The Dot Product of Two Vectors
The dot product of the vectors A and B is written A ⋅ B and is expressed as:
A ⋅B = ABcosθ
(24-3)
where θ, just as in the case of the cross product, is the angle between the two vectors after they
have been placed tail to tail.
B
θ
A
The dot product can be interpreted as either A|B (the component of A along B , times, the
magnitude of B ) or B| A (the component of B along A , times, the magnitude of A ), both of
which evaluate to one and the same value. This makes the dot product perfect for calculating the
work. Since F ⋅ r
∆ =
∆r
|
F
and
r
∆
|
F
is W, we have
W = F ⋅ r
∆
(24-4)
By means of the dot product, we can solve the example in the last section much more quickly
than we did before.
A
Find the work done on the block by
θ
d
the gravitational force when the object
moves from point A to Point B.
B
F = m
g
g
1
67