Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR ALGEBRA

a

a

a

b

b

b

c

c

c

a+(b+c)

(a+b)+c

b+c

b+c

a+b

a+b

Figure 7.2 Addition of three vectors showing the associativity relation.

−b

b

a

a

a−b

Figure 7.3 Subtraction of two vectors.

The subtraction of two equal vectors yields the zero vector, 0 , which has zero

magnitude and no associated direction.

7.3 Multiplication by a scalar

Multiplication of a vector by a scalar (not to be confused with the ‘scalar

product’, to be discussed in subsection 7.6.1) gives a vector in the same direction

as the original but of a proportional magnitude. This can be seen in figure 7.4.

The scalar may be positive, negative or zero. It can also be complex in some

applications. Clearly, when the scalar is negative we obtain a vector pointing

in the opposite direction to the original vector. Multiplication by a scalar is

associative, commutative and distributive over addition. These properties may be

summarised for arbitrary vectorsaandband arbitrary scalarsλandμby

(λμ)a=λ(μa)=μ(λa), (7.3)

λ(a+b)=λa+λb, (7.4)

(λ+μ)a=λa+μa. (7.5)