[yiyus]

You could use the exterior product.

Maxwell did not use a cross product in his original publications. In fact, the development of the cross product was a response to the longer (quaternion based) formulation of electromagnetism equations at the time.

[kbenson]

I'm confused by GP's reasoning. Is it valid to assume that since we have an equation that requires three dimensions to show how a force reacts, that if you remove a dimension that force doesn't exist?

Isn't it just as possible that there's a different equation for how that force would work in two dimensions (or to assume our current equation is a specialized version of a general equation that works in all dimensions)?

It just seems rather odd to assume that since our understanding doesn't extend to a circumstance that means something is impossible there. Or is there some aspect of this I'm missing?

Edit: Perhaps I was misinterpreting the GP contextually. Maybe they were just asking if we have an equation for it, rather than questioning how it could exist.