# What does pseudovector mean?

I am reading about vector multiplication and cross product is a pseudovector. Could any one explain what a pseudovector to me in simple, layman, idiot-proof terms please? Thanks a lot.

## 5 Answers

- A vector-like object which is invariant under inversion is called a pseudovector, also called an axial vector (as a result of such vectors frequently arising as vectors describing rotation.01
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What does pseudovector mean?

I am reading about vector multiplication and cross product is a pseudovector. Could any one explain what a pseudovector to me in simple, layman, idiot-proof terms please? Thanks a lot.00 - momentum is velocity x mass (p=mv)00
- Pseudo Vector00
- In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, i.e. a transformation that rotates vectors and pseudovectors by an arbitrary angle about an arbitrary axis, but gains an additional sign flip under an improper rotation: a transformation that can be expressed as a proper rotation of a mirror image, or equivalently as an inversion through the origin followed by a proper rotation. The conceptual opposite of a pseudovector is a (true) vector or a polar vector (more formally, a contravariant vector).

A common way of constructing a pseudovector p is by taking the cross product of two vectors a and b:

p = a × b.

Under inversion the two vectors change sign, but their cross product is invariant.A simple example of an improper rotation in 3D (but not in 2D) is a coordinate inversion: x goes to −x, y to −y and z to −z. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector.

This concept 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.

Many occurrences of pseudovectors in mathematics and physics are more naturally analyzed as bivectors, following the calculus of differential forms; the double negation is natural for a bivector. However, bivectors are "less intuitive" in some senses than ordinary vectors, and since in R3 every bivector a ʌ b has a unique dual vector a × b, it is this dual which is more often used.

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