I want to check if my understanding about the cross product is correct.
Wikipedia page on cross product says the definition of cross product depends on the orientation of the vector space. Orientation is defined in terms of equivalent class, so we fix one basis and say it's positively oriented and all other basis in the same equivalent class bucket becomes positively oriented, and the rest are negatively oriented. Usually we use standard basis to get standard orientation.
Now it seems what makes sense to me is to define the cross product $a \times b$ in a way such that $a$, $b$ and $a \times b$ form a basis that is positively oriented. However, a lot of references refer to handedness as the way to define the direction of the cross product. This leaves the impression that there's some external definition of handedness for a particular basis (an example picture).
This doesn't make much sense to me. Since vectors are abstract entities, if we fix a basis as the orientation, no matter if we draw the basis in a left handed fashion or right handed fashion, it is certainly the same basis and the same orientation?
The cross product wikipedia article states
The standard basis vectors $i$, $j$, and $k$ satisfy the following equalities in a right hand coordinate system:
$i \times j = k$
$j \times k = i$
$k \times i = j$
This makes it sound like cross product will follow a different set of rules when in left hand coordinate system. But if we use the standard basis $i$, $j$, $k$ as the orientation, certainly it will always be $i \times j = k$, no matter we call it a left handed coordinate system or a right handed one?