Geometric Method
So what does the scalar obtained in doing the dot product
u
.
v
represent? We can get an idea of
what's going on by looking at the dot product of a vector with unit vectors. In
Unit Vectors
we
defined the unit vectors
i
,
j
, and
k
for the 3dimensional case. In two dimensions we have only
i
= (1, 0) and
j
= (0, 1) . (For now we will work in two dimensions, since it is easier to represent
such vectors graphically.) The dot products of a vector
v
= (
v
1
,
v
2
) with unit vectors
i
and
j
are
given by:
v
·
i
=
v
1
1 +
v
2
0 =
v
1
v
·
j
=
v
1
0 +
v
2
1 =
v
2
In other words, the dot product of
v
with
i
picks off the component of
v
in the
x
direction, and
similarly
v
's dot product with
j
picks off the component of
v
which lies in the
y
direction. This
is the same as computing the magnitude of the projection of
v
onto the
x
 and
y
axes,
respectively.
This may not seem too exciting, since in some sense we already knew this as soon as we wrote
our vector down in terms of components. But what would happen if instead of components we
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 Fall '10
 DavidJudd
 Physics, Linear Algebra, Vector Space, Dot Product

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