Making two or more objects mutually perpendicular

Hello Blender community!

I am trying to make a visualization of a 3D coordinate system (with mesh arrows) where two of the axes are fixed to separate objects that can rotate relative to one another (thus these two axes can be NON-orthogonal), but the third axis is ALWAYS orthogonal to the others (given by their cross product). How can i set this up with armatures, bone constraints, or object transformation locks, etc.? I’m having trouble making the third axis perpendicular to the other two, no matter how the other two are oriented.

Please help! Thank you in advance.