The CrossProduct command computes the cross product of u and v.

The infix operator &x can also be used to compute the cross product.

The cross product of u and v is given by the formula $⟨u_2{v}_3-u_3{v}_2\,-u_1{v}_3+u_3{v}_1\,u_1{v}_2-u_2{v}_1⟩$.

If u and v are both row Vectors, a row Vector is returned. Otherwise, a column Vector is returned.

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$u≔⟨a\,b\,c⟩$

$u≔\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$

$v≔⟨d\,e\,f⟩$

$v≔\left[\begin{array}{c}d\\ e\\ f\end{array}\right]$

$\mathrm{CrossProduct}\left(u\,v\right)$

$\left[\begin{array}{c}bf-ce\\ -af+cd\\ ae-bd\end{array}\right]$

$v\&xu$

$\left[\begin{array}{c}-bf+ce\\ af-cd\\ -ae+bd\end{array}\right]$

$w≔⟨g\,h\,i⟩$

$w≔\left[\begin{array}{c}g\\ h\\ i\end{array}\right]$

$\mathrm{simplify}\left(\left(u\&xv\right)\&xw\right)$

$\left[\begin{array}{c}\left(-af+cd\right)i-\left(ae-bd\right)h\\ \left(-bf+ce\right)i+\left(ae-bd\right)g\\ \left(af-cd\right)g+\left(bf-ce\right)h\end{array}\right]$

$\mathrm{simplify}\left(u\&x\left(v\&xw\right)\right)$

$\left[\begin{array}{c}\left(-be-cf\right)g+\left(bh+ci\right)d\\ \left(-ad-cf\right)h+\left(ag+ci\right)e\\ \left(-ad-be\right)i+\left(ag+bh\right)f\end{array}\right]$

The Student[MultivariateCalculus][CrossProduct] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.