Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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)
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