7.6 MULTIPLICATION OF VECTORS
θO
R
P
F
rFigure 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,ofa 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 theorigin, with an angular velocityωthen we can describe this rotation by a vector
ωthat has magnitudeωand points along the axis of rotation. The direction ofω