12.3 The Dot Product
The Dot Product
DEFINITION
If a = ⟨a1 , a2 , a3 ⟩ and b = ⟨b1 , b2 , b3 ⟩ , then the dot product of a and b is the number a · b given by
a · b = a 1 b1 + a 2 b2 + a 3 b3
Note that the dot product a · b of the vectors a = ⟨a1 , a2 ⟩ and b = ⟨b1 , b2 ⟩ is given by
a · b = a 1 b1 + a 2 b2
Example 12.3.1: Find a · b if a = 2 i - 3 j + 5 k and b = 7 i + 4 j + 2 k .
PROPERTIES OF THE DOT PRODUCT
If a, b, and c are vectors in V 3 and c is a scalar, then
a 2
2.
1.
a·a =
3.
a · (b + c) = a · b + a · c
4.
5.
0·a = 0
a·b = b·a
(c a) · b = c(a · b) = a ·(c b)
An alternate definition of the dot product is given by the following theorem.
THEOREM
If θ is the angle between the vectors a and b, then a · b = a b cos θ
Note that 0 ≤ θ ≤ π.
COROLLARY
If θ is the angle between the nonzero vectors a and b, then cos θ =
a·b
a b
Example 12.3.2: Find the angle θ between the vectors a = 2 i - 3 j + 5 k and b = 7 i + 4 j + 2 k.
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| 12.3 - The Dot Product
Two vectors a and b are orthogonal if and only if a · b = 0
Example 12.3.3: Determine if the vectors are orthogonal:
a = 2 i - 3 j + k and b = 7 i + 4 j - 2 k.
Direction Angles and Direction Cosines
The direction angles of a nonzero vector a are the angles α, β, and γ (in the interval [0, π ]) that a
makes with the positive x-, y-, and z-axes. The cosines of these directions angles, cos α, cos β, and
cos γ, are called the direction cosines of a. If
If a = ⟨a1 , a2 , a3 ⟩ , then
cos α =
a1
a
cos β =
a2
a
cos γ =
Projections

b

b

a

a
Scalar projection of b onto a:
comp a b =
Vector projection of b onto a:
proja b = 
a·b
a
a·b
a

a a
=
a·b
a2
a
a3
a
12.3 - The Dot Product |
Note: The result of the dot product is a scalar.
Example 12.3.4: Let a = 2 i - 3 j + 5 k and b = 7 i + 4 j + 2 k . Find proj a b.
Recall that the definition of work is force times displacement, that is
W = F ·d
In this case, W, F, and d are all scalar quantities. This definition only applies if the force is applied in the
direction of motion.
D
P
Q
F
R
The work done by a constant force F acting through a displacement D = P Q is given by
W = F · D = F D cos θ where θ is the angle between F and D.
Example 12.3.5: A car drives 500 ft on a road that is inclined 12° to the horizontal. The car weighs
2500 lb. Find the work done by the car in overcoming gravity.
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