group theory

Pseudovectors, or axial vectors as they are sometimes referred to as, are commonly encountered, but mysterious mathematical objects. In physics they arise in many different areas, particularly when cross products are involved - e.g., magnetic fields, angular momentum, and torque. The goal of this post is to address two seemingly different definitions of a pseudovector, in 3 dimensions, and in the process unravel this mystery. Definition 1 A pseudovector is a tensor on $\mathbb{R}^3$ that transforms like a vector under proper rotations, but picks up a sign under an improper rotation, like a reflection....