Page 36 - Dynamics of Mechanical Systems

0593_C02_fm Page 17 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 17

Vector subtraction may be defined from vector addition. Specifically, the difference of
two vectors A and B, written as A – B, is simply the sum of A with the negative of B. That is,

−
ΑΑΒΒ= A + − ( ) B (2.3.3)
An item of interest in vector addition is the magnitude of the resultant, which may be
determined using the geometry of the parallelogram and the law of cosines. For example,
in Figure 2.3.5, let θ be the angle between A and B, as shown. Then, the magnitude of the
resultant R is given by:

− )
2
2
R = A + B − 2 A + B cos (πθ (2.3.4)
Example 2.3.1: Resultant Magnitude
To illustrate the use of Eq. (2.3.4), suppose the magnitude of A is 15 N, the magnitude of
B is 12 N, and the angle θ between A and B is 60˚. Then, the magnitude of the resultant R is:

/
2 2 ) 12
R = () +() −()( )( )cos (2 3π = 23 43N (2.3.5)
12
15
2 15 12
.
Observe from Eq. (2.3.4) that if we double the magnitude of both A and B, the magnitude
of the resultant R is also doubled. Indeed, if we multiply A and B by any scalar s, R will
also be multiplied by s. That is,
s A B)
sA + sB = sR = ( + (2.3.6)
This means that vector addition is distributive with respect to scalar multiplication.
Next, suppose we have three vectors A, B, and C, and suppose we wish to find their
resultant. Suppose further that the vectors are not parallel to the same plane, as, for
example, in Figure 2.3.6. The resultant R is obtained in the same manner as before. That
is, the vectors are connected head to tail, as depicted in Figure 2.3.7. Then, the resultant
R is obtained by connecting the tail of the first vector A to the head of the third vector C
as in Figure 2.3.7. That is,

+
+
R = A B C (2.3.7)

A
R R
B
B R B B

θ
A A A
FIGURE 2.3.3 FIGURE 2.3.4
Resultant (sum) R of vectors A Two ways of adding vectors A FIGURE 2.3.5
and B. and B. Vector triangle geometry.