Page 160 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 151

3. The sum or resultant of vectors A and B of Fig. 7-3(a)below is
a vector C formed by placing the initial point of B on the A
terminal point of A and joining the initial point of A to the
terminal point of B [see Fig. 7-3(b)below]. The sum C is _ A
written C ¼ A þ B. The definition here is equivalent to the
parallelogram law for vector addition as indicated in Fig.7-3(c)
below.
Extensions to sums of more than two vectors are
immediate. For example, Fig. 7-4 below shows how to obtain Fig. 7-2
the sum or resultant E of the vectors A, B, C,and D.

B
A B A
C = A + B
C = A + B
A B

(a) (b) (c)
Fig. 7-3

C
B
B
D D
A
A
C
E = A + B + C + D

Fig. 7-4

4. The difference of vectors A and B, represented by A B,is that vector C which added to B gives
A. Equivalently, A B may be defined as A þð BÞ.If A ¼ B, then A B is defined as the null
or zero vector and is represented by the symbol 0. This has a magnitude of zero but its direction
is not defined.
The expression of vector equations and related concepts is facilitated by the use of real
numbers and functions. In this context, these are called scalars. This special designation arises
from application where the scalars represent object that do not have direction, such as mass,
length, and temperature.
5. Multiplication of a vector A by a scalar m produces a vector mA with magnitude jmj times the
magnitude of A and direction the same as or opposite to that of A according as m is positive or
negative. If m ¼ 0, mA ¼ 0, the null vector.

ALGEBRAIC PROPERTIES OF VECTORS
The following algebraic properties are consequences of the geometric definition of a vector. (See
Problems 7.1 and 7.2.)