Page 174 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 165

C
1 b 1 a
b 2 2 a
D d E
A B
c
Fig. 7-19 Fig. 7-20

By the Pythagorean theorem,
2 2 2
ðOPÞ ¼ðOQÞ þðQPÞ
2
2
2
where OP denotes the magnitude of vector OP, etc. Similarly, ðOQÞ ¼ðORÞ þðRQÞ .
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2 2 2 2 2 2 2 2 2
1 2 3
Then ðOPÞ ¼ðORÞ þðRQÞ þðQPÞ or A ¼ A 1 þ A 2 þ A 3 , i.e., A ¼ A þ A þ A .
7.9. Determine the vector having initial point Pðx 1 ; y 1 ; z 1 Þ
and terminal point Qðx 2 ; y 2 ; z 2 Þ and find its magnitude.
See Fig. 7-21.
The position vector of P is r 1 ¼ x 1 i þ y 1 j þ z 1 k.
The position vector of Q is r 2 ¼ x 2 i þ y 2 j þ z 2 k.
r 1 ¼ PQ ¼ r 2 or

PQ ¼ r 2 r 1 ¼ðx 2 i þ y 2 j þ z 2 kÞ ðx 1 i þ y 1 j þ z 1 kÞ
¼ðx 2 x 1 Þi þðy 2 y 1 Þj þðz 2 z 1 Þk

Magnitude of PQ ¼ PQ
Fig. 7-21
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
ðx 2 x 1 Þ þðy 2 y 1 Þ þðz 2 z 1 Þ :
¼
Note that this is the distance between points P and Q. A
E F
THE DOT OR SCALAR PRODUCT
G H B
7.10. Prove that the projection of A on B is equal to A b, where b is a
unit vector in the direction of B. Fig. 7-22
Through the initial and terminal points of A pass planes perpen-
dicular to B at G and H respectively, as in the adjacent Fig. 7-22: then
Projection of A on B ¼ GH ¼ EF ¼ A cos ¼ A b B C

(B + C)
7.11. Prove A ðB þ CÞ¼ A B þ A C. See Fig. 7-23.
Let a be a unit vector in the direction of A;then
Projection of ðB þ CÞ on A ¼ projection of B on A þ projection
of C on A E F G A
ðB þ CÞ a ¼ B a þ C a
Fig. 7-23