Page 38 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 2] VECTORS 23

◦
equal to 90 . Since φ = θ = 90 ,
◦
◦
◦
φ = 90 − θ = 90 − 63 = 27 ◦
◦
The resultant R has a magnitude of 11.2 km, and its direction is 27 east of north.
◦

Fig. 2-11

RESOLVING A VECTOR
Just as two or more vectors can be added to yield a single resultant vector, so it is possible to break up a single
vector into two or more other vectors. If vectors A and B are together equivalent to vector C, then vector C is
equivalent to the two vectors A and B (Fig. 2-12). When a vector is replaced by two or more others, the process
is called resolving the vector, and the new vectors are known as the components of the initial vector.

Fig. 2-12

The components into which a vector is resolved are nearly always chosen to be perpendicular to one another.
Figure 2-13 shows a wagon being pulled by a man with force F. Because the wagon moves horizontally, the
entire force is not effective in influencing its motion. The force F may be resolved into two component vectors
F x and F y , where
F x = horizontal component of F

F y = vertical component of F
The magnitudes of these components are
F x = F cos θ F y = F sin θ
Evidently the component F x is responsible for the wagon’s motion, and if we were interested in working out the
details of this motion, we would need to consider only F x .
In Fig. 2-13 the force F lies in a vertical plane, and the two components F x and F y are enough to describe it.
In general, however, three mutually perpendicular components are required to completely describe the magnitude
and direction of a vector quantity. It is customary to label the directions of these components the x, y, and z axes,
as in Fig. 2-14. The components of some vector A in these directions are accordingly denoted A x , A y , and A z .Ifa
component falls on the negative part of an axis, its magnitude is considered negative. Thus if A z were downward
in Fig. 2-14 instead of upward and its length were equivalent to, say, 12 N, we would write A z =−12 N.
[The newton (N) is the SI unit of force; it is equal to 0.225 lb.]