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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 $u\times v$) is $\mathbf {w} =\langle u_{2}v_{3}-u_{3}v_{2},-(u_{1}v_{3}-u_{3}v_{1}),u_{1}v_{2}-u_{2}v_{1}\rangle$. 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