Cross Product

linear-algebra

Definition

Cross Product

The cross product of two vectors $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^3$ is the vector

$$\mathbf{a}\times \mathbf{b}=\left[\begin{array}{c}{\mathbf{a}}_1\\ {\mathbf{a}}_2\\ {\mathbf{a}}_3\end{array}\right]\times \left[\begin{array}{c}{\mathbf{b}}_1\\ {\mathbf{b}}_2\\ {\mathbf{b}}_3\end{array}\right]=\underset{\text{Symmetric Pattern}}{\underbrace{\left[\begin{array}{c}{\mathbf{a}}_2{\mathbf{b}}_3-{\mathbf{a}}_3{\mathbf{b}}_2\\ {\mathbf{a}}_3{\mathbf{b}}_1-{\mathbf{a}}_1{\mathbf{b}}_3\\ {\mathbf{a}}_1{\mathbf{b}}_2-{\mathbf{a}}_2{\mathbf{b}}_1\end{array}\right]}}$$

The resulting vector $\mathbf{a}\times \mathbf{b}$ is orthogonal to both $\mathbf{a}$ and $\mathbf{b}$, with magnitude $\Vert \mathbf{a}\times \mathbf{b}\Vert =\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \sin \theta$ equal to the area of the parallelogram they span.