Begin2.DVI

Geometric Interpretation

A geometric interpretation that can be assigned to the magnitude of the cross

product of two vectors is illustrated in figure 6-8.

Figure 6-8. Parallelogram with sides A
and B.

The area of the parallelogram having the vectors A
and B
for its sides is given

by

Area =|A
|·h=|A
||B
|sin θ=|A
×B
|. (6 .28)

Therefore, the magnitude of the cross product of two vectors represents the area of

the parallelogram formed from these vectors when their origins are made to coincide.

Vector Identities

The following vector identities are often needed to simplify various equations in

science and engineering.

1. A
×B
=−B
×A
 (6 .29)

2. A
·(B
×C
) = B
·(C
×A
) = C
·(A
×B
) (6 .30)

An identity known as the triple scalar product.

3. (A
×B
)×(C
×D
) = C

[

D
·(A
×B
)

]

−D

[

C
·(A
×B
)

]

=B

[

A
·(C
×D
)

]

−A

[

B
·(C
×D
)

]

(6 .31)

4. A
×(B
×C
) = B
(A
·C
)−C
(A
·B
) (6 .32)

The quantity A
×(B
×C
)is called a triple vector product.

5. (A
×B
)·(C
×D
) = (A
·C
)(B
·D
)−(A
·D
)(B
·C
) (6 .33)

6. The triple vector product satisfies

A
×(B
×C
) + B
×(C
×A
) + C
×(A
×B
) = 
 0 (6 .34)

Note that in the triple scalar product A
·(B
×C
) the parenthesis is sometimes

omitted because (A
·B
)×C
is meaningless and so A
·B
×C
can have only one meaning.

The parenthesis just emphasizes this one meaning.