WEBVTT
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[MUSIC PLAYING]
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When we're looking
at the plane, we've
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learned to love
Cartesian coordinates.
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They've been so useful
in so many ways.
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It turns out, though, that there
is another alternate method
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to assign coordinates to points
in the plane, which is really
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very different.
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It's called the method
of polar coordinates.
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As we'll see, it's a
beautiful subject that
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leads to beautiful diagrams.
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How does it work exactly?
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Well, polar coordinates begin
by specifying two things--
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first, a point in the plane
that we call the pole, and then
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a ray emanating from that point
that we call the polar axis.
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And that's all there is for
describing polar coordinates.
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How does it work?
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If I give myself a typical
point in the plane,
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what are its coordinates--
its polar coordinates--
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relative to this mechanism?
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Well, the first coordinate
is simply the distance
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from the pole to the point.
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Let's call that r.
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That's the first
polar coordinate.
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The second coordinate
is going to be
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the angle theta
determined by that line
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segment and the polar axis.
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And so these are the polar
coordinates of my point--
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r and theta.
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As you can see, they determine
the position of the point.
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There's something
a little different,
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though, because they wouldn't be
the only polar coordinates that
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would determine the
position of the point.
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For example, if I were to give
you the polar coordinates r
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and theta plus 2 pi, well, that
would be exactly the same point
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that you would get, because
adding 2 pi to theta
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just brings you back
to the same position.
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Similarly, minus 2 pi
would do the same thing.
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More generally, the polar
coordinates of that point
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can be thought of as
being r on the one hand
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and theta plus any
integer multiple of 2 pi,
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k being positive or
negative as an integer.
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And another kind of ambiguity
arises at the pole itself.
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The pole is when r is 0.
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But what is the angle
theta at the pole?
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There is no obvious
unique choice.
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It could be any theta,
as long as r is 0,
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and you'll be at the pole.
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So these coordinates
work somewhat differently
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from Cartesian
coordinates that were
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so well-defined for any point.
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And yet, we'll see
that they're extremely
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useful in certain circumstances.
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Just to get a hint of why
they're going to be useful,
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suppose in polar
coordinates I ask you,
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what is the level set
of the set of points
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r, theta, for which r equals 2?
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What a simple
equation, r equals 2.
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What does that describe
as a locus of points?
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Answer, to say that your
distance 2 from the pole
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is to say that you're
on a circle of radius 2
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around that pole.
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You see how easy
it is to describe
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a circle in polar coordinates.
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We'll see other examples later.
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Now, a circle could
have been described
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in Cartesian coordinates,
of course, which
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raises the issue,
what is the connection
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between polar coordinates
and Cartesian coordinates?
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Let's first think about,
given the polar coordinates,
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how do you find
the Cartesian ones?
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Well, we have to put in
the Cartesian coordinates
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if we're going to compare them.
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And the usual way is
to place the origin
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of the Cartesian
coordinates at the pole
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and to place the positive x-axis
for your Cartesian coordinates
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along the polar axis and
then the Cartesian axis
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corresponding to y goes
where it has to go.
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So this is the
canonical configuration
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when you want to speak
of both polar coordinates
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and Cartesian coordinates.
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And how do you find the
Cartesian coordinates
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of a given point with polar
coordinates as are r and theta?
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Well, you can do the following.
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You can drop a vertical down
from the point to the x-axis.
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And you see that
the sine of theta,
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by our triangle
definition of sine--
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you know, opposite
over hypotenuse--
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is y over r, and the cosine
of theta is x over r.
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Therefore, you find x and y--
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x is r cos theta,
y is r sine theta.
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That was pretty easy.
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And by the way, this will
work even if
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theta is not in the first
quadrant and if theta is negative, and so forth.