Right, with bivectors as in the article it all works. What I was saying is that summing areas and length is exactly what happens with vector product. The k vec

Right, with bivectors as in the article it all works. What I was saying is that summing areas and length is exactly what happens with vector product. The k vector (or the k imaginary unit in quaternions) is the third unit vector, so it has length 1. But when I compute i⨯j=k, I suddenly interpret it as the area of the parallelogram formed by a and j, and at the same time k is the direction perpendicular to both i and j so its coefficient must be a length. Likewise for triple product which computes a volume but it expresses it as a number (i.e. a length). We study all of these things in vector calculus and don't pay attention to these inconsistencies, but they are there.