Pseudovector

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does not transform like a vector under certain discontinuous rigid transformations such as reflections. For example, the angular velocity of a rotating object is a pseudovector because, when the object is reflected in a mirror, the reflected image rotates in such a way that its angular velocity is not the mirror image of the angular velocity of the original object. For true vectors (also known as polar vectors), the reflection vector and the original vector must be mirror images. 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. In three dimensions, the curl of a polar vector field at a point and the cross product of two polar vectors are pseudovectors. A number of vector physical quantities behave as pseudovectors rather than polar vectors, including magnetic field and torque. 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 exterior algebra and 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 Physical examples of pseudovectors include angular velocity, angular acceleration, angular momentum, torque, 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 (by the right-hand rule). If the world is reflected in a mirror which switches the left and right side of the car, the "reflection" of this angular momentum "vector" (viewed as an ordinary vector) points to the right, but the actual angular momentum vector of the wheel (which is still turning forward in the reflection) still points to the left (by the right-hand rule), corresponding to the extra sign flip in the reflection of a pseudovector. The distinction between polar vectors and pseudovectors becomes important in understanding the effect of symmetry on the solution to physical systems. Consider an electric current loop in the z = 0 plane that inside the loop generates a magnetic field oriented in the z direction. This system is symmetric (invariant) under mirror reflections through this plane, with the magnetic field unchanged by the reflection. But reflecting the magnetic field as a vector through that plane would be expected to reverse it; this expectation is corrected by realizing that the magnetic field is a pseudovector, with the extra sign flip leaving it unchanged. In physics, pseudovectors are generally the result of taking the cross product of two polar vectors or the curl of a polar vector field. The cross product and curl are defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule. The entire body of physics that deals with (right-handed) pseudovectors and the right hand rule could be replaced by using (left-handed) pseudovectors and the left hand rule without issue. The (left) pseudovectors so defined would be opposite in direction to those defined by the right-hand rule. While vector relationships in physics can be expressed in a coordinate-free manner, a coordinate system is required in order to express vectors and pseudovectors as numerical quantities. Vectors are represented as ordered triplets of numbers: e.g. a = ( a x , a y , a z ) {\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z})} , and pseudovectors are represented in this form too. When transforming between left and right-handed coordinate systems, representations of pseudovectors do not transform as vectors, and treating them as vector representations will cause an incorrect sign change, so that care must be taken to keep track of which ordered triplets represent vectors, and which represent pseudovectors. This problem does not exist if the cross product of two vectors is replaced by the exterior product of the two vectors, which yields a bivector which is a 2nd rank tensor and is represented by a 3×3 matrix. This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.

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