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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.