Mathematical Methods for Physics and Engineering - Matematica.NET

7.6 MULTIPLICATION OF VECTORSPFθRrOFigure 7.10 The moment of the force F about O is r×F. The cross representsthe direction of r × F, which is perpendicularly into the plane of the paper.From its definition, we see that the vector product has the very useful propertythat if a × b = 0 then a is parallel or antiparallel to b (unless either of them iszero). We also note thata × a = 0. (7.26)◮Show that if a = b + λc, for some scalar λ, thena × c = b × c.From (7.23) we havea × c =(b + λc) × c = b × c + λc × c.However, from (7.26), c × c = 0 and soa × c = b × c. (7.27)We note in passing that the fact that (7.27) is satisfied does not imply that a = b. ◭An example of the use of the vector product is that of finding the area, A, ofa parallelogram with sides a and b, using the formulaA = |a × b|. (7.28)Another example is afforded by considering a force F acting through a point R,whose vector position relative to the origin O is r (see figure 7.10). Its momentor torque about O is the strength of the force times the perpendicular distanceOP, which numerically is just Fr sin θ, i.e. the magnitude of r × F. Furthermore,the sense of the moment is clockwise about an axis through O that pointsperpendicularly into the plane of the paper (the axis is represented by a crossin the figure). Thus the moment is completely represented by the vector r × F,in both magnitude and spatial sense. It should be noted that the same vectorproduct is obtained wherever the point R is chosen, so long as it lies on the lineof action of F.Similarly, if a solid body is rotating about some axis that passes through theorigin, with an angular velocity ω then we can describe this rotation by a vectorω that has magnitude ω and points along the axis of rotation. The direction of ω223