Page 35 - Dynamics of Mechanical Systems

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16 Dynamics of Mechanical Systems

Z
n
z L
V

O Y
n
y
FIGURE 2.2.1
A fixed line L and a fixed vector V. X n x

2.3 Vector Addition

Vectors obey the parallelogram law of addition. This is a simple geometric algorithm for
understanding and exhibiting the powerful analytical utility of vectors. To see this, con-
sider two vectors A and B as in Figure 2.3.1. Let A and B be free vectors. To add A and
B, let them be connected “head-to-tail” (without changing their characteristics) as in Figure
2.3.2. That is, relocate B so that its tail is at the head of A. Then, the sum of A and B, called
the resultant, is the vector R connecting the tail of A to the head of B, as in Figure 2.3.3.
That is,

R = A B (2.3.1)
+
The vectors A and B are called the components of R.
The reason for the name “parallelogram law” is that the same result is obtained if the
head of B is connected to the tail of A, as in Figure 2.3.4. The two ways of adding A and
B produce a parallelogram, as shown. The order of the addition — that is, which vector
is taken first and which is taken second — is therefore unimportant; hence, vector addition
is commutative. That is,

+
=
+
AB B A (2.3.2)

B
B

A
A
FIGURE 2.3.1 FIGURE 2.3.2
Two vectors A and B to be added. Vectors A and B connected head to tail.