T
he
D
ot
P
roduct
Thus far we have discussed the addition, subtraction, and scalar multiplication of
vectors. Another operation on vectors is called the
dot product
. It is important because it
can be used to compute the angle between two vectors. There are other properties of the
dot product which are covered in a calculus course. We begin by defining the dot product.
Dot Product
Let
1
2
3
v
v
v
v
i
j
k
and
1
2
3
w
w
w
w
i
j
k
. The dot product of
v
and
w
,
denoted by
, is a
real number
given by
v w
1
1
2
2
3
3
v w
v w
v w
v w
Example 1:
Compute
if
v w
4
3
0
v
i
j
k
and
2
2
5
w
i
j
k
.
Solution:
.
4( 2)
( 3)(2)
(0)(5)
14
v w
As we mentioned above, the dot product can be used to find the angle between two
vectors. The relationship is given by the following:
Angle Between Two Vectors
If
v
and
w
are nonzero vectors, then
cos
v w
v
w
Thus, the angle
q
between
v
and
w
is given by
1
cos
v w
v
w
The vectors
v
and
w
are perpendicular if and only if
.
0
v w
To see why the first formula holds, consider the following diagram. For the sake of
simplicity, the vectors shown are only in two dimensions.
1

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*