Page 172 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 163

Solved Problems

VECTOR ALGEBRA
7.1. Show that addition of vectors is commutative, i.e., A þ B ¼ B þ A. See Fig. 7-15 below.

OP þ PQ ¼ OQ or A þ B ¼ C;
and OR þ RQ ¼ OQ or B þ A ¼ C:

Then A þ B ¼ B þ A.

Q
P B Q
B
P
C = A + B A A C

A C = B + A (A + B)
R (B + C)
O
B D
O R
Fig. 7-15 Fig. 7-16

7.2. Show that the addition of vectors is associative, i.e., A þðB þ CÞ¼ ðA þ BÞþ C. See Fig. 7-16
above.
and
OP þ PQ ¼ OQ ¼ðA þ BÞ PQ þ QR ¼ PR ¼ðB þ CÞ
Since OP þ PR ¼ OR ¼ D; i:e:; A þðB þ CÞ¼ D
OQ þ QR ¼ OR ¼ D; i:e:; ðA þ BÞþ C ¼ D
we have A þðB þ CÞ¼ ðA þ BÞþ C.
Extensions of the results of Problems 7.1 and 7.2 show that the order of addition of any number of
vectors is immaterial.

7.3. An automobile travels 3 miles due north, then 5 miles N Q
northeast as shown in Fig. 7-17. Represent these displace-
ments graphically and determine the resultant displacement
(a) graphically, (b) analytically. B A
Vector OP or A represents displacement of 3 mi due north.
45° C = A + B R
Vector PQ or B represents displacement of 5 mi northeast. P
Vector OQ or C represents the resultant displacement or sum 135°
of vectors A and B, i.e., C ¼ A þ B. This is the triangle law of
A B
vector addition.
The resultant vector OQ can also be obtained by constructing
the diagonal of the parallelogram OPQR having vectors OP ¼ A W O E
and OR (equal to vector PQ or B)as sides. This is the parallelo-
gram law of vector addition. Unit = 1 mile
S
(a) Graphical Determination of Resultant. Lay off the 1 mile unit
on vector OQ to find the magnitude 7.4 mi (approximately). Fig. 7-17