TITLE: Why is cross product not commutative?
QUESTION [9 upvotes]: Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why it makes logical sense for the resultant vector to be either going in the negative or positive direction depending on the order of the cross operation. I don't have any formal experience in linear algebra, so I would appreciate if an answer would take that into account (I'm merely learning vectors as part of a 3D game math education).

REPLY [2 votes]: Because $\vec{a}$, $\vec{b}$ and $\vec{a}\wedge\vec{b}$ are a right hand triple of vectors.
And, if $(\vec{a},\vec{b},\vec{c})$ is a right hand (ordered) triple of vectors, then also $(\vec{b},\vec{a},-\vec{c})$ is, while $(\vec{b},\vec{a},\vec{c})$ is not. Indeed not only cross product is not commutative, further it is anticommutative.
See, also, the cross product tag wiki.