Mathematical Methods for Physics and Engineering : A Comprehensive Guide

7.6 MULTIPLICATION OF VECTORS

If we introduce a set of basis vectors that are mutually orthogonal, such asi,j,

k, we can write the components of a vectora, with respect to that basis, in terms

of the scalar product ofawith each of the basis vectors, i.e.ax=a·i,ay=a·jand

az=a·k. In terms of the componentsax,ayandazthe scalar product is given by

a·b=(axi+ayj+azk)·(bxi+byj+bzk)=axbx+ayby+azbz, (7.21)

where the cross terms such asaxi·byjare zero because the basis vectors are

mutually perpendicular; see equation (7.18). It should be clear from (7.15) that

the value ofa·bhas a geometrical definition and that this value is independent

of the actual basis vectors used.

Find the angle between the vectorsa=i+2j+3kandb=2i+3j+4k.

From (7.15) the cosine of the angleθbetweenaandbis given by

cosθ=

a·b

|a||b|

.

From (7.21) the scalar producta·bhas the value

a·b=1×2+2×3+3×4=20,

and from (7.13) the lengths of the vectors are

|a|=

√

12 +2^2 +3^2 =

√

14 and |b|=

√

22 +3^2 +4^2 =

√

29.

Thus,

cosθ=

20

√

14

√

29

≈ 0. 9926 ⇒ θ=0.12 rad.

We can see from the expressions (7.15) and (7.21) for the scalar product that if

θis the angle betweenaandbthen

cosθ=

ax

a

bx

b

+

ay

a

by

b

+

az

a

bz

b

whereax/a,ay/aandaz/aare called thedirection cosinesofa, since they give the

cosine of the angle made byawith each of the basis vectors. Similarlybx/b,by/b

andbz/bare the direction cosines ofb.

If we take the scalar product of any vectorawith itself then clearlyθ= 0 and

from (7.15) we have

a·a=|a|^2.

Thus the magnitude ofacan be written in a coordinate-independent form as

|a|=

√

a·a.

Finally, we note that the scalar product may be extended to vectors with

complex components if it is redefined as

a·b=a∗xbx+a∗yby+a∗zbz,

where the asterisk represents the operation of complex conjugation. To accom-