Calculus-Based Physics by Jeffrey W. Schnick - HTML preview

Chapter 24 Work and Energy

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

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