Page 44 - Dynamics of Mechanical Systems

0593_C02_fm Page 25 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 25

Z

n
z
A
L
Y
n
y
n x θ n
X

FIGURE 2.6.1 FIGURE 2.6.2
Mutually perpendicular unit vectors. Vector A, line L, and unit vector n.

where A is:

A = ( A n n ) = ( A cos ) θ n (2.6.11)
⋅
||
To further develop these components, let a vector C be the resultant (sum) of vectors A
and B. That is,

C = A B (2.6.12)
+
Let L be a line passing through the tail O of C, as in Figure 2.6.3, and let n be a unit vector
parallel to L. Let A and B be the projection points of the heads of A and B onto L, as
shown. Then, from Eq. (2.6.9), the lengths of the line segments OA, AB, and OB may be
expressed as:

Β
⋅
⋅
OA = nA AB = n ⋅ OB = n C (2.6.13)
However, from Figure 2.6.3, we see that:
=
+
OB OA AB (2.6.14)
Hence,

⋅
⋅
+
⋅
nC = n A n B (2.6.15)
Therefore, from Eq. (2.6.12), we have the distributive law:
⋅(
+
⋅
+
⋅
nA B) = n A n B (2.6.16)
Continuing in this manner, suppose s is a scalar. Then, from the definition of Eq. (2.6.1),
we have:
sA B A ⋅(
sAB = ( ) ⋅+ sB) = A ⋅ sB = AB s (2.6.17)
⋅
⋅