The Handy Math Answer Book

A row vector is usually used for problem solving in which a vector is described as

V(x, y, z) during the specification of a problem. But note: Row vectors should not

really be used with any mathematical descriptions.

How is the lengthof a vectorcalculated?

The length of a vector is calculated by taking the square root of the sum of the squares

of each coordinate. For example, if the vector is defined by (x, y, z), then the length (L)

of the vector is calculated this way:

L  x^2 y^2 z^2

What is a normalized vector?

A normalized (or unit) vector is one in which the sum of the squares of all coordinates

is equal to one. For example, the vector (2, 2, 0) is not normalized; the vectors (0.707,

0.707, 0.0) and (1.0, 0.0, 0.0) are normalized. (An outward normalis another name for

a normalized vector; it represents the direction that a polygon surface or vertex [end-

point] is facing.) Normalized (or unit) vectors are often seen written as x><, but more

usually as xˆ(the ˆ is often referred to as a “hat”).

A vector can be normalized by calculating the magnitude or length of the vector

and dividing each coordinate by this value. For example, consider the following vector:

The length of this vector is 5.0 (see above to determine how to solve for length of

the vector); or |V| 5.0. Thus, the value of the normalized vector is given by:

.

.

.

.

.

.

V

V

30

40

00

5

1

06

08

00

==:

.

.

.

V

30

40

00

=

238

How are vectors used?

M

any physical quantities—especially those in association with mathematics

and science—can be represented by vectors, such as force, velocity, and

momentum. In specifying these quantities, one must state not only how large it

is but also in what direction it acts. Even more complex is the use of multi-

dimensional vectors for such problems as relativity, wind velocities in an atmos-

phere, and in determining electromagnetic fields.