Editing Cross product (section)

== Handedness ==<!-- This section is linked from [[Vector calculus]] -->
{{Original research section|date=September 2021}}

=== Consistency ===
When physics laws are written as equations, it is possible to make an arbitrary choice of the coordinate system, including handedness. One should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two [[polar vector]]s, one must take into account that the result is an [[Pseudovector|axial vector]]. Therefore, for consistency, the other side must also be an axial vector.{{Citation needed|date=April 2008}} More generally, the result of a cross product may be either a polar vector or an axial vector, depending on the type of its operands (polar vectors or axial vectors). Namely, polar vectors and axial vectors are interrelated in the following ways under application of the cross product:

* polar vector &times; polar vector = axial vector
* axial vector &times; axial vector = axial vector
* polar vector &times; axial vector = polar vector
* axial vector &times; polar vector = polar vector
or symbolically
* polar &times; polar = axial 
* axial &times; axial = axial 
* polar &times; axial = polar 
* axial &times; polar = polar

Because the cross product may also be a polar vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a polar vector and the other one is an axial vector (e.g., the cross product of two polar vectors). For instance, a [[vector triple product]] involving three polar vectors is a polar vector.

A handedness-free approach is possible using exterior algebra.

=== The paradox of the orthonormal basis ===
Let ('''i''', '''j''', '''k''') be an orthonormal basis. The vectors '''i''', '''j''' and '''k'''  do not depend on the orientation of the space. They can even be defined in the absence of any orientation. They can not therefore be axial vectors. But if '''i''' and '''j''' are polar vectors, then '''k''' is an axial vector for '''i''' × '''j''' = '''k''' or '''j''' × '''i''' = '''k'''. This is a paradox.

"Axial" and "polar" are ''physical'' qualifiers for ''physical'' vectors; that is, vectors which represent ''physical'' quantities such as the velocity or the magnetic field. The vectors '''i''', '''j''' and '''k''' are mathematical vectors, neither axial nor polar. In mathematics, the cross-product of two vectors is a vector. There is no contradiction.