How would electromagnetism work in a world without a cross product?
And the generalization of the cross product, the wedge product works just fine in 2d. If you take the wedge of two 1-forms (vectors) in 2d you get a 2-form. In 3d that is equivalent to a (3-2)-form or in other words a (pseudo-)vector. In 2d you get a (2-2)-form or in other words a pseudo-scalar 0-form. That is of course nothing but the z component of the resulting pseudo-vector if you had done it in 3d. So EM in 2d might get flat landers started on weird group theory a bit early, but it is not fundamentally incompatible with life.
Edit: the reason 2d doesn't have a cross product is because you can no longer misinterpret a bivector <<xy,xz,yz>> as a (1-)vector <yz,xz,xy> by confusing each basis bivector with the basis vector[1] orthogonal to it.
0: https://en.wikipedia.org/wiki/Wedge_product
1: Actually one of the two (positive or negative) possible basis vectors; this is why cross products have a right-hand-rule versus left-hand-rule ambiguity.
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.
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.