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So -- So, yesterday we learned
about the questions of planes
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00:00:28 --> 00:00:33
and how to think of 3x3 linear
systems in terms of
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00:00:33 --> 00:00:38
intersections of planes and how
to think about them
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00:00:38 --> 00:00:42
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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00:00:47 --> 00:00:49
the system,
but maybe we have no solutions
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00:00:49 --> 00:00:53
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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00:01:02 --> 00:01:08
looking at the equations of
lines.
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00:01:08 --> 00:01:18
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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00:01:34 --> 00:01:37
And, we know what equations of
planes look like.
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00:01:37 --> 00:01:42
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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00:01:46 --> 00:01:48
But that's not the most
convenient way to think about
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00:01:48 --> 00:01:51
the line usually,
though, because when you have
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00:01:51 --> 00:01:53
these two questions,
have you solve them?
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00:01:53 --> 00:01:57
Well, OK, you can,
but it takes a bit of effort.
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00:01:57 --> 00:02:03
So, instead,
there is another representation
28
00:02:03 --> 00:02:07
of a line.
So, if you have a line in
29
00:02:07 --> 00:02:12
space, well, you can imagine may
be that you have a point on it.
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00:02:12 --> 00:02:14
And, that point is moving in
time.
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00:02:14 --> 00:02:19
And, the line is the trajectory
of a point as time varies.
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00:02:19 --> 00:02:33
So, think of a line as the
trajectory of a moving point.
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00:02:33 --> 00:02:44
And, so when we think of the
trajectory of the moving point,
34
00:02:44 --> 00:02:51
that's called a parametric
equation.
35
00:02:51 --> 00:03:01
OK, so we are going to learn
about parametric equations of
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00:03:01 --> 00:03:07
lines.
So, let's say for example that
37
00:03:07 --> 00:03:13
we are looking at the line.
So, to specify a line in space,
38
00:03:13 --> 00:03:18
I can do that by giving you two
points on the line or by giving
39
00:03:18 --> 00:03:22
you a point and a vector
parallel to the line.
40
00:03:22 --> 00:03:28
For example,
so let's say I give you two
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00:03:28 --> 00:03:35
points on the line:
(-1,2,2), and the other point
42
00:03:35 --> 00:03:40
will be (1,3,-1).
So, OK, it's pretty good
43
00:03:40 --> 00:03:43
because we have two points in
that line.
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00:03:43 --> 00:03:46
Now, how do we find all the
other points?
45
00:03:46 --> 00:03:50
Well, the other points in
between these guys and also on
46
00:03:50 --> 00:03:54
either side.
Let's imagine that we have a
47
00:03:54 --> 00:04:00
point that's moving on the line,
and at time zero,
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00:04:00 --> 00:04:03
it's here at Q0.
And, in a unit time,
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00:04:03 --> 00:04:05
I'm not telling you what the
unit is.
50
00:04:05 --> 00:04:08
It could be a second.
It could be an hour.
51
00:04:08 --> 00:04:12
It could be a year.
At t=1, it's going to be at Q1.
52
00:04:12 --> 00:04:14
And, it moves at a constant
speed.
53
00:04:14 --> 00:04:17
So, maybe at time one half,
it's going to be here.
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00:04:17 --> 00:04:19
Times two, it would be over
there.
55
00:04:19 --> 00:04:21
And, in fact,
that point didn't start here.
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00:04:21 --> 00:04:24
Maybe it's always been moving
on that line.
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00:04:24 --> 00:04:29
At time minus two,
it was down there.
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00:04:29 --> 00:04:51
So, let's say Q(t) is a moving
point, and at t=0 it's at Q0.
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00:04:51 --> 00:04:55
And, let's say that it moves.
Well, we couldn't make it move
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00:04:55 --> 00:04:58
in any way we want.
But, probably the easiest to
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00:04:58 --> 00:05:02
find, so our role is going to
find formulas for a position of
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00:05:02 --> 00:05:06
this moving point in terms of t.
And, we'll use that to say,
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00:05:06 --> 00:05:08
well,
any point on the line is of
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00:05:08 --> 00:05:12
this form where you have to plug
in the current value of t
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00:05:12 --> 00:05:16
depending on when it's hit by
the moving point.
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00:05:16 --> 00:05:24
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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00:05:31 --> 00:05:36
that at time one,
it's at Q1.
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00:05:36 --> 00:05:45
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00:05:45 --> 00:05:56
So, the question we want to
answer is, what is the position
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00:05:56 --> 00:06:03
at time t, so,
the point Q(t)?
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00:06:03 --> 00:06:08
Well, to answer that we have an
easy observation,
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00:06:08 --> 00:06:15
which is that the vector from
Q0 to Q of t is proportional to
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00:06:15 --> 00:06:23
the vector from Q0 to Q1.
And, what's the proportionality
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00:06:23 --> 00:06:27
factor here?
Yeah, it's exactly t.
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00:06:27 --> 00:06:34
At time one,
Q0 Q is exactly the same.
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00:06:34 --> 00:06:36
Maybe I should draw another
picture again.
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00:06:36 --> 00:06:43
I have Q0.
I have Q1, and after time t,
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00:06:43 --> 00:06:56
I'm here at Q of t where this
vector from Q0 Q(t) is actually
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00:06:56 --> 00:07:04
going to be t times the vector
Q0 Q1.
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00:07:04 --> 00:07:09
So, when t increases,
it gets longer and longer.
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00:07:09 --> 00:07:15
So, does everybody see this now?
Is that OK?
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00:07:15 --> 00:07:24
Any questions about that?
Yes?
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00:07:24 --> 00:07:26
OK, so I will try to avoid
using blue.
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00:07:26 --> 00:07:37
Thanks for, that's fine.
So, OK, I will not use blue
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00:07:37 --> 00:07:42
anymore.
OK, well, first let me just
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00:07:42 --> 00:07:46
make everything white just for
now.
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00:07:46 --> 00:07:49
This is the vector from Q0 to
Q(t).
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00:07:49 --> 00:07:56
This is the point Q(t).
OK, is it kind of visible now?
90
00:07:56 --> 00:08:03
OK, thanks for pointing it out.
I will switch to brighter
91
00:08:03 --> 00:08:09
colors.
So, OK, so apart from that,
92
00:08:09 --> 00:08:13
I claim now we can find the
position of its moving point
93
00:08:13 --> 00:08:15
because,
well, this vector,
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00:08:15 --> 00:08:19
Q0Q1 we can find from the
coordinates of Q0 and Q1.
95
00:08:19 --> 00:08:26
So, we just subtract the
coordinates of Q0 from those of
96
00:08:26 --> 00:08:30
Q1 will get that vector Q0 Q1 is
97
00:08:30 --> 00:08:32
98
00:08:32 --> 00:08:36
OK,
so, if I look at it,
99
00:08:36 --> 00:08:44
well, so let's call x(t),
y(t), and z(t) the coordinates
100
00:08:44 --> 00:08:50
of the point that's moving on
the line.
101
00:08:50 --> 00:09:00
Then we get x of t minus,
well, actually plus one equals
102
00:09:00 --> 00:09:07
t times two.
I'm writing the components of
103
00:09:07 --> 00:09:13
Q0Q(t).
And here, I'm writing t times
104
00:09:13 --> 00:09:19
Q0Q1.
y(t) minus two equals t,
105
00:09:19 --> 00:09:28
and z(t) minus two equals -3t.
So, in other terms,
106
00:09:28 --> 00:09:34
the more familiar way that we
used to write these equations,
107
00:09:34 --> 00:09:42
let me do it that way instead,
minus one plus 2t,
108
00:09:42 --> 00:09:53
y(t) = 2 t, z(t) = 2 - 3t.
And, if you prefer,
109
00:09:53 --> 00:10:02
I can just say Q(t) is Q0 plus
t times vector Q0Q1.
110
00:10:02 --> 00:10:07
OK, so that's our first
parametric equation of a line in
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00:10:07 --> 00:10:10
this class.
And, I hope you see it's not
112
00:10:10 --> 00:10:13
extremely hard.
In fact, parametric equations
113
00:10:13 --> 00:10:17
of lines always look like that.
x, y, and z are functions of t
114
00:10:17 --> 00:10:22
but are of the form a constant
plus a constant times t.
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00:10:22 --> 00:10:26
The coefficients of t tell us
about a vector along the line.
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00:10:26 --> 00:10:33
Here, we have a vector,
Q0Q1, which is .
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00:10:33 --> 00:10:37
And, the constant terms tell us
about where we are at t=0.
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00:10:37 --> 00:10:41
If I plug t=0 these guys go
away, I get minus 1,2,
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00:10:41 --> 00:10:46
2.
That's my starting point.
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00:10:46 --> 00:10:59
OK, so, any questions about
that?
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00:10:59 --> 00:11:05
No?
OK, so let's see,
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00:11:05 --> 00:11:12
now, what we can do with these
parametric equations.
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00:11:12 --> 00:11:26
So, one application is to think
about the relative position of a
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00:11:26 --> 00:11:36
line and a plane with respect to
each other.
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00:11:36 --> 00:11:44
So, let's say that we take
still the same line up there,
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00:11:44 --> 00:11:53
and let's consider the plane
with the equation x 2y 4z = 7.
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00:11:53 --> 00:11:55
OK, so I'm giving you this
plane.
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00:11:55 --> 00:11:58
And, the questions that we are
going to ask ourselves are,
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00:11:58 --> 00:12:00
well, does the line intersect
the plane?
130
00:12:00 --> 00:12:02
And, where does it intersect
the plane?
131
00:12:02 --> 00:12:22
132
00:12:22 --> 00:12:28
So, let's start with the first
primary question that maybe we
133
00:12:28 --> 00:12:32
should try to understand.
We have these points.
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00:12:32 --> 00:12:35
We have this plane,
and we have these points,
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00:12:35 --> 00:12:38
Q0 and Q1.
I'm going to draw them in
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00:12:38 --> 00:12:42
completely random places.
Well, are Q0 and Q1 on the same
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00:12:42 --> 00:12:47
side of a plane or on different
sides, on opposite sides of the
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00:12:47 --> 00:12:50
planes?
Could it be that maybe one of
139
00:12:50 --> 00:12:59
the points is in the plane?
So, I think I'm going to let
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00:12:59 --> 00:13:05
you vote on that.
So, is that readable?
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00:13:05 --> 00:13:08
Is it too small?
OK, so anyway,
142
00:13:08 --> 00:13:12
the question says,
relative to the plane,
143
00:13:12 --> 00:13:16
x 2y 4z = 7.
This point, Q0 and Q1,
144
00:13:16 --> 00:13:22
are they on the same side,
on opposite sides,
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00:13:22 --> 00:13:29
is one of them on the plane,
or we can't decide?
146
00:13:29 --> 00:13:41
OK, that should be better.
So, I see relatively few
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00:13:41 --> 00:13:46
answers.
OK, it looks like also a lot of
148
00:13:46 --> 00:13:51
you have forgotten the cards
and, so I see people raising two
149
00:13:51 --> 00:13:55
fingers, I see people raising
three fingers.
150
00:13:55 --> 00:13:57
And, I see people raising four
fingers.
151
00:13:57 --> 00:14:01
I don't see anyone answering
number one.
152
00:14:01 --> 00:14:03
So, the general idea seems to
be that either they are on
153
00:14:03 --> 00:14:07
opposite sides.
Maybe one of them is on the
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00:14:07 --> 00:14:10
plane.
Well, let's try to see.
155
00:14:10 --> 00:14:14
Is one of them on the plane?
Well, let's check.
156
00:14:14 --> 00:14:20
OK, so let's look at the point,
sorry.
157
00:14:20 --> 00:14:25
I have one blackboard to use
here.
158
00:14:25 --> 00:14:31
So, I take the point Q0,
which is at (-1,2,2).
159
00:14:31 --> 00:14:37
Well, if I plug that into the
plane equation,
160
00:14:37 --> 00:14:44
so, x 2y 4z will equal minus
one plus two times two plus four
161
00:14:44 --> 00:14:48
times two.
That's, well,
162
00:14:48 --> 00:14:52
four plus eight,
12 minus one,
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00:14:52 --> 00:14:54
11.
That, I think,
164
00:14:54 --> 00:15:01
is bigger than seven.
OK, so Q0 is not in the plane.
165
00:15:01 --> 00:15:07
Let's try again with Q1.
(1,3, - 1) well,
166
00:15:07 --> 00:15:15
if we plug that into x 2y 4z,
we'll have one plus two times
167
00:15:15 --> 00:15:20
three makes seven.
But, we add four times negative
168
00:15:20 --> 00:15:23
one.
We add up with three less than
169
00:15:23 --> 00:15:25
seven.
Well, that one is not in the
170
00:15:25 --> 00:15:27
plane, either.
So, I don't think,
171
00:15:27 --> 00:15:32
actually, that the answer
should be number three.
172
00:15:32 --> 00:15:37
So, let's get rid of answer
number three.
173
00:15:37 --> 00:15:42
OK, let's see,
in light of this,
174
00:15:42 --> 00:15:50
are you willing to reconsider
your answer?
175
00:15:50 --> 00:15:53
OK, so I think now everyone
seems to be interested in
176
00:15:53 --> 00:15:57
answering number two.
And, I would agree with that
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00:15:57 --> 00:16:00
answer.
So, let's think about it.
178
00:16:00 --> 00:16:02
These points are not in the
plane, but they are not in the
179
00:16:02 --> 00:16:05
plane in different ways.
One of them somehow overshoots;
180
00:16:05 --> 00:16:08
we get 11.
The other one we only get 3.
181
00:16:08 --> 00:16:12
That's less than seven.
If you think about how a plan
182
00:16:12 --> 00:16:15
splits space into two half
spaces on either side,
183
00:16:15 --> 00:16:22
well, one of them is going to
be the point where x 2y 4z is
184
00:16:22 --> 00:16:27
less than seven.
And, the other one will be,
185
00:16:27 --> 00:16:32
so, that's somehow this side.
And, that's where Q1 is.
186
00:16:32 --> 00:16:43
And, the other side is where x
2y 4z is actually bigger than
187
00:16:43 --> 00:16:47
seven.
And, to go from one to the
188
00:16:47 --> 00:16:53
other, well, x 2y 4z needs to go
through the value seven.
189
00:16:53 --> 00:16:57
If you're moving along any path
from Q0 to Q1,
190
00:16:57 --> 00:17:02
this thing will change
continuously from 11 to 3.
191
00:17:02 --> 00:17:05
At some time,
it has to go through 7.
192
00:17:05 --> 00:17:09
Does that make sense?
So, to go from Q0 to Q1 we need
193
00:17:09 --> 00:17:12
to cross P at some place.
So, they're on opposite sides.
194
00:17:12 --> 00:17:31
195
00:17:31 --> 00:17:37
OK, now that doesn't quite
finish answering the question
196
00:17:37 --> 00:17:43
that we had, which was,
where does the line intersect
197
00:17:43 --> 00:17:46
the plane?
But, why can't we do the same
198
00:17:46 --> 00:17:48
thing?
Now, we know not only the
199
00:17:48 --> 00:17:51
points Q0 and Q1,
we know actually any point on
200
00:17:51 --> 00:17:55
the line because we have a
parametric equation up there
201
00:17:55 --> 00:17:57
telling us,
where is the point that's
202
00:17:57 --> 00:18:04
moving on the line at time t?
So, what about the moving
203
00:18:04 --> 00:18:08
point, Q(t)?
Well, let's plug its
204
00:18:08 --> 00:18:10
coordinates into the plane
equation.
205
00:18:10 --> 00:18:24
So, we'll take x(t) 2y(t) 4z(t).
OK, that's equal to,
206
00:18:24 --> 00:18:34
well, (-1 2t) 2( 2 t) 4( 2 -
3t).
207
00:18:34 --> 00:18:41
So, if you simplify this a bit,
you get 2t 2t -12t.
208
00:18:41 --> 00:18:46
That should be -8t.
And, the constant term is minus
209
00:18:46 --> 00:18:54
one plus four plus eight is 11.
OK, and we have to compare that
210
00:18:54 --> 00:18:57
with seven.
OK, the question is,
211
00:18:57 --> 00:19:07
is this ever equal to seven?
Well, so, Q(t) is in the plane
212
00:19:07 --> 00:19:16
exactly when -8t 11 equals
seven.
213
00:19:16 --> 00:19:20
And, that' the same.
If you manipulate this,
214
00:19:20 --> 00:19:27
you will get t equals one half.
In fact, that's not very
215
00:19:27 --> 00:19:30
surprising.
If you look at these values,
216
00:19:30 --> 00:19:32
11 and three,
you see that seven is actually
217
00:19:32 --> 00:19:35
right in between.
It's the average of these two
218
00:19:35 --> 00:19:39
numbers.
So, it would make sense that
219
00:19:39 --> 00:19:44
it's halfway in between Q0 and
Q1, but we will get seven.
220
00:19:44 --> 00:19:50
OK, and that at that time,
Q at time one half,
221
00:19:50 --> 00:19:59
well, let's plug the values.
So, minus one plus 2t will be
222
00:19:59 --> 00:20:04
zero.
Two plus t will be two and a
223
00:20:04 --> 00:20:11
half of five halves,
and two minus three halves will
224
00:20:11 --> 00:20:15
be one half, OK?
So, this is where the line
225
00:20:15 --> 00:20:16
intersects the plane.
226
00:20:16 --> 00:20:43
227
00:20:43 --> 00:20:47
So, you see that's actually a
pretty easy way of finding where
228
00:20:47 --> 00:20:49
a line on the plane intersects
each other.
229
00:20:49 --> 00:20:52
If we can find a parametric
equation of a line and an
230
00:20:52 --> 00:20:55
equation of a plane,
but we basically just plug one
231
00:20:55 --> 00:20:59
into the other,
and see at what time the moving
232
00:20:59 --> 00:21:04
point hits the plane so that we
know where this.
233
00:21:04 --> 00:21:23
OK, other questions about this?
Yes?
234
00:21:23 --> 00:21:30
Sorry, can you say that?
Yes, so what if we don't get a
235
00:21:30 --> 00:21:32
solution?
What happens?
236
00:21:32 --> 00:21:36
So, indeed our line could have
been parallel to the plane or
237
00:21:36 --> 00:21:38
maybe even contained in the
plane.
238
00:21:38 --> 00:21:42
Well, if the line is parallel
to the plane then maybe what
239
00:21:42 --> 00:21:46
happens is that what we plug in
the positions of the moving
240
00:21:46 --> 00:21:48
point,
we actually get something that
241
00:21:48 --> 00:21:50
never equals seven because maybe
we get actually a constant.
242
00:21:50 --> 00:21:53
Say that we had gotten,
I don't know,
243
00:21:53 --> 00:21:56
13 all the time.
Well, when is 13 equal to seven?
244
00:21:56 --> 00:21:59
The answer is never.
OK, so that's what would tell
245
00:21:59 --> 00:22:02
you that the line is actually
parallel to the plane.
246
00:22:02 --> 00:22:06
You would not find a solution
to the equation that you get at
247
00:22:06 --> 00:22:13
the end.
Yes?
248
00:22:13 --> 00:22:16
So, if there's no solution at
all to the equation that you
249
00:22:16 --> 00:22:19
get, it means that at no time is
the traveling point going to be
250
00:22:19 --> 00:22:22
in the plane.
That means the line really does
251
00:22:22 --> 00:22:25
not have the plane ever.
So, it has to be parallel
252
00:22:25 --> 00:22:27
outside of it.
On the other hand,
253
00:22:27 --> 00:22:30
if a line is inside the plane,
then that means that no matter
254
00:22:30 --> 00:22:33
what time you choose,
you always get seven.
255
00:22:33 --> 00:22:37
OK, that's what would happen if
a line is in the plane.
256
00:22:37 --> 00:22:44
You always get seven.
So, maybe I should write this
257
00:22:44 --> 00:22:54
down.
So, if a line is in the plane
258
00:22:54 --> 00:23:10
then plugging x(t),
y(t), z(t) into the equation,
259
00:23:10 --> 00:23:18
we always get,
well, here in this case seven
260
00:23:18 --> 00:23:22
or whatever the value should be
for the plane,
261
00:23:22 --> 00:23:34
If the line is parallel to the
plane -- -- in fact,
262
00:23:34 --> 00:23:45
we, well, get,
let's see, another constant.
263
00:23:45 --> 00:23:49
So, in fact,
you know, when you plug in
264
00:23:49 --> 00:23:51
these things,
normally you get a quantity
265
00:23:51 --> 00:23:54
that's of a form,
something times t plus a
266
00:23:54 --> 00:23:57
constant because that's what you
plug into the equation of a
267
00:23:57 --> 00:23:59
plane.
And so, in general,
268
00:23:59 --> 00:24:01
you have an equation of the
form, something times t plus
269
00:24:01 --> 00:24:05
something equals something.
And, that usually has a single
270
00:24:05 --> 00:24:08
solution.
And, the special case is if
271
00:24:08 --> 00:24:11
this coefficient of t turns out
to be zero in the end,
272
00:24:11 --> 00:24:14
and that's actually going to
happen,
273
00:24:14 --> 00:24:20
exactly when the line is either
parallel or in the plane.
274
00:24:20 --> 00:24:24
In fact, if you think this
through carefully,
275
00:24:24 --> 00:24:26
the coefficient of t that you
get here,
276
00:24:26 --> 00:24:30
see, it's one times two plus
two times one plus four times
277
00:24:30 --> 00:24:33
minus three.
It's the dot product between
278
00:24:33 --> 00:24:37
the normal vector of a plane and
the vector along the line.
279
00:24:37 --> 00:24:41
So, see, this coefficient
becomes zero exactly when the
280
00:24:41 --> 00:24:44
line is perpendicular to the
normal vector.
281
00:24:44 --> 00:24:46
That means it's parallel to the
plane.
282
00:24:46 --> 00:24:51
So, everything makes sense.
OK, if you're confused about
283
00:24:51 --> 00:24:55
what I just said,
you can ignore it.
284
00:24:55 --> 00:25:03
OK, more questions? No?
OK, so if not,
285
00:25:03 --> 00:25:09
let's move on to linear
parametric equations.
286
00:25:09 --> 00:25:13
So, I hope you've seen here
that parametric equations are a
287
00:25:13 --> 00:25:18
great way to think about lines.
There are also a great way to
288
00:25:18 --> 00:25:22
think about actually any curve,
any trajectory that can be
289
00:25:22 --> 00:25:34
traced by a moving point.
So -- -- more generally,
290
00:25:34 --> 00:26:00
we can use parametric equations
-- -- for arbitrary motions --
291
00:26:00 --> 00:26:15
-- in the plane or in space.
So, let's look at an example.
292
00:26:15 --> 00:26:20
Let's take, so,
it's a famous curve called a
293
00:26:20 --> 00:26:23
cycloid.
A cycloid is something that you
294
00:26:23 --> 00:26:27
can actually see sometimes at
night when people are biking If
295
00:26:27 --> 00:26:31
you have something that reflects
light on the wheel.
296
00:26:31 --> 00:26:33
So, let me explain what's the
definition of a cycloid.
297
00:26:33 --> 00:27:05
298
00:27:05 --> 00:27:07
So, I should say,
I've seen a lecture where,
299
00:27:07 --> 00:27:10
actually, the professor had a
volunteer on a unicycle to
300
00:27:10 --> 00:27:13
demonstrate how that works.
But, I didn't arrange for that,
301
00:27:13 --> 00:27:17
so instead I will explain it to
you using more conventional
302
00:27:17 --> 00:27:23
means.
So, let's say that we have a
303
00:27:23 --> 00:27:31
wheel that's rolling on a
horizontal ground.
304
00:27:31 --> 00:27:34
And, as it's rolling of course
it's going to turn.
305
00:27:34 --> 00:27:40
So, it's going to move forward
to a new position.
306
00:27:40 --> 00:27:45
And, now, let's mention that we
have a point that's been painted
307
00:27:45 --> 00:27:47
red on the circumference of the
wheel.
308
00:27:47 --> 00:27:51
And, initially,
that point is here.
309
00:27:51 --> 00:27:53
So, as the wheel stops
rotating, well,
310
00:27:53 --> 00:27:57
of course, it moves forward,
and so it turns on itself.
311
00:27:57 --> 00:28:02
So, that point starts falling
back behind the point of contact
312
00:28:02 --> 00:28:07
because the wheel is rotating at
the same time as it's moving
313
00:28:07 --> 00:28:12
forward.
And so, the cycloid is the
314
00:28:12 --> 00:28:21
trajectory of this moving point.
OK, so the cycloid is obtained
315
00:28:21 --> 00:28:27
by considering,
so we have a wheel,
316
00:28:27 --> 00:28:38
let's say, of radius a.
So, this height here is (a)
317
00:28:38 --> 00:28:47
rolling on the floor which is
the x axis.
318
00:28:47 --> 00:28:53
And, let's -- And,
we have a point,
319
00:28:53 --> 00:29:01
P, that's painted on the wheel.
Initially, it's at the origin.
320
00:29:01 --> 00:29:04
But, of course,
as time goes by,
321
00:29:04 --> 00:29:13
it moves on the wheel.
P is a point on the rim of the
322
00:29:13 --> 00:29:21
wheel, and it starts at the
origin.
323
00:29:21 --> 00:29:27
So, the question is,
what happens?
324
00:29:27 --> 00:29:32
In particular,
can we find the position of
325
00:29:32 --> 00:29:37
this point, x(t),
y(t), as a function of time?
326
00:29:37 --> 00:29:42
So, that's the reason why I
have this computer.
327
00:29:42 --> 00:29:48
So, I'm not sure it will be
very easy to visualize,
328
00:29:48 --> 00:29:54
but so we have a wheel,
well, I hope you can vaguely
329
00:29:54 --> 00:30:00
see that there's a circle that's
moving.
330
00:30:00 --> 00:30:05
The wheel is green here.
And, there's a radius that's
331
00:30:05 --> 00:30:09
been painted blue in it.
And, that radius rotates around
332
00:30:09 --> 00:30:12
the wheel as the wheel is moving
forward.
333
00:30:12 --> 00:30:23
So, now, let's try to paint,
actually, the trajectory of a
334
00:30:23 --> 00:30:26
point.
[LAUGHTER]
335
00:30:26 --> 00:30:30
OK, so that's what the cycloid
looks like.
336
00:30:30 --> 00:30:37
[APPLAUSE]
OK, so -- So the cycloid,
337
00:30:37 --> 00:30:47
well, I guess it doesn't quite
look like what I've drawn.
338
00:30:47 --> 00:30:52
It looks like it goes a bit
higher up, which will be the
339
00:30:52 --> 00:30:57
trajectory of this red point.
And, see, it hits the bottom
340
00:30:57 --> 00:31:01
once in a while.
It forms these arches because
341
00:31:01 --> 00:31:04
when the wheel has rotated by a
full turn,
342
00:31:04 --> 00:31:07
then you're basically back at
the same situation,
343
00:31:07 --> 00:31:09
except a bit further along the
route.
344
00:31:09 --> 00:31:13
So, if we do it once more,
you see the point now is at the
345
00:31:13 --> 00:31:18
top, and now it's at the bottom.
And then we start again.
346
00:31:18 --> 00:31:23
It's at the top,
and then again at the bottom.
347
00:31:23 --> 00:31:40
OK.
No.
348
00:31:40 --> 00:31:48
[LAUGHTER]
OK, so the question that we
349
00:31:48 --> 00:31:58
want to answer is what is the
position x(t),
350
00:31:58 --> 00:32:05
y(t), of the point P?
OK, so actually,
351
00:32:05 --> 00:32:07
I'm writing x(t),
y(t).
352
00:32:07 --> 00:32:10
That means that I have,
maybe I'm expressing the
353
00:32:10 --> 00:32:13
position in terms of time.
Let's see, is time going to be
354
00:32:13 --> 00:32:15
a good thing to do?
Well, suddenly,
355
00:32:15 --> 00:32:20
the position changes over time.
But doesn't actually matter how
356
00:32:20 --> 00:32:24
fast the wheel is rolling?
No, because I can just play the
357
00:32:24 --> 00:32:27
motion fast-forward.
The wheel will be going faster,
358
00:32:27 --> 00:32:29
but the trajectory is still the
same.
359
00:32:29 --> 00:32:32
So, in fact,
time is not the most relevant
360
00:32:32 --> 00:32:36
thing here.
What matters to us now is how
361
00:32:36 --> 00:32:39
far the wheel has gone.
So, we could use as a
362
00:32:39 --> 00:32:44
parameter, for example,
the distance by which the wheel
363
00:32:44 --> 00:32:46
has moved.
We can do even better because
364
00:32:46 --> 00:32:49
we see that, really,
the most complicated thing that
365
00:32:49 --> 00:32:50
happens here is really the
rotation.
366
00:32:50 --> 00:32:55
So, maybe we can actually use
the angle by which the wheel has
367
00:32:55 --> 00:32:57
turned to parameterize the
motion.
368
00:32:57 --> 00:33:02
So, there's various choices.
You can choose whichever one
369
00:33:02 --> 00:33:04
you prefer.
But, I think here,
370
00:33:04 --> 00:33:07
we will get the simplest answer
if we parameterize things by the
371
00:33:07 --> 00:33:10
angle.
So, in fact,
372
00:33:10 --> 00:33:23
instead of t I will be using
what's called theta as a
373
00:33:23 --> 00:33:36
function of the angle,
theta, by which the wheel has
374
00:33:36 --> 00:33:50
rotated.
So, how are we going to do that?
375
00:33:50 --> 00:33:57
Well, because we are going to
try to use our new knowledge,
376
00:33:57 --> 00:34:03
let's try to do it using
vectors in a smart way.
377
00:34:03 --> 00:34:07
So, let me draw a picture of
the wheel after things have
378
00:34:07 --> 00:34:12
rotated by a certain amount.
So, maybe my point,
379
00:34:12 --> 00:34:18
P, now, is here.
And, so the wheel has rotated
380
00:34:18 --> 00:34:21
by this angle here.
And, I want to find the
381
00:34:21 --> 00:34:23
position of my point,
P, OK?
382
00:34:23 --> 00:34:29
So, the position of this point,
P, is going to be the same as
383
00:34:29 --> 00:34:35
knowing the vector OP from the
origin to this moving point.
384
00:34:35 --> 00:34:39
So, I haven't really simplify
the problem yet because we don't
385
00:34:39 --> 00:34:43
really know about vector OP.
But, maybe we know about
386
00:34:43 --> 00:34:47
simpler vectors where some will
be OP.
387
00:34:47 --> 00:34:50
So, let's see,
let's give names to a few of
388
00:34:50 --> 00:34:52
our points.
For example,
389
00:34:52 --> 00:34:54
let's say that this will be
point A.
390
00:34:54 --> 00:34:58
A is the point where the wheel
is touching the road.
391
00:34:58 --> 00:35:02
And, B will be the center of
the wheel.
392
00:35:02 --> 00:35:07
Then, it looks like maybe I
have actually a chance of
393
00:35:07 --> 00:35:12
understanding vectors like maybe
OA doesn't look quite so scary,
394
00:35:12 --> 00:35:16
or AB doesn't look too bad.
BP doesn't look too bad.
395
00:35:16 --> 00:35:27
And, if I sum them together,
I will obtain OP.
396
00:35:27 --> 00:35:35
So, let's do that.
So, now we've greatly
397
00:35:35 --> 00:35:39
simplified the problem.
We had to find one vector that
398
00:35:39 --> 00:35:42
we didn't know.
Now we have to find three
399
00:35:42 --> 00:35:47
vectors which we don't know.
But, you will see each of them
400
00:35:47 --> 00:35:50
as fairly easy to think about.
So, let's see.
401
00:35:50 --> 00:35:56
Should we start with vector OA,
maybe?
402
00:35:56 --> 00:36:04
So, OA has two components.
One of them should be very easy.
403
00:36:04 --> 00:36:06
Well, the y component is just
going to be zero,
404
00:36:06 --> 00:36:10
OK?
It's directed along the x axis.
405
00:36:10 --> 00:36:15
What about the x component?
So, OA is the distance by which
406
00:36:15 --> 00:36:21
the wheel has traveled to get to
its current position.
407
00:36:21 --> 00:36:23
Yeah.
I hear a lot of people saying R
408
00:36:23 --> 00:36:25
theta.
Let me actually say a(theta)
409
00:36:25 --> 00:36:28
because I've called a the radius
of the wheel.
410
00:36:28 --> 00:36:33
So, this distance is a(theta).
Why is it a(theta)?
411
00:36:33 --> 00:36:36
Well, that's because the wheel,
well, there's an assumption
412
00:36:36 --> 00:36:38
which is that the wheel is
rolling on something normal like
413
00:36:38 --> 00:36:40
a road,
and not on, maybe,
414
00:36:40 --> 00:36:45
ice, or something like that.
S So, it's rolling without
415
00:36:45 --> 00:36:48
slipping.
So, that means that this
416
00:36:48 --> 00:36:53
distance on the road is actually
equal to the distance here on
417
00:36:53 --> 00:36:57
the circumference of the wheel.
This point, P,
418
00:36:57 --> 00:37:01
was there, and the amount by
which the things have moved can
419
00:37:01 --> 00:37:06
be measured either here or here.
These are the same distances.
420
00:37:06 --> 00:37:15
OK, so, that makes it a(theta),
and maybe I should justify by
421
00:37:15 --> 00:37:22
saying amount by which the wheel
has rolled,
422
00:37:22 --> 00:37:30
has moved, is equal to the,
so, the distance from O to A is
423
00:37:30 --> 00:37:37
equal to the arc length on the
circumference of the circle from
424
00:37:37 --> 00:37:40
A to P.
And, you know that if you have
425
00:37:40 --> 00:37:42
a sector corresponding to an
angle, theta,
426
00:37:42 --> 00:37:45
then its length is a times
theta, provided that,
427
00:37:45 --> 00:37:48
of course, you express the
angel in radians.
428
00:37:48 --> 00:37:58
That's the reason why we always
used radians in math.
429
00:37:58 --> 00:38:01
Now, let's think about vector
AB and vector BP.
430
00:38:01 --> 00:38:30
431
00:38:30 --> 00:38:39
OK, so AB is pretty easy,
right, because it's pointing
432
00:38:39 --> 00:38:45
straight up, and its length is
a.
433
00:38:45 --> 00:38:55
So, it's just zero, a.
Now, the most serious one we've
434
00:38:55 --> 00:39:00
kept for the end.
What about vector BP?
435
00:39:00 --> 00:39:04
So, vector BP,
we know two things about it.
436
00:39:04 --> 00:39:17
We know actually its length,
so, the magnitude of BP -- --
437
00:39:17 --> 00:39:23
a.
And, we know it makes an angle,
438
00:39:23 --> 00:39:29
theta, with the vertical.
So, that should let us find its
439
00:39:29 --> 00:39:34
components.
Let's draw a closer picture.
440
00:39:34 --> 00:39:40
Now, in the picture I'm going
to center things at B.
441
00:39:40 --> 00:39:44
So, I have my point P.
Here I have theta.
442
00:39:44 --> 00:39:49
This length is A.
Well, what are the components
443
00:39:49 --> 00:39:57
of BP?
Well, the X component is going
444
00:39:57 --> 00:39:59
to be?
Almost.
445
00:39:59 --> 00:40:03
I hear people saying things
about a, but I agree with a.
446
00:40:03 --> 00:40:04
I hear some cosines.
I hear some sines.
447
00:40:04 --> 00:40:07
I think it's actually the sine.
Yes.
448
00:40:07 --> 00:40:10
It's a(sin(theta)),
except it's going to the left.
449
00:40:10 --> 00:40:18
So, actually it will have a
negative a(sin(theta)).
450
00:40:18 --> 00:40:23
And, the vertical component,
well, it will be a(cos(theta)),
451
00:40:23 --> 00:40:27
but also negative because we
are going downwards.
452
00:40:27 --> 00:40:46
So, it's negative a(cos(theta)).
So, now we can answer the
453
00:40:46 --> 00:40:52
initial question because vector
OP, well, we just add up OA,
454
00:40:52 --> 00:40:57
AB, and BP.
So, the X component will be
455
00:40:57 --> 00:41:09
a(theta) - a(sin(theta)).
And, a-a(cos(theta)).
456
00:41:09 --> 00:41:25
OK.
So, any questions about that?
457
00:41:25 --> 00:41:29
OK, so, what's the answer?
Because this thing here is the
458
00:41:29 --> 00:41:35
x coordinate as a function of
theta, and that one is the y
459
00:41:35 --> 00:41:39
coordinate as a function of
theta.
460
00:41:39 --> 00:41:44
So, now, just to show you that
we can do a lot of things when
461
00:41:44 --> 00:41:48
we have a parametric equation,
here is a small mystery.
462
00:41:48 --> 00:41:54
So, what happens exactly near
the bottom point?
463
00:41:54 --> 00:41:57
What does the curve look like?
The computer tells us,
464
00:41:57 --> 00:41:59
well, it looks like it has some
sort of pointy thing,
465
00:41:59 --> 00:42:02
but isn't that something of a
display?
466
00:42:02 --> 00:42:12
Is it actually what happens?
So, what do you think happens
467
00:42:12 --> 00:42:19
near the bottom point?
Remember, we had that picture.
468
00:42:19 --> 00:42:24
Let me show you once more,
where you have these
469
00:42:24 --> 00:42:28
corner-like things at the
bottom.
470
00:42:28 --> 00:42:31
Well, actually,
is it indeed a corner with some
471
00:42:31 --> 00:42:34
angle between the two
directions?
472
00:42:34 --> 00:42:38
Does it make an angle?
Or, is it actually a smooth
473
00:42:38 --> 00:42:42
curve without any corner,
but we don't see it because
474
00:42:42 --> 00:42:46
it's too small to be visible on
the computer screen?
475
00:42:46 --> 00:42:50
Does it actually make a loop?
Does it actually come down and
476
00:42:50 --> 00:42:55
then back up without going to
the left or to the right and
477
00:42:55 --> 00:43:01
without making an angle?
So, yeah, I see the majority
478
00:43:01 --> 00:43:05
votes for answers number two or
four.
479
00:43:05 --> 00:43:08
And, well, at this point,
we can't quite tell.
480
00:43:08 --> 00:43:10
So, let's try to figure it out
from these formulas.
481
00:43:10 --> 00:43:17
The way to answer that for sure
is to actually look at the
482
00:43:17 --> 00:43:23
formulas.
OK, so question that we are
483
00:43:23 --> 00:43:34
trying to answer now is what
happens near the bottom point?
484
00:43:34 --> 00:43:52
485
00:43:52 --> 00:43:58
OK, so how do we answer that?
Well, we should probably try to
486
00:43:58 --> 00:44:03
find simpler formulas for these
things.
487
00:44:03 --> 00:44:06
Well, to simplify,
let's divide everything by a.
488
00:44:06 --> 00:44:08
Let's rescale everything by a.
If you want,
489
00:44:08 --> 00:44:12
let's say that we take the unit
of length to be the radius of
490
00:44:12 --> 00:44:15
our wheel.
So, instead of measuring things
491
00:44:15 --> 00:44:18
in feet or meters,
we'll just measure them in
492
00:44:18 --> 00:44:25
radius.
So, take the length unit to be
493
00:44:25 --> 00:44:32
equal to the radius.
So, that means we'll have a=1.
494
00:44:32 --> 00:44:35
Then, our formulas are slightly
simpler.
495
00:44:35 --> 00:44:45
We get x(theta) is theta -
sin(theta), and y equals 1 - cos
496
00:44:45 --> 00:44:49
(theta).
OK, so, if we want to
497
00:44:49 --> 00:44:52
understand what these things
look like, maybe we should try
498
00:44:52 --> 00:44:56
to take some approximation.
OK, so what about
499
00:44:56 --> 00:45:00
approximations?
Well, probably you know that if
500
00:45:00 --> 00:45:07
I take the sine of a very small
angle, it's close to the actual
501
00:45:07 --> 00:45:12
angle itself if theta is very
small.
502
00:45:12 --> 00:45:18
And, you know that the cosine
of an angle that's very small is
503
00:45:18 --> 00:45:21
close to one.
Well, that's pretty good.
504
00:45:21 --> 00:45:23
If we use that,
we will get theta minus theta,
505
00:45:23 --> 00:45:26
one minus one,
it looks like it's not precise
506
00:45:26 --> 00:45:29
enough.
We just get zero and zero.
507
00:45:29 --> 00:45:31
That's not telling us much
about what happens.
508
00:45:31 --> 00:45:39
OK, so we need actually better
approximations than that.
509
00:45:39 --> 00:45:50
So -- So, hopefully you have
seen in one variable calculus
510
00:45:50 --> 00:45:57
something called Taylor
expansion.
511
00:45:57 --> 00:46:14
That's [GROANS].
I see that -- OK,
512
00:46:14 --> 00:46:17
so if you have not seen Taylor
expansion,
513
00:46:17 --> 00:46:21
or somehow it was so traumatic
that you've blocked it out of
514
00:46:21 --> 00:46:24
your memory,
let me just remind you that
515
00:46:24 --> 00:46:27
Taylor expansion is a way to get
a better approximation than just
516
00:46:27 --> 00:46:32
looking at the function,
its derivative.
517
00:46:32 --> 00:46:42
So -- And, here's an example of
where it actually comes in handy
518
00:46:42 --> 00:46:52
in real life.
So, Taylor approximation says
519
00:46:52 --> 00:47:01
that if t is small,
then the value of the function,
520
00:47:01 --> 00:47:04
f(t), is approximately equal
to,
521
00:47:04 --> 00:47:07
well, our first guess,
of course, would be f(0).
522
00:47:07 --> 00:47:12
That's our first approximation.
If we want to be a bit more
523
00:47:12 --> 00:47:15
precise, we know that when we
change by t,
524
00:47:15 --> 00:47:17
well, t times the derivative
comes in,
525
00:47:17 --> 00:47:23
that's for linear approximation
to how the function changes.
526
00:47:23 --> 00:47:28
Now, if we want to be even more
precise, there's another term,
527
00:47:28 --> 00:47:32
which is t^2 over two times the
second derivative.
528
00:47:32 --> 00:47:37
And, if we want to be even more
precise, you will have t^3 over
529
00:47:37 --> 00:47:41
six times the third derivative
at zero.
530
00:47:41 --> 00:47:43
OK, and you can continue,
and so on.
531
00:47:43 --> 00:47:49
But, we won't need more.
So, if you use this here,
532
00:47:49 --> 00:47:53
it tells you that the sine of a
smaller angle,
533
00:47:53 --> 00:47:57
theta, well,
yeah, it looks like theta.
534
00:47:57 --> 00:48:01
But, if we want to be more
precise, then we should add
535
00:48:01 --> 00:48:06
minus theta cubed over six.
And, cosine of theta,
536
00:48:06 --> 00:48:12
well, it's not quite one.
It's close to one minus theta
537
00:48:12 --> 00:48:16
squared over two.
OK, so these are slightly
538
00:48:16 --> 00:48:21
better approximations of sine
and cosine for small angles.
539
00:48:21 --> 00:48:28
So, now, if we try to figure
out, again, what happens to our
540
00:48:28 --> 00:48:31
x of theta, well,
it would be,
541
00:48:31 --> 00:48:36
sorry, theta minus theta cubed
over six.
542
00:48:36 --> 00:48:44
That's theta cubed over six.
And y, on the other hand,
543
00:48:44 --> 00:48:53
is going to be one minus that.
That's about theta squared over
544
00:48:53 --> 00:48:57
two.
So, now, which one of them is
545
00:48:57 --> 00:49:01
bigger when theta is small?
Yeah, y is much larger.
546
00:49:01 --> 00:49:03
OK, if you take the cube of a
very small number,
547
00:49:03 --> 00:49:06
it becomes very,
very, very small.
548
00:49:06 --> 00:49:09
So, in fact,
we can look at that.
549
00:49:09 --> 00:49:15
So, x, an absolute value,
is much smaller than y.
550
00:49:15 --> 00:49:17
And, in fact,
what we can do is we can look
551
00:49:17 --> 00:49:21
at the ratio between y and x.
That tells us the slope with
552
00:49:21 --> 00:49:27
which we approach the origin.
So, y over x is,
553
00:49:27 --> 00:49:35
well, let's take the ratio of
this, too.
554
00:49:35 --> 00:49:38
That gives us three divided by
theta.
555
00:49:38 --> 00:49:45
That tends to infinity when
theta approaches zero.
556
00:49:45 --> 00:49:53
So, that means that the slope
of our curve,
557
00:49:53 --> 00:50:00
the origin is actually
infinite.
558
00:50:00 --> 00:50:05
And so, the curve picture is
really something like this.
559
00:50:05 --> 00:50:07
So, the instantaneous motion,
if you had to describe what
560
00:50:07 --> 00:50:09
happens very,
very close to the origin is
561
00:50:09 --> 00:50:12
that your point is actually not
moving to the left or to the
562
00:50:12 --> 00:50:17
right along with the wheel.
It's moving down and up.
563
00:50:17 --> 00:50:20
I mean, at the same time it is
actually moving a little bit
564
00:50:20 --> 00:50:24
forward at the same time.
But, the dominant motion,
565
00:50:24 --> 00:50:29
near the origin is really where
it goes down and back up,
566
00:50:29 --> 00:50:33
so answer number four,
you have vertical tangent.
567
00:50:33 --> 00:50:37
OK, I think I'm at the end of
time.
568
00:50:37 --> 00:50:44
So, have a nice weekend.
And, I'll see you on Tuesday.
569
00:50:44 --> 00:50:47
So, on Tuesday I will have
practice exams for next week's
570
00:50:47 --> 00:50:50
test.
571
00:50:50 --> 00:50:55