Page 178 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 178
CHAP. 7] VECTORS 169
TRIPLE PRODUCTS
7.23. Show that A ðB CÞ is in absolute value equal to the volume
of a parallelepiped with sides A, B, and C. See Fig. 7-28.
Let n be a unit normal to parallelogram I, having the direction of
B C, and let h be the height of the terminal point of A above the
parallelogram I. Fig. 7-28
Volume of a parallelepiped ¼ðheight hÞðarea of parallelogram IÞ
¼ðA nÞðjB CjÞ
¼ A fjB Cjng¼ A ðB CÞ
If A, B and C do not form a right-handed system, A n < 0 and the volume =jA ðB CÞj.
7.24. 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, C ¼ C 1 i þ C 2 j þ C 3 k show that
A 1 A 2
A 3
A ðB CÞ¼ B 1 B 2 B 3
C 1 C 2 C 3
i j k
A ðB CÞ¼ A B 1 B 2 B 3
C 1 C 2 C 3
¼ðA 1 i þ A 2 j þ A 3 kÞ ½ðB 2 C 3 B 3 C 2 Þi þðB 3 C 1 B 1 C 3 Þj þðB 1 C 2 B 2 C 1 Þk
A 1 A 2
A 3
¼ A 1 ðB 2 C 3 B 3 C 2 Þþ A 2 ðB 3 C 1 B 1 C 3 Þþ A 3 ðB 1 C 2 B 2 C 1 Þ¼ B 1 B 2 B 3 :
C 1 C 2 C 3
7.25. Find the volume of a parallelepiped with sides A ¼ 3i j; B ¼ j þ 2k; C ¼ i þ 5j þ 4k.
3 1
0
1
By Problems 7.23 and 7.24, volume of parallelepiped ¼jA ðB CÞj ¼ j 0 2 j
1 5 4
¼j 20j¼ 20:
7.26. Prove that A ðB CÞ¼ðA BÞ C, i.e., the dot and cross can be interchanged.
A 1 A 2 C 1 C 2
A 3 C 3
B 3 ;
By Problem 7.24: A ðB CÞ¼ B 1 B 2 ðA BÞ C ¼ C ðA BÞ¼ A 1 A 2 A 3
C 1 C 2 C 3 B 1 B 2 B 3
Since the two determinants are equal, the required result follows.
7.27. Let r 1 ¼ x 1 i þ y 1 j þ z 1 k, r 2 ¼ x 2 i þ y 2 j þ z 2 k and r 3 ¼ x 3 i þ y 3 j þ z 3 k be the position vectors of
points P 1 ðx 1 ; y 1 ; z 1 Þ, P 2 ðx 2 ; y x ; z 2 Þ and P 3 ðx 3 ; y 3 ; z 3 Þ. Find an equation for the plane passing
through P 1 , P 2 ; and P 3 . See Fig. 7-29.
We assume that P 1 , P 2 , and P 3 do not lie in the same straight line; hence, they determine a plane.
Let r ¼ xi þ yj þ zk denote the position vectors of any point Pðx; y; zÞ in the plane. Consider vectors
P 1 P 2 ¼ r 2 r 1 , P 1 P 3 ¼ r 3 r 1 and P 1 P ¼ r r 1 which all lie in the plane. Then
P 1 P P 1 P 2 P 1 P 3 ¼ 0
