Cross product

Remarks

For the standard basis vectors $e_1$, $e_2$, $e_3$ of $\mathbb{R}^3$: $e_1 \times e_2 = e_3$, $e_2 \times e_3 = e_1$ and $e_3 \times e_1 = e_2$. If $\theta$ is the angle between nonzero vectors $a$ and $b$, then $\|a \times b\| = \|a\| \, \|b\| \sin \theta$. This is the area of the parallelogram with the sides $a$ and $b$. The cross product is perpendicular to its factors and anticommutative. It is bilinear, but it is not associative. In the usual right-handed coordinates, $a$, $b$ and $a \times b$ follow the right-hand rule. The physics notation $a \wedge b$ is also used.