There are many sorts of vector quantities in physics whose magnitude is reversed when the mirror image of the physical system is considered. The mass of the Earth remains positive but its angular momentum is reversed in a mirror. These vectors are often called “axial vectors”. Note that velocity is not axial and that the 2D form of velocity is a 2D vector. Note that angular momentum in 2D problems is an important concept but such is a scalar, not a vector in 2D! It takes 1 number to measure angular momentum in 2D, 3 in 3D and 6 in 4D!
In n dimensions n(n–1)/2 numbers are required to specify a general angular velocity. Having chosen an orthogonal coördinate system each distinct pair of coördinates has a spin that must be specified to capture all possible ways a rigid body can be moving while its origin remains fixed. Tensor notation collects this information best as an antisymmetric tensor thus specifying a number for all pairs, but requiring that ωij = –ωji. When the angular velocity of a rigid body is ωij then the velocity of a point on the body at xi is vj = xiωij. (Beware the summation convention in the previous expression.)
In 3D it will be seen that axial vector <x, y, z> describes the same angular velocity as the tensor
There is a nice equation relating angular momentum L to angular velocity ω using axial vectors for both:
L = Iω where I is the tensor of inertia which is symmetric but in this equation is taken as a symmetric matrix. To generalize equations like this to other dimensions we need the tensor equivalent—we need a form which automatically yields antisymmetric results.