Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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-

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