Editing Euclidean vector (section)

==Vectors, pseudovectors, and transformations==
{{Multiple issues|section=yes|
{{Unreferenced section|date=December 2021}}
{{Technical|section|date=December 2021}}
}}
An alternative characterization of Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under a [[coordinate system|coordinate transformation]]. A ''contravariant vector'' is required to have components that "transform opposite to the basis" under changes of [[Basis (linear algebra)|basis]]. The vector itself does not change when the basis is transformed; instead, the components of the vector make a change that cancels the change in the basis. In other words, if the reference axes (and the basis derived from it) were rotated in one direction, the component representation of the vector would rotate in the opposite way to generate the same final vector. Similarly, if the reference axes were stretched in one direction, the components of the vector would reduce in an exactly compensating way. Mathematically, if the basis undergoes a transformation described by an [[invertible matrix]] ''M'', so that a coordinate vector '''x''' is transformed to {{nowrap|1='''x'''′ = ''M'''''x'''}}, then a contravariant vector '''v''' must be similarly transformed via {{nowrap|1='''v'''′ = ''M''<math>^{-1}</math>'''v'''}}. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. For example, if ''v'' consists of the ''x'', ''y'', and ''z''-components of [[velocity]], then ''v'' is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an abstract [[vector space|vector]], but this vector would not be contravariant, since rotating the box does not change the box's length, width, and height. Examples of contravariant vectors include [[displacement (vector)|displacement]], [[velocity]], [[electric field]], [[momentum]], [[force]], and [[acceleration]].

In the language of [[differential geometry]], the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a ''contravariant vector'' to be a [[tensor]] of [[Covariance and contravariance of vectors|contravariant]] rank one. Alternatively, a contravariant vector is defined to be a [[tangent space|tangent vector]], and the rules for transforming a contravariant vector follow from the [[chain rule]].

Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip {{em|and}} gain a minus sign. A transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change the ''[[orientation (space)|orientation]]'' of space. A vector which gains a minus sign when the orientation of space changes is called a ''[[pseudovector]]'' or an ''axial vector''. Ordinary vectors are sometimes called ''true vectors'' or ''polar vectors'' to distinguish them from pseudovectors. Pseudovectors occur most frequently as the [[cross product]] of two ordinary vectors.

One example of a pseudovector is [[angular velocity]]. Driving in a [[car]], and looking forward, each of the [[wheel]]s has an angular velocity vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the ''reflection'' of this angular velocity vector points to the right, but the {{em|actual}} angular velocity vector of the wheel still points to the left, corresponding to the minus sign. Other examples of pseudovectors include [[magnetic field]], [[torque]], or more generally any cross product of two (true) vectors.

This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying [[symmetry]] properties.