Math 200, Spring
2010
Handout 2
Section 123
Dot Product and the Angle Between Vectors
Section 124
The Cross Product
Definition of Dot Product (or Scalar Product or Inner Product)
If
1
2
3
,
,
a
a
a
=
a
and
1
2
3
,
,
b b
b
=
b
, then the
dot product
of
a
and
b
is the number
⋅
a b
given by
1 1
2
2
3 3
a b
a b
a b
⋅
=
+
+
a b
In two dimensions,
1
2
1
2
1 1
2
2
,
,
a
a
b b
a b
a b
⋅
=
⋅
=
+
a b
1.
Find
⋅
a b
.
(a)
2,3 ,
3,7
= −
= −
a
b
(b)
4
3 ,
2
4
6
=
−
=
−
+
a
j
k
b
i
j
k
Properties of the Dot Product
If
a
,
b
, and
c
are vectors in
3
V
and
λ
is a scalar, then
1.
2
⋅
a a= a
Relation with Length
2.
⋅
=
⋅
a b
b a
Commutativity
3.
(
)
⋅
=
⋅
⋅
a
b+c
a b+a c
Distributive Law
4.
(
)
(
)
(
)
λ
λ
λ
⋅
=
⋅
⋅
a
b
a b =a
b
Pulling out scalars
5.
0
⋅
=
0 a
2. Simplify the expression
(
) (
)
2
+
⋅
+
−
⋅
v
w
v
w
v w
.
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Theorem
The angle
θ
between two vectors is chosen to satisfy 0
θ
π
≤
≤
.
If
θ
is the angle between the nonzero vectors
a
and
b
then
cos
θ
⋅
=
a b
a
b
or
cos
θ
⋅
=
a b
a
b
1
cos
θ
−
⋅
=
a b
a
b
3.
Find
⋅
a b
if
3,
6
=
=
a
b
, the angle between
a
and
b
is 45
°
.
4.
Find the angle between the vectors
4,0,2
and
2,
1,0
=
=
−
a
b
.
Orthogonal Vectors
Two nonzero vectors
a
and
b
are called
orthogonal
(perpendicular) if the angle between them is
2
θ
π
=
.
Two vectors
a
and
b
are orthogonal if and only if
0
⋅
=
a b
Interpretation of Dot Product
We can think of
⋅
a b
as measuring the extent to which
a
and
b
point in the same direction.
Let
θ
be the angle between nonzero vectors
a
and
b
.
If
2
θ
π
<
,
then
0
⋅
>
a b
.
(acute angle between vectors)
If
2
θ
π
=
,
then
0
⋅
=
a b
.
(orthogonal vectors)
If
2
θ
π
>
,
then
0
⋅
<
a b
.
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 Spring '10
 JAMESDCAMPBELL
 Vectors, Scalar, Dot Product, scalar triple product

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