Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

7.6 MULTIPLICATION OF VECTORS


θ

O


R


P


F


r

Figure 7.10 The moment of the forceFaboutOisr×F. The cross represents
the direction ofr×F, which is perpendicularly into the plane of the paper.

From its definition, we see that the vector product has the very useful property


that ifa×b= 0 thenais parallel or antiparallel tob(unless either of them is


zero). We also note that


a×a= 0. (7.26)

Show that ifa=b+λc,for some scalarλ,thena×c=b×c.

From (7.23) we have


a×c=(b+λc)×c=b×c+λc×c.

However, from (7.26),c×c= 0 and so


a×c=b×c. (7.27)

We note in passing that the fact that (7.27) is satisfied doesnotimply thata=b.


An example of the use of the vector product is that of finding the area,A,of

a parallelogram with sidesaandb, using the formula


A=|a×b|. (7.28)

Another example is afforded by considering a forceFacting through a pointR,


whose vector position relative to the originOisr(see figure 7.10). Itsmoment


ortorqueaboutOis the strength of the force times the perpendicular distance


OP, which numerically is justFrsinθ, i.e. the magnitude ofr×F.Furthermore,


the sense of the moment is clockwise about an axis throughOthat points


perpendicularly into the plane of the paper (the axis is represented by a cross


in the figure). Thus the moment is completely represented by the vectorr×F,


in both magnitude and spatial sense. It should be noted that the same vector


product is obtained wherever the pointRis chosen, so long as it lies on the line


of action ofF.


Similarly, if a solid body is rotating about some axis that passes through the

origin, with an angular velocityωthen we can describe this rotation by a vector


ωthat has magnitudeωand points along the axis of rotation. The direction ofω

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