# Why are vectors considered to have odd/negative/- parity while pseudovectors are even/positive/+ in parity?

Most places I read say that true/polar vectors are of odd or - parity, while axial/pseudovectors are of even + parity.

But, pseudovectors gain an 'extra' sign flip after a reflection/parity transformation... Also, they are the ones called -pseudo'...

I thought that 'objects' of whatever type that gained extra sign flip(s) after a 'mirror reflection' (parity transformation) were, by definition, of negative/odd parity...

In short, I am confused about the parity of vectors and pseudovectors...

• It's the same definition in mathematics, for example an odd function is one such that $f(-x)=-f(x)$, while even functions follow $f(-x)=f(x)$. Jan 21, 2021 at 20:05
• "Most places I read say that true/polar vectors are of odd or - parity" If there are any that say otherwise they are wrong. Jan 21, 2021 at 20:15
• The extra minus sign of a pseudo-vector is added to the built-in minus sign of a vector. A vector is said to have odd parity because it has a built-in minus sign. It's not extra, it's built-in. Jan 21, 2021 at 20:24

The cross product of two polar vector forms an axial vector. Say vector $$\vec{A}$$ and $$\vec{B}$$ under inversion operation [ $$P_i ( \vec{r} ) \rightarrow -\vec{r}$$].
$$P_i (\vec{A}) = -\vec{A}; \text{ and } P_i (\vec{B}) = -\vec{B}$$
Then we examine the $$\vec{C} = \vec{A} \times \vec{B}$$
$$P_i(\vec{C}) = P_i( \vec{A} \times \vec{B}) =P_i (\vec{A}) \times P_i (\vec{B}) = \vec{C} .$$
This make $$\vec{C}$$ an axial vector.