Slide 1
The Dot Product of Two Vectors
Section 10.3
Slide 2
Definition of Dot Product
The
dot product
of
and
is
The
dot product
of
and
is
2
1
,
u
u
u
=
2
1
,
v
v
v
=
2
2
1
1
v
u
v
u
v
u
+
=
•
3
2
1
,
,
u
u
u
u
=
3
2
1
,
,
v
v
v
v
=
3
3
2
2
1
1
v
u
v
u
v
u
v
u
+
+
=
•
Slide 3
Properties of Dot Product
Let
u
,
v
, and
w
be vectors in the plane or in space
and let
c
be a scalar.
u
v
v
u
•
=
•
(
29
w
u
v
u
w
v
u
•
+
•
=
+
•
(
29
v
c
u
v
u
c
v
u
c
•
=
•
=
•
0
0
=
•
v
2
v
v
v
=
•
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Slide 4
Angle Between Two Vectors
If
θ
is the angle between two nonzero vectors
u
and
v
, then
Note that you can rewrite the formula above to
get an alternative form of the dot product.
v
u
v
u
•
=
θ
cos
θ
cos
v
u
v
u
=
•
Slide 5
Orthogonal Vectors
The vectors
u
and
v
are orthogonal if
u
•
v
= 0.
What is the difference between perpendicular,
orthogonal, and normal?
Slide 6
Direction Cosines
The direction angle for a vector in the plane is
measured counterclockwise from the positive x
axis to the vector.
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 Spring '12
 PamelaSatterfield
 Linear Algebra, Vectors, Dot Product, Inner product space

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