Pseudovector

In physics and mathematics, a pseudovector (or axial vector)^[2] 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 (also known as polar vectors).^[3]

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,^[4] 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. In three dimensions, the curl of a polar vector field at a point and the cross product of two polar vectors are pseudovectors.^[5]

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.