ratio between y and x.
That tells us the slope with
00:49:21.000 --> 00:49:27.000
which we approach the origin.
So, y over x is,
00:49:27.000 --> 00:49:35.000
well, let's take the ratio of
this, too.
00:49:35.000 --> 00:49:38.000
That gives us three divided by
theta.
00:49:38.000 --> 00:49:45.000
That tends to infinity when
theta approaches zero.
00:49:45.000 --> 00:49:53.000
So, that means that the slope
of our curve,
00:49:53.000 --> 00:50:00.000
the origin is actually
infinite.
00:50:00.000 --> 00:50:05.000
And so, the curve picture is
really something like this.
00:50:05.000 --> 00:50:07.000
So, the instantaneous motion,
if you had to describe what
00:50:07.000 --> 00:50:09.000
happens very,
very close to the origin is
00:50:09.000 --> 00:50:12.000
that your point is actually not
moving to the left or to the
00:50:12.000 --> 00:50:17.000
right along with the wheel.
It's moving down and up.
00:50:17.000 --> 00:50:20.000
I mean, at the same time it is
actually moving a little bit
00:50:20.000 --> 00:50:24.000
forward at the same time.
But, the dominant motion,
00:50:24.000 --> 00:50:29.000
near the origin is really where
it goes down and back up,
00:50:29.000 --> 00:50:33.000
so answer number four,
you have vertical tangent.
00:50:33.000 --> 00:50:37.000
OK, I think I'm at the end of
time.
00:50:37.000 --> 00:50:44.000
So, have a nice weekend.
And, I'll see you on Tuesday.
00:50:44.000 --> 00:50:47.000
So, on Tuesday I will have
practice exams for next week's
00:50:47.000 --> 00:50:50.000
test.