Mathematical Methods for Physics and Engineering : A Comprehensive Guide

7.2 ADDITION AND SUBTRACTION OF VECTORS

a

a

b

b a+b

b+a

Figure 7.1 Addition of two vectors showing the commutation relation. We

make no distinction between an arrowhead at the end of the line and one

along the line’s length, but rather use that which gives the clearer diagram.

end of the line and one along the line’s length but, rather, use that which gives the

clearer diagram. Furthermore, even though we are considering three-dimensional

vectors, we have to draw them in the plane of the paper. It should not be assumed

that vectors drawn thus are coplanar, unless this is explicitly stated.

7.2 Addition and subtraction of vectors

Theresultantorvector sumof two displacement vectors is the displacement vector

that results from performing first one and then the other displacement, as shown

in figure 7.1; this process is known as vector addition. However, the principle

of addition has physical meaning for vector quantities other than displacements;

for example, if two forces act on the same body then the resultant force acting

on the body is the vector sum of the two. The addition of vectors only makes

physical sense if they are of a like kind, for example if they are both forces

acting in three dimensions. It may be seen from figure 7.1 that vector addition is

commutative, i.e.

a+b=b+a. (7.1)

The generalisation of this procedure to the addition of three (or more) vectors is

clear and leads to the associativity property of addition (see figure 7.2), e.g.

a+(b+c)=(a+b)+c. (7.2)

Thus, it is immaterial in what order any number of vectors are added.

The subtraction of two vectors is very similar to their addition (see figure 7.3),

that is,

a−b=a+(−b)

where−bis a vector of equal magnitude but exactly opposite direction to vectorb.