Mathematical Methods for Physics and Engineering : A Comprehensive Guide

7.3 MULTIPLICATION BY A SCALAR

a

a

λ

Figure 7.4 Scalar multiplication of a vector (forλ>1).

O

A

B

P

a

b

p

μ

λ

Figure 7.5 An illustration of the ratio theorem. The pointPdivides the line

segmentABin the ratioλ:μ.

Having defined the operations of addition, subtraction and multiplication by a

scalar, we can now use vectors to solve simple problems in geometry.

A pointPdivides a line segmentABin the ratioλ:μ(see figure 7.5). If the position

vectors of the pointsAandBareaandb, respectively, find the position vector of the

pointP.

As is conventional for vector geometry problems, we denote the vector from the pointA
to the pointBbyAB. If the position vectors of the pointsAandB, relative to some origin
O,areaandb, it should be clear thatAB=b−a.
Now, from figure 7.5 we see that one possible way of reaching the pointPfromOis
first to go fromOtoAand to go along the lineABfor a distance equal to the the fraction
λ/(λ+μ) of its total length. We may express this in terms of vectors as

OP=p=a+

λ

λ+μ

AB

=a+

λ

λ+μ

(b−a)

=

(

1 −

λ

λ+μ

)

a+

λ

λ+μ

b

=

μ

λ+μ

a+

λ

λ+μ

b, (7.6)

which expresses the position vector of the pointPin terms of those ofAandB.Wewould,
of course, obtain the same result by considering the path fromOtoBand then toP.