Page 45 - Dynamics of Mechanical Systems

0593_C02_fm Page 26 Monday, May 6, 2002 1:46 PM

26 Dynamics of Mechanical Systems

C

B

A L
B
FIGURE 2.6.3 A
A vector sum projected onto a line L. O

That is, the scalar may be placed anywhere in the product. The parentheses are unneces-
sary. The dot, however, must remain between the vectors.
Next, suppose n , n , and n are mutually perpendicular unit vectors parallel to X-, Y-,
x
y
z
and Z-axes, and suppose V is a nonzero vector, as in Figure 2.6.4. Let θ , θ , and θ be the
z
y
x
angles between V and the X-, Y-, and Z-axes. Then, the projections of V onto X, Y, and Z are:
⋅
⋅
⋅
Vn = V cosθ V n = V cosθ Vn = V cosθ (2.6.18)
x x y y z z
Then, from Eq. (2.6.11), if V , V , and V are mutually perpendicular components of V
y
x
z
parallel to X, Y, and Z, we can express V , V , and V as:
y
z
x
y (
V = ( V n n ) V = ⋅ y) V = ( V n n )
⋅
⋅
x x x V n n z z z (2.6.19)
y
Then, we also may express V as:
)
V = V + V + V = ( Vn n ) + ( V n n +( Vn n )
⋅
⋅
⋅
x y z x x y y z z
(2.6.20)
= V n + V n + V n
x x y y z z
where V , V , and V are the scalar components of V along X, Y, and Z.
z
x
y
As a final example, suppose n , n , and n are mutually perpendicular unit vectors, and
1
2
3
suppose A and B can be expressed as:
A = A n + A n + A n (2.6.21)
1 1 2 2 3 3
=
BB n + B n + B n (2.6.22)
1 1 2 2 3 3
Then, by using Eqs. (2.6.8), (2.6.16), and (2.6.17), we can express A · B as:
AB ( A n + A n + n B n + B n + B )
⋅=
n
3
3
1 1 2 2 A ) ⋅( 1 1 2 2 3 3
⋅
⋅
= AB nn + AB nn + AB nn
⋅
11 1 1 1 2 1 2 1 3 1 3
+ AB n ⋅n + AB n ⋅n + AB n ⋅n (2.6.23)
21 2 1 3 2 3 2 2 3 2 3
+
+ AB n ⋅n + AB n ⋅n + AB n ⋅n
31 3 1 3 2 3 2 3 3 3 3
= A B + AB + AB
1 1 2 2 3 3