My understanding of pseudovectors vs vectors is pretty basic. Both transform in the same way under a rotation, but differently upon reflection. I might even be able to summarize that using an equation, but that's about it.

Similarly, I can follow arguments that pseudovectors behave differently in "mirrors" than vectors. But my response to this is always: Okay, so what? When would I ever "do physics" in a mirror?

The usefulness eludes me. I'd like to gain a better understanding of the importance of this difference.


When is it useful for an experimental physicist to distinguish between the two?
When is it useful for a theoretical physicist to distinguish between the two?


I believe symmetry is important to at least one of these, but would appreciate a practical rather than abstract argument of when one has to be careful about the distinction.

[Disclaimer: I'm not providing an argument where the distinction would be useful. I am providing an argument that pseudovectors and vectors describe intrinsically different geometrical concepts, and should, for clarify of argument, never be conflated just because they look so similar]

The point is that pseudovectors, by their very nature, are not the same objects as vectors:

A vector, as commonly understood in physics, is an element of the vector space $\mathbb{R}^n$ spanned by the standard basis $e_i$. It points in a direction, and is geometrically connected to a line, i.e. a one-dimensional subspace of $\mathbb{R}^n$.

A pseudovector, as almost no one ever explicitly will tell you, is an element of the sub-top degree of the exterior algebra $\Lambda^{n-1}\mathbb{R}^n$, the space spanned by $e_{i_1} \wedge \dots \wedge e_{i_{n-1}}$. This does not directly point into a direction, but is geometrically the $n-1$-dimensional hyperplane spanned by the vectors $e_{i_1},\dots,e_{i_{n-1}}$, and can then be interpreted as pointing in the direction perpendicular to that hyperplane. Formally, this translation from hyperplanes into normal vectors is the Hodge dual mapping $\Lambda^k\mathbb{R}^n$ to $\Lambda^{n-k}\mathbb{R}^n$. 

And there you see why pseudovectors are different from vectors under reflection, geometrically: In $\mathbb{R}^3$, i.e. our ordinary world, the planes are spanned by two vectors - if both change their signs, the pseudovector described by them will not (since the wedge $\wedge$ is linear and anticommutative).

One importance of these considerations is when you want to step from $\mathbb{R}^3$ to higher dimensions. You lose the cross product (which is really just the concatenation of the wedge and the Hodge), and your former pseudovectors are now suddenly no vectors in the ordinary sense at all anymore, since $\Lambda^2 \mathbb{R}^n$ (the "space of planes") does not map to unique normal vectors by the Hodge dual in dimensions that are not three. Now you need to genuinely tell your former pseudovectors and vectors apart, since they now have a different number of independent coordinate entries.
