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CHRISTINE BREINER: Welcome
back to recitation.

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In this video, what I'd like us
to do is answer some questions

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I've posed here in a
really long problem.

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So let me take us through it.

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So we're going to consider
a position vector that's

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described by x of t, y of t, 0.

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I want to have it
in three-space so I

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can take a cross product later.

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And then we're going to
suppose that it actually

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has constant length.

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And that when I look at the
acceleration vector at t,

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it's actually equal to
a constant times r of t,

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where the constant is not 0.

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OK?

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So I can, these
are the two things

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I know about this
position vector.

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It has constant length.

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For all t, it has
constant length.

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And the acceleration is always
equal to some constant times

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the position.

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OK, that's what I'm giving you.

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And now I want you to use
those things and vector

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differentiation to show
that r dot v is equal to 0,

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where v is the velocity.

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And then to show that
r cross v is constant.

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So you're going to have to
figure out-- essentially,

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the thing you have
to figure out is

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what relationship do you
want to differentiate

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to show these two things.

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OK?

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That's the hard part
of this problem.

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And then, I went
to see if you can

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give an example of such an r.

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So if you can give an
example of a position vector

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that has these properties.

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And maybe if you're
having a hard time,

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the first thing for you to do
might be to think about this,

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and to see if you can figure
out an example of that,

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and then see kind of how things
work together in that example.

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That may actually
proved helpful.

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So why don't you work on this
problem, pause the video,

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and then when you're feeling
good about your answer,

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you can bring the video back up
and I'll show you how I do it.

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OK, welcome back.

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So there was a lot to
do in this problem,

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but let me just remind
you what the framework is.

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We have a position vector, and
we know two things about it.

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We know that it
has constant length

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and we know that the
acceleration is always

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equal to some constant
times the position.

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I didn't give you
the constant, but we

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know it's always equal to some
constant times the position.

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And then we wanted to
show two things using

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vector differentiation.

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We wanted to show
that r dot v was 0.

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And we wanted to show that
r cross v was constant.

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And then we want to talk
about what is an example.

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So let's start off and
see if we can figure out

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how to show that r
dot v is equal to 0.

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And, you know, as you're
thinking about this problem,

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something that you
want to remember

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as you're thinking
about this is, well,

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what are the things that I know?

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I know that r dot r
is constant, so I'm

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going to write that down.

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r dot r-- I'm not
going to say is

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equal to c, because that's a
different constant-- so I'll

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just, let me just call it c_1.

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OK?

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That's a different
constant than my c, maybe.

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OK, I know that r dot
r is some constant,

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and I want to show something
about r dot v. Right?

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So if I'm looking
at this and I say,

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well, I know this thing here.

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So it's the only dot product
relationship that I have.

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Because right now, I know a
relationship between a and r,

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and I know r has
constant length.

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Since that's all
I gave you, if I

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want to look at a
dot product, this

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is the relationship I know.

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The constant length thing.

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And so I know I somehow have
to use this one to figure out

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something about r dot v.

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OK.

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Well, what's the point
that we should realize?

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What is v?

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v is d/dt of r.

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So I could take the
derivative-- if I

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could take the derivative
of just one of these things,

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then I would get
r dot v down here.

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Right?

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If I took d/dt of
just one of these,

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I would get the r dot v.
And d/dt of this is 0.

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But I can't do that, right?

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Because if I take d/dt
of this whole thing,

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I'm going to end up having
to differentiate this r once

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and leave this
alone, and I'm going

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to have to differentiate
this r and leave this alone.

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But if you heard what
I was just saying,

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maybe you see that that's
actually still going to be OK.

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So let's look at what happens.

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I mean, this is all really
I have to work with,

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so I'm going to explore.

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Let's look at what happens
when I take d/dt of r dot r.

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And I'm going to start
leaving off the hats here,

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because I'm going to
leave them off somewhere,

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so we'll just
leave them off now,

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and then I won't leave
some off and put some on.

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So from here on out, I'm just
going to write r, v, and a

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without that hats,
but they're vectors.

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OK.

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So d/dt of r dot r, well,
I have to take d/dt of r,

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and then I dot that
with r, and then

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I take r dotted with--
sorry-- d/dt of r.

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Right?

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Well, what do I get here?
d/dt of r we said was v,

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so that's v dot r.

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And what do I get here?

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This is r dot-- there's d/dt
of r again-- so I get r dot v.

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Now the great thing
about the dot product

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is that if I switch
the order of these two,

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it's still the same thing.

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So I can just write
this as-- well, I'll

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switch the order of this one.

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So they both look like
r dot v plus r dot v,

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and that means I get
2 r dot v. Right?

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So d/dt of r dot r
is actually 2 of r

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dotted with v-- the position
dotted with the velocity.

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Now why is this
going to help me?

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Because what do I know
about this quantity r dot r?

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I know it's constant, right?

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So what is d/dt of a constant?

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d/dt of a constant is 0.

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So I actually started off
with knowing this was 0.

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So if I go through
the whole chain,

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I see 0 is equal to-- let me
put the 0 down here again-- 0 is

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equal to 2 r dot v, and so r
dot v I know is equal to 0.

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What does that
mean geometrically?

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That means r and
v are orthogonal.

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And where is v going to sit?

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Well, if I come back over
to how I described r,

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r is in the xy-plane, right?

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The z-component is 0.

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So if I differentiate--
if I take

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d/dt of r-- I'm going to
have the derivative of x

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as a function of t.

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And then whatever y
is as a function of t,

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I take that derivative.

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And this is still 0.

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So v is going to
sit in the xy-plane,

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and based on what
we know so far,

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we know it's actually
orthogonal to r.

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So they make a 90-degree
angle at all times t.

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OK, and how did
we do that again?

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I just want to
remind you, we knew

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one dot product
relationship, that

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was r dot r was a constant.

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So we differentiated that and
tried to see what happened.

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And the main point
at the end of it,

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was that when I had a v dot r, I
could rewrite it as an r dot v,

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and so I just end up
with two of something

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that I want to know about.

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So that's the main
idea of the first part.

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Now the second part was
I asked you to figure out

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something over here.

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We wanted to know that
r cross v is constant.

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OK?

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And t, it's always the same.

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All right?

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So let's think about
if I want to show

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that for every t
something is constant,

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I could show-- actually, I've
sort of seen it already--

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I could show that
its derivative is 0.

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OK.

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So if r cross v is constant--
or if its derivative is 0,

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I should say-- then r
cross v was constant.

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Right?

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If its derivative is 0 for all
t, then r cross v is constant.

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Right?

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So that's really what
we want to exploit here.

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So let's look at the idea.

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So we want to-- again,
let me write it down--

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show r cross v is constant.

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And the strategy
we're going to use

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is to do this we're going to
show that d/dt of r cross v

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is equal to 0.

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Right?

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If we can show that, then
this means that for all t

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it's the same.

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It's not changing in t.

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Right?

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So for all t, r cross v
is going to be the same,

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so r cross v is
going to be constant.

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So the difference
between the two problems

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was in the first problem
you didn't quite know maybe

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what expression you needed
to differentiate to find

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what you were looking for.

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Here, we know what we
need to differentiate,

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but we have to make
sure we understand--

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to show this is constant--
when we differentiate,

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we should get 0.

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OK?

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So that's sort of a slightly
different type of problem.

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And I'm asking you maybe
from the other side here.

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So let's now--
let's just see what

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we get on the left-hand side.

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So what's d/dt of r cross v?

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Well, d/dt of r is v, right?

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So we get v cross v
for the first term.

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So take d/dt of r, we get
v. We leave this v alone,

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and then we add to that, r cross
d/dt of v-- what's d/dt of v?

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That's a.

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Right?

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Now, let's take a look at this.

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Well, v cross v, v is
pointing in the same direction

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as itself.

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So when you try and take
a cross product of that,

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you know that the
length of your vector

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should be the area of
the parallelogram formed

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by these two vectors.

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But v is pointing in the
same direction as itself,

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so there's no area there.

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That's a geometric
interpretation

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of why this thing should be 0.

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Another reason is that
remember that your v cross

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v is going to
include a sine theta

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term, where theta is the angle
between the two vectors, right?

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That's another formula you have.

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And so when you look at the
angle between this vector

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and itself, it's 0.

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And sine 0 is 0.

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So this is, in fact,
0 in that part.

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So if this is 0, then
we get what we want.

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Well, I only gave
you one other bit

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of information in the problem.

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And if you remember,
it was that a is always

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equal to a constant times r.

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So I can rewrite
this right-hand side

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as r cross a constant times r.

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And because of properties
of these cross products,

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I can pull out the constant.

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Or I can actually,
I guess I don't

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need to pull it out
to talk about it,

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but it's nicer if I pull it out.

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And look at what I have here.

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I have the exact same
situation as v cross v. I mean,

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00:10:08,716 --> 00:10:10,590
this is still pointing
in the same direction.

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00:10:10,590 --> 00:10:14,880
Constant times r and r still
point in the same direction

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00:10:14,880 --> 00:10:16,750
as if I were to compare r and r.

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00:10:16,750 --> 00:10:18,500
So I didn't have to
pull out the constant,

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00:10:18,500 --> 00:10:20,220
but then right
here it's very easy

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00:10:20,220 --> 00:10:22,370
to see that this is also 0.

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00:10:22,370 --> 00:10:25,120
So I had 0 plus
this, so I get 0.

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00:10:25,120 --> 00:10:27,860
So, I've shown through
this process-- maybe

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00:10:27,860 --> 00:10:30,070
I should have written
equal signs here--

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00:10:30,070 --> 00:10:34,720
that d/dt of r cross v
is actually equal to 0.

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00:10:34,720 --> 00:10:38,870
And so you see that this cross
product between r and v--

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00:10:38,870 --> 00:10:41,640
which we know are orthogonal
sitting in the xy plane--

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00:10:41,640 --> 00:10:45,100
that it's always the same,
it's always the same vector.

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00:10:45,100 --> 00:10:47,232
And now I asked you
to give an example,

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00:10:47,232 --> 00:10:48,690
and maybe you
thought of an example

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00:10:48,690 --> 00:10:50,930
first, and then thought
of how it worked.

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00:10:50,930 --> 00:10:56,750
And so the easiest example is if
you let r of t equal cosine t,

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00:10:56,750 --> 00:10:58,100
sine t.

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00:10:58,100 --> 00:11:04,400
So the easiest example--
there are others, obviously--

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00:11:04,400 --> 00:11:07,480
is if you let r of t equal
cosine t, sine t comma 0.

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00:11:07,480 --> 00:11:11,920
Sorry, I was thinking about
it in three-space, right?

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00:11:11,920 --> 00:11:15,100
And in fact, if I were to scale
this, it would still work.

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00:11:15,100 --> 00:11:17,070
I could put any
constant in front.

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00:11:17,070 --> 00:11:19,950
This carves out-- if I
let t go between 0 and 2

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00:11:19,950 --> 00:11:22,080
pi, or even minus
infinity to infinity,

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00:11:22,080 --> 00:11:24,020
I'm just carving
out vectors that

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00:11:24,020 --> 00:11:27,790
are-- the position vector
is always on the unit

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00:11:27,790 --> 00:11:28,950
circle on this case.

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00:11:28,950 --> 00:11:29,590
Right?

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00:11:29,590 --> 00:11:32,220
If I put a constant in front,
it's on another circle.

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00:11:32,220 --> 00:11:36,580
So for whatever values of t
I'm letting myself vary over,

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00:11:36,580 --> 00:11:41,160
all the vectors are going to
lie on some part of a circle.

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00:11:41,160 --> 00:11:41,790
OK?

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00:11:41,790 --> 00:11:44,600
And so this is maybe
the easy example.

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00:11:44,600 --> 00:11:46,060
Maybe you want to
calculate, just

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00:11:46,060 --> 00:11:50,290
to give yourself some practice,
what v of t is and what a of t

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00:11:50,290 --> 00:11:50,870
actually is.

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00:11:50,870 --> 00:11:53,710
What these two quantities
actually are, and then look

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00:11:53,710 --> 00:11:54,380
at what happens.

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00:11:54,380 --> 00:11:59,245
What happens with r and v, and
see why r dot v is equal to 0.

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00:11:59,245 --> 00:12:02,970
What you should see-- I'll try
and give you a picture of it

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00:12:02,970 --> 00:12:03,600
geometrically.

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00:12:03,600 --> 00:12:07,730
What you should see
is that, you know,

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00:12:07,730 --> 00:12:10,995
if this is-- we'll see
if I can effectively

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00:12:10,995 --> 00:12:13,970
do a two-dimensional drawing
in three-space-- if this is

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00:12:13,970 --> 00:12:17,910
the r I'm looking at, that v
has to be coming in this way,

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00:12:17,910 --> 00:12:19,370
out this way.

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00:12:19,370 --> 00:12:21,980
So there's r and there's v.
And this angle-- because I'm

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00:12:21,980 --> 00:12:24,070
trying to squash what
was a circle-- I guess

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00:12:24,070 --> 00:12:26,460
I'll look from above first.

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00:12:26,460 --> 00:12:28,400
My picture, from
above, is looking

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00:12:28,400 --> 00:12:29,900
like something like this.

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00:12:29,900 --> 00:12:33,140
There's r and there's v,
and that's a right angle.

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00:12:33,140 --> 00:12:33,640
Right?

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00:12:33,640 --> 00:12:37,500
So if I look from-- coming
down on to the xy-plane,

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00:12:37,500 --> 00:12:40,450
here's a position vector,
here's its velocity,

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00:12:40,450 --> 00:12:41,372
they're orthogonal.

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00:12:41,372 --> 00:12:43,205
And as I move all the
way around the circle,

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00:12:43,205 --> 00:12:44,930
that position vector
and the velocity

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00:12:44,930 --> 00:12:46,430
are going to keep
that relationship.

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00:12:46,430 --> 00:12:50,810
When I look at this-- I'm trying
to insert in the z-axis here.

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00:12:50,810 --> 00:12:52,570
When I look at this,
if I look at what

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00:12:52,570 --> 00:12:54,320
is the cross product
of these two vectors,

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00:12:54,320 --> 00:12:56,714
well it's always going to
point in the z-direction.

315
00:12:56,714 --> 00:12:59,130
It's going to point straight
in the z-direction from here.

316
00:12:59,130 --> 00:13:01,060
Because it's orthogonal
to both of these,

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00:13:01,060 --> 00:13:04,740
it's going to point
straight in the z-direction.

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00:13:04,740 --> 00:13:06,910
I know r is constant length.

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00:13:06,910 --> 00:13:08,530
I can then see v
is constant length

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00:13:08,530 --> 00:13:09,750
from the example I have here.

321
00:13:09,750 --> 00:13:11,124
And as I rotate,
I'm going to get

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00:13:11,124 --> 00:13:12,850
that this is constant length.

323
00:13:12,850 --> 00:13:13,350
OK?

324
00:13:13,350 --> 00:13:18,450
So this is where
the picture of what

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00:13:18,450 --> 00:13:21,050
we were actually describing
very much more generally

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00:13:21,050 --> 00:13:22,930
in the first part of this.

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00:13:22,930 --> 00:13:25,950
So the main point that I want
us to see in this problem

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00:13:25,950 --> 00:13:29,590
is that when we want to find
information about relationships

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00:13:29,590 --> 00:13:33,055
between r, v, and a--
this position and velocity

330
00:13:33,055 --> 00:13:35,970
and acceleration--
what we can do

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00:13:35,970 --> 00:13:37,640
is differentiate
these vector fields.

332
00:13:37,640 --> 00:13:39,912
We saw an example when
you were looking at I

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00:13:39,912 --> 00:13:44,070
think Kepler's second law
you saw this in lecture.

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00:13:44,070 --> 00:13:46,960
But I just want to show
you that you can use this

335
00:13:46,960 --> 00:13:50,010
if you don't necessarily
know explicitly

336
00:13:50,010 --> 00:13:52,330
things about a position vector.

337
00:13:52,330 --> 00:13:55,340
You can still find out
things about its relationship

338
00:13:55,340 --> 00:13:57,702
if you're given
some information.

339
00:13:57,702 --> 00:13:59,410
You don't actually
have to have a formula

340
00:13:59,410 --> 00:14:02,910
to find out some information
about these relationships.

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00:14:02,910 --> 00:14:05,000
So I think that's
where I'll stop.

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00:14:05,000 --> 00:14:05,646