Cross Product

$\vec{A}\times\vec{B} = \|\vec{A}\|\times\|\vec{B}\|\times\sin\theta\times\widehat{\vec{N}}$
Where $\vec{N}$ is perpendicular to the $\vec{A}$, $\vec{B}$ plane.

$\vec{A}\times\vec{B}$ produces a vector perpendicular to both vectors, in the direction of the chirality of your coordinate system. For example, if $\vec{A}$ is thumb and $\vec{B}$ is index, then $\vec{A}\times\vec{B}$ will point in the direction of the middle finger. The magnitude of $\vec{A}\times\vec{B}$ is the area of the parallelogram of $\vec{A}\vec{B}\vec{A}\vec{B}$, i.e. two vectors along the positive $X$ and $Y$ axes of lengths $2$ and $3$ would produce a cross product along the positive $Z$ axis of length $6$.

For normalised vectors, the magnitude of the cross product is equal to the sine of the angle between them.

The cross product follows the following rules:

• $\vec{A}\times\vec{B}=-\vec{B}\times\vec{A} = \vec{B}\times-\vec{A}$
• $\vec{A}\times\left(\vec{B}+\vec{C}\right)=\vec{A}\times\vec{B}+\vec{A}\times\vec{C}$