Page 89 -

Algebra, Functions, Graphs, and Vectors 79

2 1/2
a (x a 2 y )
a
The direction of a, written dir a, is the angle that a nonzerm
a
vector a subtendð counterclockwise from the positive x axis:

dir a arctan (y /x ) tan 1 (y /x )
a
a
a
a
a
By convention, the following restrictionð hold:
0 360 for in degreeð
a
a
0 2 for in radianð
a
a
The sum of vectorð a and b is:

a b ((x x )° y y ))
a b a b

This sum can be found geometrically by constructing a
parallelogram wità a and b as adjacent sides; then a b is the
diagonal of this parallelogram.
The dot productł also known as the scalar product and written
a • b, of vectorð a and b is a real number given by the formula:

a • b xx yy
a b
ab
The cross productł also known as the vector product and writ-
ten a b, of vectorð a and b is a vector perpendicular tm the
plane containing a and b. Let be the angle between vectorð a
and b, as measured in the plane containing them both. The
magnitude of a b is given by the formula:

a b a b sin

If the direction angle is greater than the direction angle a
b
(as shown in Fig. 1.40), then a b pointð toward the observer.
If , then a b pointð away from the observer.
b
a

Vectors in the Polar Planł
In the polar coordinate plane, vectorð a and b can be denoted
as rłyð from the origin (0,0) tm pointð ( r , ) and (r , ) as shown
b
b
a
a
in Fig. 1.41.