Page 43 - Dynamics of Mechanical Systems

0593_C02_fm Page 24 Monday, May 6, 2002 1:46 PM

24 Dynamics of Mechanical Systems

A
B A B

θ

FIGURE 2.5.1 FIGURE 2.5.2
Two nonzero vectors. Vectors A and B connected tail to tail.

However, the angle that a vector makes with itself is zero (see Problem P2.5.1); hence, we
have:

⋅
2
AA = A A = A = A 2 (2.6.4)
2
where the last equality is a definition of A .
Suppose in this last case that A is a unit vector, say n. Then,

2
⋅
nn = n = n = 1 (2.6.5)
2
Next, suppose that n , n , and n are mutually perpendicular unit vectors parallel to the
1
2
3
X-, Y-, and Z-axes as shown in Figure 2.6.1. Then, the various scalar products of these unit
vectors may be expressed as:
⋅
⋅
⋅
n n = 1, nn = 1, nn = 1 (2.6.6)
2
3
2
1
3
1
⋅
⋅
n n = n ⋅ n = n n = n ⋅ n = n ⋅ n = n ⋅ n = 0 (2.6.7)
1 2 2 1 1 3 3 1 2 3 3 2
These results may be expressed in the compact form:
1 i = j

nn = δ =  (2.6.8)
⋅
i j ij
 i ≠ j
0
where δ is often called Kronecker’s delta function.
ij
Next, suppose that n is a unit vector parallel to a line L and that A is a nonzero vector,
as in Figure 2.6.2. Then, A • n is:
⋅
An = A n cosθ = A cosθ (2.6.9)
We can interpret this result as the “projection” of A onto L. Indeed, suppose we express
A in terms of two components: one parallel to L, called A , and the other perpendicular

to L, called A . Then,
⊥
A = A + A (2.6.10)
|| ⊥