Vector and scalar products

It all goes back to Hamilton and quaterions, which are a nifty mathematical structure but not one especially well suited to the uses of physicists and engineers. Quaternions are four dimensional, and Hamilton talked of the "vector of a product" and "scalar of a product". When Gibbs came along, he took a practical engineer's look at it and replaced it with a "vector product" and "scalar product", both of which were defined for three dimensional real vector space only. He didn't know about the sophisticated work of Grassmann, and might not have cared if he had, as space-time hadn't been invented yet and Gibbs had engineering and physics in mind. It all produced a "vector analysis" which pure mathematicians are not in love with but which still is widely used in physics and engineering, despite the fact that we know all about space time these days and wedge products work there also.

The vector product or cross product, if it comes up in math, is either a very special case of a Lie algebra product or more likely the dual of a wedge product in the three-dimensional case. The "x" notation would not be used in either case, nor the terminology. The scalar product comes up all the time but mathematicians don't call it that, they call that an inner product or dot product, and of course it isn't restricted to three dimensions. The product of a scalar with a vector is called "scalar multiplication", and probably shouldn't be called a scalar product due to the conflict of definitions but sometimes is. I think we would be well advised never to use the term "scalar product" for anything. It would make me a lot happier and it would be clearer.