I wonder if the cross product was "invented" because it would be handy in physics to calculate quantities like torque = r x F and magnetic force = qv x B. Or is there some other history behind it?
BTW - the book cited in that article, A History of Vector Analysis; The Evolution of the Idea of a Vectorial System, Michael J. Crowe, 1967, republished in 1985, is EXCELLENT!!! See here for details.
I alway thought that vector calculs in general was developed for the analysis of fluid flow. The concept of tourque induced on a paddle wheel in a fluid indicating that there is Curl in a region of flow was the first physical concept that brought all the vector calculus theory to reality for me. To this day I find it easier to visualize E and H as fliud fields.