Page 48 - Dynamics of Mechanical Systems

0593_C02_fm Page 29 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 29

Z
V
P(2,1,3)
k
L
Q(1,6,1)
n
R
Y
FIGURE 2.6.6 j
A vector V and line L and a Cartesian i
coordinate system R. X

where, as before, θ is the angle between A and B and n is a unit vector perpendicular to
both A and B. The sense of n is determined by the “right-hand rule”: if A is rotated toward
B so that θ is diminished, the sense of n is in the direction of advance of a right-hand
screw, with axis perpendicular to A and B, and rotated in the same way (A to B). Because
a cross is placed between the vectors, the operation is often called the cross product.
Observe in the definition of n that the sense of n depends upon which vector occurs
first in the product; hence, the vector product is anticommutative. That is,

AB = − B A (2.7.2)
×
×
Observe further from the definition that, if s is a scalar, then:

sAB = ( sA) × B A ×( sB) = AB s (2.7.3)
×
=
×
That is, the scalar can be placed at any position in the operation; the parentheses are
unnecessary. The cross, however, must remain between the vectors.
Consider some examples: First, the vector product of a vector with itself is zero because
the angle a vector makes with itself is zero. That is,

×
AA = 0 (2.7.4)
Next, consider the vector product of mutually perpendicular unit vectors: Let n , n ,
1
2
and n be parallel to the X-, Y-, and Z-axes, as shown in Figure 2.7.1. Then, from the
3
definition Eq. (2.7.1), we have the following results:
n × n = 0 n × n = n n × n = − n
1 1 1 2 3 1 3 2
n × n = − n n × n = 0 n × n = n (2.7.5)
2 1 3 2 2 2 3 1
n × n = n n × n = − n n × n = 0
3 1 2 3 2 1 3 3

Observe that these results may be summarized by the expression:

j ∑
n × n = 3 e n (2.7.6)
i ijk k
i=1