How do I determine whether the orientation of a basis is positive or negative using the cross product

Question

I know that if I have an orthonormal base in $\mathbb{R}^3$, namely $e_1$, $e_2$ & $e_3$, then it is positively oriented if

$$e_1 \times e_2 = e_3$$ $$e_2 \times e_3 = e_1$$ $$e_3 \times e_1 = e_2$$

It would make sense if it was negatively oriented when

$$e_1 \times e_2 = -e_3$$ $$e_2 \times e_3 = -e_1$$ $$e_3 \times e_1 = -e_2$$

But I would like to be sure. Am I correct?

Answer 1

A basis $u$, $v$, $w$ of $\mathbb{R}^3$ is positively oriented if $(u\times v)\cdot w > 0$, and negatively oriented if $(u \times v) \cdot w < 0$. In the case of orthonormal vectors, that is probably equivalent to what you have.