structural:reformatting, linguistic:voice_change
The paradox of the orthonormal basis

Let (i, j, k) be an orthonormal basis. The vectors i, j and k do not depend on the orientation of the space. They can even be defined in the absence of any orientation. They can not therefore be axial vectors. But if i and j are polar vectors, then k is an axial vector for i × j = k or j × i = k. This is a paradox.

"Axial" and "polar" are physical qualifiers for physical vectors; that is, vectors which represent physical quantities such as the velocity or the magnetic field. The vectors i, j and k are mathematical vectors, neither axial nor polar. In mathematics, the cross-product of two vectors is a vector. There is no contradiction.