Page 164 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 155

6. jA Bj¼ the area of a parallelogram with sides A and B.
7. If A B ¼ 0 and neither A nor B is a null vector, then A and B are parallel.

TRIPLE PRODUCTS
Dot and cross multiplication of three vectors, A, B, and C may produce meaningful products of the
form ðA BÞC; A ðB CÞ; and A ðB CÞ. The following laws are valid:
1. ðA BÞC 6¼ AðB CÞ in general
2. A ðB CÞ¼ B ðC AÞ¼ C ðA BÞ¼ volume of a parallelepiped having A, B, and C as
edges, or the negative of this volume according as A, B, and C do or do not form a right-
handed system. If A ¼ A 1 i þ A 2 j þ A 3 k, B ¼ B 1 i þ B 2 j þ B 3 k and C ¼ C 1 i þ C 2 j þ C 3 k, then

A 1 A 2

A 3
A ðB CÞ¼ B 1 B 2 B 3 ð6Þ

C 1 C 2 C 3

3. A ðB CÞ 6¼ðA BÞ C (Associative Law for Cross Products Fails)
4. A ðB CÞ¼ ðA CÞB ðA BÞC
ðA BÞ C ¼ðA CÞB ðB CÞA
The product A ðB CÞ is called the scalar triple product or box product and may be denoted by
½ABCŠ. The product A ðB CÞ is called the vector triple product.
In A ðB CÞ parentheses are sometimes omitted and we write A B C. However, parentheses
must be used in A ðB CÞ (see Problem 7.29). Note that A ðB CÞ¼ðA BÞ C. This is often
expressed by stating that in a scalar triple product the dot and the cross can be interchanged without
affecting the result (see Problem 7.26).

AXIOMATIC APPROACH TO VECTOR ANALYSIS
From the above remarks it is seen that a vector r ¼ xi þ yj þ zk is determined when its 3 components
ðx; y; zÞ relative to some coordinate system are known. In adopting an axiomatic approach, it is thus
quite natural for us to make the following

Definition. A three-dimensional vector is an ordered triplet of real numbers with the following
properties. If A ¼ðA 1 ; A 2 ; A 3 Þ and B ¼ðB 1 ; B 2 ; B 3 Þ then

1. A ¼ B if and only if A 1 ¼ B 1 ; A 2 ¼ B 2 ; A 3 ¼ B 3
2. A þ B ¼ðA 1 þ B 1 ; A 2 þ B 2 ; A 3 þ B 3 Þ
3. A B ¼ðA 1 B 1 ; A 2 B 2 ; A 3 B 3 Þ
4. 0 ¼ð0; 0; 0Þ
5. mA ¼ mðA 1 ; A 2 ; A 3 Þ¼ ðmA 1 ; mA 2 ; mA 3 Þ
In addition, two forms of multiplication are established.
6. A B ¼ A 1 B 1 þ A 2 B 2 þ A 3 B 3
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffi
p 2 2 2
7. Length or magnitude of A ¼jAj¼ A A ¼ A þ A þ A 3
1
2
8. A B ¼ðA 2 B 3 A 3 B 2 ; A 3 B 1 A 1 B 3 ; A 1 B 2 A 2 B 1 Þ
Unit vectors are defined to be ð1; 0; 0Þ; ð0; 1; 0Þ; ð0; 0; 1Þ and then designated by i; j; k, respectively,
thereby identifying the components axiomatically introduced with the geometric orthonormal basis
elements.
If one wishes, this axiomatic formulation (which provides a component representation for vectors)
can be used to reestablish the fundamental laws previously introduced geometrically; however, the