Editing Pseudovector (section)

==Physical examples==
Physical examples of pseudovectors include [[angular velocity]],<ref name=FeynmanLectures/> [[angular acceleration]], [[angular momentum]],<ref name=FeynmanLectures/> [[torque]],<ref name=FeynmanLectures/> [[magnetic field]],<ref name=FeynmanLectures/> and [[magnetic dipole moment]].

[[Image:Impulsmoment van autowiel onder inversie.svg|thumb|Each wheel of the car on the left driving away from an observer has an angular momentum pseudovector pointing left. The same is true for the mirror image of the car. The fact that the arrows point in the same direction, rather than being mirror images of each other, indicates that they are pseudovectors.]]

Consider the pseudovector [[angular momentum]] {{nowrap|1='''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 [[Symmetry in physics|the effect of symmetry on the solution to physical systems]]. Consider an electric current loop in the {{nowrap|1=''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 (mathematics)|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. <math>\mathbf{a}=(a_x,a_y,a_z)</math>, 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.