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Math 224 – Calculus III
1
12.3 The Dot Product
Recommended Homework: # 1-47, 51, 55(odds)
In 12.1 we saw addition, subtraction and scalar multiplication of vectors.
Here we look at “multiplying”
two vectors.
We actually define two types of product:
(i) The Dot Product
(ii) The Cross Product (section 12.4)
The Dot Product (or Inner Product)
Given vectors
1
2
3
,,
a a a
a
and
1
2
3
b b b
b
in component form their
dot product
is given by:
1 1
2 2
3 3
a b
a b
a b
ab
Note
:
The result of a dot product is a real number
!
The dot product of two vectors does not have a convenient interpretation, but it is related to the
angle between the two vectors.
The equivalent rule also works for vectors in
2
Properties of the Dot Product
The familiar arithmetic properties (commutative, associative etc) hold for the dot product (try them!)
Given the vectors
a
,
b
and
c
and suppose
c
is a scalar.
(i)
2
a a = a
(ii)
a b = b a
(iii)
(
a b +c) = a b a c
(iv)
((
c
c
c
( a) b =
a b) = a
b)
(v)
0
0 a =
Geometric Interpretation of the Dot Product
:
Theorem
:
If
is the angle between vectors
a
,
b

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