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Concept
Pseudovector
Summary
In physics and mathematics, a pseudovector (or axial vector) is a quantity that behaves like a vector in many situations, but its direction does not conform when the object is rigidly transformed by rotation, translation, reflection, etc. This can also happen when the orientation of the space is changed. For example, the angular momentum is a pseudovector because it is often described as a vector, but by just changing the position of reference (and changing the position vector), angular momentum can reverse direction, which is not supposed to happen with true vectors.
In three dimensions, the curl of a polar vector field at a point and the cross product of two polar vectors are pseudovectors.
One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, that span the plane. The vector a × b is a normal to the plane (there are two normals, one on each side – the right-hand rule will determine which), and is a pseudovector. This has consequences in computer graphics where it has to be considered when transforming surface normals.
A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. In mathematics, in three-dimensions, pseudovectors are equivalent to bivectors, from which the transformation rules of pseudovectors can be derived. More generally in n-dimensional geometric algebra pseudovectors are the elements of the algebra with dimension n − 1, written ⋀n−1Rn. The label "pseudo" can be further generalized to pseudoscalars and pseudotensors, both of which gain an extra sign flip under improper rotations compared to a true scalar or tensor.
Physical examples of pseudovectors include torque, angular velocity, angular momentum, magnetic field, and magnetic dipole moment.
Consider the pseudovector angular momentum L = Σ(r × p). Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left.
Official source
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