Page 177 - Schaum's Outline of Theory and Problems of Advanced Calculus

168 VECTORS [CHAP. 7

and the distributive law holds. Multiplying by 1, using Problem 7.16, this becomes ðB þ CÞ A ¼
B A þ C A. Note that the order of factors in cross products is important. The usual laws of algebra
apply only if proper order is maintained.

i j k

.

7.19. (a)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 A 2 A 3

B 1 B 2 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

i j
k

¼ðA 2 B 3 A 3 B 2 Þi þðA 3 B 1 A 1 B 3 Þj þðA 1 B 2 A 2 B 1 Þk ¼ A 1 A 2 A 3

B 1 B 2 B 3

(b)Use the determinant representation to prove the result of Problem 7.18.
7.20. If A ¼ 3i j þ 2k and B ¼ 2i þ 3j k, find A B.

j
i k

1
2 3 2 3 1
A B ¼ 3 1 2 ¼ i j þ k

3 1 2 1 2 3
2 3 1
¼ 5i þ 7j þ 11k
7.21. Prove that the area of a parallelogram with sides A and B is jA Bj. See Fig. 7-27.
Area of parallelogram ¼ hjBj
¼jAj sin jBj
¼jA Bj
A h
Note that the area of the triangle with sides A and
1
B ¼ jA Bj.
2
B
Fig. 7-27
7.22. Find the area of the triangle with vertices at
Pð2; 3; 5Þ; Qð4; 2; 1Þ; Rð3; 6; 4Þ.
PQ ¼ð4 2Þi þð2 3Þj þð 1 5Þk ¼ 2i j 6k
PR ¼ð3 2Þi þð6 3Þj þð4 5Þk ¼ i þ 3j k

1 1
2 2
Area of triangle ¼ jPQ PRj¼ jð2i j6kÞ ði þ 3j kÞj

j
i k

1
1
2 2
¼ 2 1 6 ¼ j19i 4j þ 7kj

1 3 1
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 2 2 1 p ffiffiffiffiffiffiffiffi
426
2 2
¼ ð19Þ þð 4Þ þð7Þ ¼