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Let's consider a two
dimensional motion.
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Suppose we have something
like projectile motion.
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And we have an object moving.
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Let's now describe how we
can describe this motion
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with vectors.
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So the first thing
we always want
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to do, and let's remind
ourselves of the steps,
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is we want to choose
a coordinate system.
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Now what does a coordinate
system consist of?
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It consists of an origin.
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It consists of two axes.
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In this case, we'll identify
the positive direction
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for each axis as
plus x and plus y.
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And at every single
point in space--
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so if we had any arbitrary point
P here-- let's call this P1.
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We have a choice of unit
vectors, i-hat 1 and j-hat 1.
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Now, what makes
Cartesian coordinates
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unique is that no matter
what point we're at,
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the unit vectors
are all the same.
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So we could erase
all these indices
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for that particular
point and just
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have an abstract set of unit
vectors, i-hat and j-hat.
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Now normally what we'll do
is we'll just put those off
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to the side.
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So in our Cartesian
coordinates, we now
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want to define the
position vector.
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And the position
vector is a vector
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from the origin to
where the object is.
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So we'll write that
position vector.
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We'll denote it by r of t.
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Because as this object
moves along its trajectory
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that position
vector is changing.
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And we'll write
down r of t in terms
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of two coordinate functions,
x of t and y of t.
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And so our vector-- position
vector of the object r of t
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is equal to x of t,
i-hat plus y of t, j-hat.
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And one of our main
goals is to figure out
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what these position functions
are for the motion of objects.
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So this is how we
describe an object
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in a Cartesian
coordinate system,
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undergoing two
dimensional motion.
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What we want to
analyze next is what
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is the velocity of that object.