Page 162 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 153

COMPONENTS OF A VECTOR
Any vector A in 3 dimensions can be represented with
initial point at the origin O of a rectangular coordinate
system [see Fig. 7-7]. Let ðA 1 ; A 2 ; A 3 Þ be the rectangular
coordinates of the terminal point of vector A with initial
point at O. The vectors A 1 i; A 2 j; and A 3 k are called the
rectangular component vectors,or simply component vec-
tors, of A in the x, y; and z directions respectively. A 1 ; A 2 ;
and A 3 are called the rectangular components,or simply
components,of A in the x, y; and z directions respectively.
The vectors of the set fi; j; kg are perpendicular to one
another, and they are unit vectors. The words orthogonal
and normal, respectively, are used to describe these charac-
teristics; hence, the set is what is called an orthonormal basis. Fig. 7-7
It is easily shown to be linearly independent. In an n-dimensional space, any set of n linearly indepen-
dent vectors is a basis. The further characteristic of a basis is that any vector of the space can be
expressed through it. It is the basis representation that provides the link between the geometric and
algebraic expressions of vectors and vector concepts.
The sum or resultant of A 1 i; A 2 j; and A 3 k is the vector A,so that we can write

A ¼ A 1 i þ A 2 j þ A 3 k ð1Þ
The magnitude of A is
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
A þ A þ A 2
A ¼jAj¼ 1 2 3 ð2Þ
In particular, the position vector or radius vector r from O to the point ðx; y; zÞ is written
r ¼ xi þ yj þ zk ð3Þ
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2
2
x þ y þ z :
and has magnitude r ¼jrj¼
DOT OR SCALAR PRODUCT

The dot or scalar product of two vectors A and B, denoted by A B (read A dot B)is defined as the
product of the magnitudes of A and B and the cosine of the angle between them. In symbols,
A B ¼ AB cos ; 0 @ @ ð4Þ

Assuming that neither A nor B is the zero vector, an immediate consequence of the definition is that
A B ¼ 0if and only if A and B are perpendicular. Note that A B is a scalar and not a vector.
The following laws are valid:

1. A B ¼ B A Commutative Law for Dot Products
2. A ðB þ CÞ¼ A B þ A C Distributive Law
3. mðA BÞ¼ðmAÞ B ¼ A ðmBÞ¼ ðA BÞm, where m is a scalar.
4. i i ¼ j j ¼ k k ¼ 1; i j ¼ j k ¼ k i ¼ 0
5. If A ¼ A 1 i þ A 2 j þ A 3 k and B ¼ B 1 i þ B 2 j þ B 3 k, then
A B ¼ A 1 B 1 þ A 2 B þ A 3 B 3
2
The equivalence of this component form the dot product with the geometric definition 4 follows
from the law of cosines. (See Fig. 7-8.)