Page 52 - Dynamics of Mechanical Systems

0593_C02_fm Page 33 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 33

Example 2.7.1: Vector Product Computation and Geometric Properties of
the Vector Product
Let vectors A and B be expressed in terms of mutually perpendicular dextral unit vectors
n , n , and n as:
3
1
2
=
A = 7 n − 2 n + 4 n and B n + 3 n − 8 n (2.7.24)
1 2 3 1 2 3
Let C be the vector product A × B.

a. Find C.
b. Show that C is perpendicular to both A and B.
c. Show that B × A = –C.
Solution:

a. From Eq. (2.7.24), C is:

n 1 n 2 n 3
C = 7 −2 4 = 4 n + 60 n + 23 n 3 (2.7.25)
2
1
1 3 −8
b. If C is perpendicular to A, with the angle θ between C and A being 90˚, C • A
is zero because cosθ is zero. Conversely, if C • A is zero, and neither C nor A is
zero, then cosθ is zero, making C perpendicular to A. From Eq. (2.6.21), C • A is:

⋅
7
2
4
2
4
60
CA = ()( ) + ( ) − ( ) + ( )( ) = 0 (2.7.26)
3
Similarly, C · B is
⋅
1
60
3
8
CB = ()( ) + ( )( ) + ( ) − ( ) = 0 (2.7.27)
2
3
4
c. From Eq. (2.7.23), B × A is:
n n n
1 2 3
×
BA = 1 3 − =−4 n − 60 n − 23 n (2.7.28)
8
1 2 3
7 −2 4
which is seen to be from Eq. (2.7.25).
2.8 Vector Multiplication: Triple Products
On many occasions it is necessary to consider the product of three vectors. Such products
are called “triple” products. Two triple products that will be helpful to use are the scalar
triple product and the vector triple product.