Everything I have read indicates that the cross product is simply defined as a x b = i( ay*bz - az*by) - j( ax*bz - az*bx ) + k( ax*by - ay*bx ) and that it just so happens that there is a shorthand notation of cross product in matrix determinant form. How is the cross product formulated??? Is it worked out geometrically, or does it reflect some property of the determinant? Or either? The most revealing thing I have run into so far is that on mathworld it says the determinant of a square matrix has the interpretation of the "content of the parallel piped spanning the column vectors". As I recall the magnitude of the cross product is equal to the area of the parallogram associated with the 2 vectors so there must be some sort of link here.