The Cross Product

Cross_product_parallelogram

The cross product is an operation between two 3-dimensional vectors that returns a third vector orthogonal (i.e., perpendicular) to the first two. For vectors $\mathbf {u} =\langle u_{1},u_{2},u_{3}\rangle$ and $\mathbf {v} =\langle v_{1},v_{2},v_{3}\rangle$, the cross product of $\mathbf {u}$ and $\mathbf {v}$ (notated as $\mathbf {u} \times \mathbf {v}$) is $\mathbf {w} =\langle u_{2}v_{3}-u_{3}v_{2},-(u_{1}v_{3}-u_{3}v_{1}),u_{1}v_{2}-u_{2}v_{1}\rangle$. The magnitude of the cross product, $|\mathbf {u} \times \mathbf {v} |$, equals the area of the parallelogram created by adding vector $\mathbf {u}$ to vector $\mathbf {v}$, and adding $\mathbf {v}$ to $\mathbf {u}$ (see image).

One way to remember the cross product of $\mathbf {u}$ and $\mathbf {v}$ is to calculate it with the following determinant:

$\mathbf {w} ={\textrm {det}}{\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\\end{vmatrix}}=(u_{2}v_{3}-u_{3}v_{2})\mathbf {i} -(u_{1}v_{3}-u_{3}v_{1})\mathbf {j} +(u_{1}v_{2}-u_{2}v_{1})\mathbf {k}$

Resources

• The Cross Product, OpenStax
• Cross Product, Cornell University