The multiplication of quaternions was defined by their inventor Maxwell as follows: Given two quaternions: q = a +b.i +c.j + d.k r = e +f.i +g.j + h.k where the coefficients a through h are real, the product is given by the rules: i.i = j.j =k.k = i.j.k = -1 i.j=k j.k=i k.i=j j.i=-k k.j=-i i.k=-j Thus q.r = a.e -b.f - h.c - hd + (e.b +b.f +h.c -g.d).i + (e.c +a.g +d.f -b.h).j+ (e.d +a.h +b.g -f.c ).k