How can you identify a pseudo vector from polar vector?
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A loop of wire (black), carrying a current I, creates a magnetic field B (blue). If the position and current of the wire are reflected across the plane indicated by the dashed line, the magnetic field it generates would not be reflected: Instead, it would be reflected and reversed. The position of the wire and its current are "true" vectors, but the magnetic field B is a pseudovector.[1]
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a true vector, also known, in this context, as a polar vector, which on reflection matches its mirror image.
In three dimensions, the pseudovector p is associated with the curl of a polar vector or with the cross product of two polar vectors a and b:[2]
The vector p calculated this way is a pseudovector. One example is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b,[3] 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, pseudovectors are equivalent to three-dimensional 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.