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, 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} \times \mathbf{v}| } , equals the area of the parallelogram created by adding vector 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} } to vector 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{v} } , and adding 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{v} } to 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} } (see image).

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

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{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_2v_3 - u_3v_2)\mathbf{i} - (u_1v_3 - u_3v_1)\mathbf{j} + (u_1v_2 - u_2v_1)\mathbf{k} }

Resources

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