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0593_C02_fm Page 19 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 19

Z Z
n
z
R R
C C
Y Y
A A
n
y
B B
n
X X x
FIGURE 2.4.1 FIGURE 2.4.2
Vector R with mutually perpendicular components. Vectors A, B, and C and unit vectors n x , n y , and n z .

2.4 Vector Components

Consider again Eq. (2.3.7) where we have the vector sum:

+
R = A B C (2.4.1)
+
Instead of thinking of this expression as a sum of components, consider it as a represen-
tation of the vector R. Suppose further that the components A, B, and C happen to be
mutually perpendicular and parallel to coordinate axes, as shown in Figure 2.4.1. Then,
by the Pythagoras theorem, the magnitude of R is simply:

/
2
2
R = ( A + B + C 2 ) 12 (2.4.2)
To develop these ideas still further, suppose that n , n , and n are unit vectors parallel
x y z
to X, Y, and Z, as in Figure 2.4.2. Then, from our discussion in Chapter 1, we see that A,
B, and C can be expressed in the forms:

A = An = a n
x x
B = Bn = b n (2.4.3)
y y
C = Cn = c n
z z

where a, b, and c are scalars representing the magnitudes of A, B, and C. Hence, R may
be expressed as:

R = a n + b n + c n (2.4.4)
x y z