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400 Mastering Technical Mathematics
Now work out a problem “the other way.” Consider the vector b ( b
,r b
) (200,4.55)
in mathematician’s polar coordinates. You want to convert this to an equivalent vector
b (x b
,y b
) in rectangular coordinates to the nearest tenth of a unit. First, solve for x b
:
Then solve for y b
:
x b
r b
cos b
4.55 cos 200
4.55 · (–0.9397)
–4.3
y b
r b
sin b
4.55 sin 200
4.55 · (–0.3420)
–1.6
Therefore, b (x b
,y b
) (–4.3, –1.6), with rectangular coordinate values accurate to the
nearest tenth of a linear unit.
MULTIPLICATION OF A VECTOR BY A SCALAR IN THE POLAR PLANE
In mathematician’s polar coordinates, imagine some vector a defined as the ordered
pair (,r) as shown in Fig. 26-3. Suppose a is multiplied by a positive real scalar k. The
result is a longer vector if k > 1 and a shorter vector if 0 < k < 1. The following equation
can be used:
ka (,kr)
Figure 26-3
Multiplication of a
polar-plane vector
a by a positive real
number k, and by
a negative real
number –k. Angles
in this drawing are
expressed in
radians.