Page 175 - Schaum's Outline of Theory and Problems of Advanced Calculus

166 VECTORS [CHAP. 7

Multiplying by A,
ðB þ CÞ Aa ¼ B Aa þ C Aa
and ðB þ CÞ A ¼ B A þ C A
Then by the commutative law for dot products,
A ðB þ CÞ¼ A B þ A C
and the distributive law is valid.

7.12. Prove that ðA þ BÞ ðC þ DÞ¼ A C þ A D þ B C þ B D.
By Problem 7.11, ðA þ BÞ ðC þ DÞ¼ A ðC þ DÞþ B ðC þ DÞ¼ A C þ A D þ B C þ B D.
The ordinary laws of algebra are valid for dot products where the operations are defined.

7.13. Evaluate each of the following.
ðaÞ i i ¼jijjij cos 08 ¼ð1Þð1Þð1Þ¼ 1
ðbÞ i k ¼jijjkj cos 908 ¼ð1Þð1Þð0Þ¼ 0
ðcÞ k j ¼jkjjjj cos 908 ¼ð1Þð1Þð0Þ¼ 0
j ð2i 3j þ kÞ¼ 2j i 3j j þ j k ¼ 0 3 þ 0 ¼ 3
ðdÞ
ðeÞð2i jÞ ð3i þ kÞ¼ 2i ð3i þ kÞ j ð3i þ kÞ¼ 6i i þ 2i k 3j i j k ¼ 6 þ 0 0 0 ¼ 6
7.14. 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, prove that A B ¼ A 1 B 1 þ A 2 B 2 þ A 3 B 3 .

A B ¼ðA 1 i þ A 2 j þ A 3 kÞ ðB 1 i þ B 2 j þ B 3 kÞ
¼ A 1 i ðB 1 i þ B 2 j þ B 3 kÞþ A 2 j ðB 1 i þ B 2 j þ B 3 kÞþ A 3 k ðB 1 i þ B 2 j þ B 3 kÞ
¼ A 1 B 1 i i þ A 1 B 2 i j þ A 1 B 3 i k þ A 2 B 1 j i þ A 2 B 2 j j þ A 2 B 3 j k
þ A 3 B 1 k i þ A 3 B 2 k j þ A 3 B 3 k k
¼ A 1 B 1 þ A 2 B 2 þ A 3 B 3
since i j ¼ k k ¼ 1 and all other dot products are zero.

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffi
p 2 2 2
A þ A þ A .
7.15. If A ¼ A 1 i þ A 2 j þ A 3 k, show that A ¼ A A ¼ 1 2 3
2
A A.
A A ¼ðAÞðAÞ cos 08 ¼ A . Then A ¼ p ffiffiffiffiffiffiffiffiffiffiffi
Also, A A ¼ðA 1 i þ A 2 j þ A 3 kÞ ðA 1 i þ A 2 j þ A 3 kÞ
2 2 2
¼ðA 1 ÞðA 1 ÞþðA 2 ÞðA 2 Þþ ðA 3 ÞðA 3 Þ¼ A 1 þ A 2 þ A 3
By Problem 7.14, taking B ¼ A.
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p ffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2
A þ A þ A is the magnitude of A. Sometimes A A is written A .
1 2 3
Then A ¼ A A ¼
THE CROSS OR VECTOR PRODUCT
7.16. Prove A B ¼ B A.
A B ¼ C has magnitude AB sin and direction such that A, B, and C form a right-handed system [Fig.
7-24(a)].
B A ¼ D has magnitude BA sin and direction such that B, A, and D form a right-handed system
[Fig. 7-24(b)].
Then D has the same magnitude as C but is opposite in direction, i.e., C ¼ D or A B ¼ B A.
The commutative law for cross products is not valid.

7.17. Prove that A ðB þ CÞ¼ A B þ A C for the case where A is perpendicular to B and also
to C.