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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...

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    $\begingroup$ It's the same definition in mathematics, for example an odd function is one such that $f(-x)=-f(x)$, while even functions follow $f(-x)=f(x)$. $\endgroup$
    – Triatticus
    Jan 21, 2021 at 20:05
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    $\begingroup$ "Most places I read say that true/polar vectors are of odd or - parity" If there are any that say otherwise they are wrong. $\endgroup$
    – my2cts
    Jan 21, 2021 at 20:15
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    $\begingroup$ 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. $\endgroup$
    – garyp
    Jan 21, 2021 at 20:24

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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.

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