"Cartesian product" actually refers to independent sets (real number lines with independent metrics, which are actually vector spaces). Such "isomorphic" operations work on scalars in a single dimension, not vectors).
i.e., in the space (where the 's refer to scalars in the same dimension), not
or (whichever is greater in magnitude for the individual vectors under the operation; i.e., the projection of the smaller vector onto the larger).
That is the difference between arithmetic sums and products and direct (vector) sums and vector products ("dot" and "cross" products).