> Traditional vector math gives the output of the cross-product of two vectors the same type. (That's why it's only meaningful in 3d.)

That is not really the reason. Relevant trivia: that cross product is defined only in dimensions 3 and 7 (not only in 3) has to do with existence of certain algebraic objects — division algebras — which only exist in dimensions 1 (reals), 2 (complex), 4 (quaternions), and 8 (octonions).

https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc...

https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...

You could say there is cross product in 1d by analogy, but of course it is always equal to 0.