The Cross Product

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 $u\times v$ ) is $\mathbf {w} =\langle u_{2}v_{3}-u_{3}v_{2},-(u_{1}v_{3}-u_{3}v_{1}),u_{1}v_{2}-u_{2}v_{1}\rangle$ .