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 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} = \langle v_1, v_2, v_3 \rangle } , 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 $\mathbf {v}$ (notated as 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 u \times v } ) is $\mathbf {w} =\langle u_{2}v_{3}-u_{3}v_{2},-(u_{1}v_{3}-u_{3}v_{1}),u_{1}v_{2}-u_{2}v_{1}\rangle$. The cross product of $\mathbf {u}$ 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} } can also be written in determinant form like so:

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} = det\begin{vmatrix} i & j & k\\ u_1 & u_2 & u_3\\ v_1 & v_2 & v_3\\ \end{vmatrix} }

Resources

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