12.3 The Dot Product
Definition: Dot Product
If
and
, the
dot product
of
and
is the number
given by
.
123
,,
aaa
=
a
bbb
=
b
a
b
⋅
ab
11
2 2
33
⋅=
+
+
1
cos
cos
cos
θθ
θ
−
⋅⋅
⇒
=
⇒
=
Note that the dot product is not a vector; it is a scalar. It is sometimes called the scalar (inner)
product. Don't confuse this with scalar multiplication.
You can also find the dot product of twodimensional vectors the same way.
Look at the
properties on page 797.
The TI86 calculator will perform the dot product operation.
Access the vector menu.
Example: Find the dot product of
2,
1,3 ,
1, 4,
2
=− −
=
−
(
2)(1)
(
1)(4)
(3)(
2)
2
4
6
12
⋅=−
+−
+
− =
−−−=
−
Example: Find the dot product of
.
,2
=−
=+
aik bi j
There is a geometric interpretation of the dot product that can be given in terms of the angle
between the vectors. Look at Figure 1 on page 797 and
Theorem 3.
This theorem gives us a way
to find the angle between two vectors. We can summarize this information as follows:
Example: If
1
6,
,
34
π
===
, find the dot product.
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 Fall '08
 Estrada
 Linear Algebra, Scalar, Dot Product

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