I noticed something when I was trying to solve a problem today. The scalar triple product is the same as the determinant or a three by three matrix with three vectors as rows:
A = [a, b, c]
det(A) = (a X b) * c
I came across this in Real Timer Rendering, and I can’t really figure out why this is, or if its even useful. It seems sort of related to the short cut method of computing the cross product using a determinate where you write the unit vectors along the top of the matrix, but I always thought that was more of a mnemonic and not actually sound math.
Is there a real relationship here, or is this just some kind of happy coincidence?
Up to a sign, the determinant of an n-by-n matrix is the volume of the parallelepiped spanned by its n n-dimensional row (or column) vectors (or the volume of a unit cube linearly transformed by that matrix). The (axb).c product does, in three dimensions, exactly the same; axb gives a vector perpendicular to a and b and of length equal to the area of the parallelogram spanned by a and b; (axb).c gives the height of c over that parallelogram, times its area. So, no, it’s no coincidence.