system
Given the following post, respond as if you're commenting on Reddit:
human
What gives rise to the right-hand rule in electric currents?

Why does the magnetic field wrap clockwise around a traveling electron?
gpt
I wrote this in response to a similar question a week ago:

>A magnetic field is not really a vector, like the electric field, it's really a bivector, pseudo-vector, or 2-form. As a vector has a direction, a 2-form lives in a plane. The thing is, in 3 dimensions, there are exactly as many dimensions (x, y, z) as there are planes (x-y, y-z, z-x). When we say a magnetic field is pointing in the z-direction, it's actually a bivector in the x-y plane. You can get the direction of a vector associated to a bivector by taking the vector product of 2 vectors in the associated plane.

>So essentially, the cross product appears as an artifact of us pretending that the magnetic field is a vector instead of a bi-vector.

>In d space dimensions, while an electric field would always have d components, a magnetic field would have d(d-1)/2, so in general we couldn't talk about the "direction" of a magnetic field.