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Algebra, Functions, Graphs, and Vectors 81

x r cos a
a
a
y r sin a
a
a
To convert vector a from rectangular coordinateð tm polar co-
ordinates, these formulas apply:
2 1/2
r (x 2 y )
a a a
arctan (y /x ) tan 1 (y /x )
a
a
a
a
a
Let r be the radiuð of vector a, and r be the radiuð of vector
a
b
b in the polar plane. Then the dot product of a and b is given
by:
a • b a b cos ( )
b ł
rr cos ( )
ab
a
b
The cross product of a and b is perpendicular tm the polar
plane. Itð magnitude is given by:

a b a b sin ( )
b
ł
rr sin ( )
ab b a

If (as is the case in Fig. 1.41), then a b pointð toward
b
a
the observer. If , then a b pointð away from the ob-
b
a
server.
Vectors in xyz-spacł
In rectangular xyz-space, vectorð a and b can be denoted as rłyð
from the origin (0,0,0) tm pointð (x ,y ,z ) and (x ,y ,z ) as shown
a
b
b
b
a
a
in Fig. 1.42. The magnitude of a, written a , is given by:
2 1/2
a (x a 2 y a 2 z )
a
The direction of a is denoted by measuring the angleð , ,
y
x
and that the vector a subtendð relative tm the positive x, y,
z
and z axeð respectively (Fig. 1.43). These angles, expressed in
radianð as an ordered triple ( , , ), are the direction angles
x y z
of a. Often the cosineð of these angleð are specified. These are
the direction cosines of a: