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We've so far
described abstractly

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what we mean by a dot
product, by a definition of AB

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equals the magnitude of
A times cosine theta,

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times the magnitude of B. But
many times in physics problems,

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we actually have
vectors in space

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and we want to
see how to do this

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in terms of a Cartesian
or any coordinate system

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in particular.

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So let's set up a coordinate
system, we'll call it

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Cartesian, a and j.

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And now this is very important.

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Let's define a vector A here,
and let's have another vector

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in a completely different point,
B. We still can take the dot

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product of these two vectors.

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But in order-- this
is our plus x, plus y.

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And I'm only doing
things in two dimensions.

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So how do we calculate
the dot product?

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Well, the first thing
that we want to look at

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is our unit vectors.

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What are the dot
products of unit vectors?

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Well, if we take i hat dot i
hat, that's the magnitude of i

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hat times cosine of the angle
0, times the magnitude of i hat.

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And cosine of 0 is 1 and the
magnitude of unit vectors

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are 1.

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So when you dot product the unit
vector with itself, you get 1.

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And therefore, it's also
true for the j hat. j

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hat dot j hat is 1.

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What happens when you
dot product two vectors

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that are perpendicular?

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Well, in this case, this is
0 because the angle theta

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is 90 degrees.

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And remember that cosine
of 90 degrees is 0.

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And anyway, when two
vectors are perpendicular,

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there's no component
of one vector

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that's parallel to the other.

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So these are the
essential facts that we're

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going to need to calculate the
dot product of two vectors that

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are separated in space.

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So the way we do that is we'll
begin by drawing, writing down

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the vectors in Cartesian
coordinates, where

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Ax A is a scalar and the vector
part is and the unit vector.

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And we have Ay j hat.

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And likewise, we can write
B as Bx i hat plus By j hat.

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And now when we take the dot
product of these two vectors,

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we're going to write
out all the terms

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here so that you see them.

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So here is our scalar
or dot product.

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We use those words
interchangeably.

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Now notice that
we've already shown

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that the dot product distributes
over vector addition.

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And also, if you multiply
a scalar by a vector,

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you can pull the scalar out.

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So there's four terms here--
Ax i hat dot Bx i hat plus ax

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i hat dot By j hat plus
Ay j hat dot Bx i hat.

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This is a little
tedious to write out.

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By j hat.

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And now because
these are scalars,

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we can pull them out and the
only part of a dot product

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that matters is how
the unit vectors dot.

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And that's why we have
these two results.

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i hat dot i hat is 1,
j hat dot j hat is 1,

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and i hat dot j hat is 0.

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So the first turn is Ax Bx.

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i hat dot j hat is 0,
so we don't need that.

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j hat dot i hat is 0.

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And finally j hat
dot j hat is 1.

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And so we get plus Ay By.

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And that's how we
define the dot product

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in Cartesian coordinates
of two vectors.

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Now, notice that if
we dotted a vector,

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A dot with itself, that would
just be Ax Ax plus Ay Ay, which

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is the components squared.

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And that's equal
to the magnitude

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of the vector squared.

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And so we can say that the
magnitude of the vector,

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of any vector, is you take
its dot product with itself.

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You take the square
root, but remember

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we always take the positive
square root, because magnitudes

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are positive.

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And that's how we calculate
the scalar product for vectors.

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And many times in the
application of physics,

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when we have physical
quantities that

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are vectors in different places,
we use vector decomposition

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and we use this procedure.

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This is the Cartesian
picture, we'll

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learn how to do that
in polar coordinates

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when we need it later on.