Cross product

Definition

Let $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ be vectors in $\mathbb{R}^3$. The cross product of $a$ and $b$ is the vector $$a \times b = (a_2 b_3 - a_3 b_2, \; a_3 b_1 - a_1 b_3, \; a_1 b_2 - a_2 b_1).$$

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.

Explanations

L1 human

The cross product of two arrows in space gives a third arrow that sticks out at right angles to both. For example, $(1, 0, 0) \times (0, 1, 0) = (0, 0, 1)$. The longer and the more perpendicular the two arrows are, the longer their cross product.