Page 163 - Schaum's Outline of Theory and Problems of Advanced Calculus

154 VECTORS [CHAP. 7

In particular,
2
2
2
jCj ¼jAj þjBj 2jAjjBj cos
Since C ¼ B A its components are B 1 A 1 ; B 2 A 2 ; B 3 A 3 and
the square of its magnitude is
2 2 2 2 2 2
ðB 1 þ B 2 þ B 3 Þþ ðA 1 þ A 2 þ A 3 Þ 2ðA 1 B 1 Þþ A 2 B 2 þ A 3 B 3 Þ
or
2 2
Fig. 7-8
jBj þjAj 2ðA 1 B 1 þ A 2 B 2 þ A 3 B 3 Þ
2
When this representation for jC j is placed in the original equation and cancellations are made, we
obtain
A 1 B 1 þ A 2 B 2 þ A 3 B 3 ¼jAjjBj cos :

CROSS OR VECTOR PRODUCT
The cross or vector product of A and B is a vector C ¼ A B (read A cross B). The magnitude of
A B is defined as the product of the magnitudes of A and B and the sine of the angle between them.
The direction of the vector C ¼ A B is perpendicular to the plane of A and B and such that A, B, and C
form a right-handed system. In symbols,
A B ¼ AB sin u; 0 @ @ ð5Þ

where u is a unit vector indicating the direction of A B.If A ¼ B or if A is parallel to B, then sin ¼ 0
and A B ¼ 0.
The following laws are valid:

1. A B ¼ B A (Commutative Law for Cross Products Fails)
2. A ðB þ CÞ¼ A B þ A C Distributive Law
3. mðA BÞ¼ ðmAÞ B ¼ A ðmBÞ¼ ðA BÞm, where m is a scalar.

Also the following consequences of the definition are important:
4. i i ¼ j j ¼ k k ¼ 0, i j ¼ k; j k ¼ i; k i ¼ j
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

i j k

A B ¼ A 1 A 2 A 3

B 1 B 2 B 3
The equivalence of this component representation
(5) and the geometric definition may be seen as follows.
Choose a coodinate system such that the direction of the
x-axis is that of A and the xy plane is the plane of the
vectors A and B. (See Fig. 7-9.)

i j k

Then 0 0 ¼ A 1 B 2 k ¼jAjjBjsine K

A B ¼ A 1

0
B 1 B 2
Since this choice of coordinate system places no
restrictions on the vectors A and B, the result is general
and thus establishes the equivalence. Fig. 7-9