Another way to consider the scalar product is to consider how it is defined in terms of the magnitudes and directions of two vectors. Consider again the vectors and
in Figure 4.29. We want to find out how much vector
will grow vector
. So again we want to identify how much the vectors interact – and one way to do this is to determine how much vector
points in the direction of vector
.
In Figure 4.31, the vectors are arranged so that their tails meet, and this makes it possible to compare their magnitudes and directions. To make this explicit, we can draw the components of , not in terms of horizontal and vertical directions, but in terms of the direction where
is pointing, as illustrated in Figure 4.32(a). Formally, the component of
that points in the direction of
is called the projection of
onto
, and if the angle between
and
is
, then the length of the projection of
onto
is
, as shown in Figure 4.32(b).
Comparing the components of with
will give us a measure of how much
and
interact, as illustrated in Figure 4.33. In the direction of
,
has a component of magnitude
, and
has a component of magnitude
, so the contribution to the value of
is
. Perpendicular to
,
has a component of magnitude
, and
has a component of magnitude 0, so the contribution to the value of
is 0.
So
and this is a measure of how much the scalar quantity grows the scalar quantity
.
and
are parallel, so
and the scalar product is given by
The scalar product of two vectors and
is
where is the angle between
and
.
Suppose that ,
and
are vectors with magnitudes 4, 3 and 2 respectively, and directions as shown in the following figure.
Find the following scalar products.
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