Cross Product

Cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both vectors. Vector products are also called cross products.

$$\begin{array}{rl}\vec{c}=\vec{a}\times \vec{b}& =\left[\begin{array}{ccc}i& j& k\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right]\\ & \\ \left[\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right]\times \left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]& =\left[\begin{array}{c}{a}_2{b}_3-a_3{b}_2\\ a_3{b}_1-a_1{b}_3\\ a_1{b}_2-a_2{b}_1\end{array}\right]\end{array}$$

Use case §

Area of Parallelogram §

$|\vec{x}\times \vec{y}|=|\vec{x}||\vec{y}|\sin \theta$

Volume of Parallelepiped §

A parallelepiped is the three dimensional analogue of a parallelogram. $V=|\vec{w}\cdot (\vec{x}\times \vec{y})|$