The CrossProduct(v1, v2) function (vector product) computes the cross product of v1 and v2, where v1 and v2 can be either three dimensional free Vectors, rooted Vectors, position Vectors, vector fields, Del or Nabla.

The function  and can be accessed through  &x or CrossProduct exports.

The behavior of cross product is contained in the following table

$\mathrm{restart}$

$\mathrm{with}\left(\mathrm{VectorCalculus}\right)\:$

$⟨1\,0\,0⟩\&x⟨0\,-1\,0⟩$

$\mathrm{v1}≔\mathrm{Vector}\left(\left[1\,4\,0\right]\,\mathrm{coordinates}=\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{v1}≔\left[\begin{array}{c}1\\ 4\\ 0\end{array}\right]$

$\mathrm{v2}≔\mathrm{Vector}\left(\left[1\,1\,1\right]\,\mathrm{coordinates}=\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{v2}≔\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

$\mathrm{CrossProduct}\left(\mathrm{v1}\,\mathrm{v2}\right)$

$\mathrm{GetCoordinates}\left(\mathrm{CrossProduct}\left(\mathrm{v1}\,\mathrm{v2}\right)\right)$

${\mathrm{cartesian}}_{x,y,z}$

$\mathrm{vs}≔\mathrm{VectorSpace}\left(\left[1\,\frac{\mathrm{\pi }}{3}\,\frac{\mathrm{\pi }}{3}\right]\,\mathrm{spherical}\left[r,p,t\right]\right)$

$\mathrm{vs}≔\mathbf{module}\left(\right)\mathbf{local}\mathrm{\_origin}\,\mathrm{\_coords}\,\mathrm{\_coords\_dim}\;\mathbf{export}\mathrm{GetCoordinates}\,\mathrm{GetRootPoint}\,\mathrm{Vector}\,\mathrm{eval}\;\mathbf{end\; module}$

$\mathrm{v1}≔\mathrm{RootedVector}\left(\mathrm{root}=\mathrm{vs}\,\left[1\,1\,1\right]\right)$

$\mathrm{v1}≔\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

$\mathrm{v2}≔\mathrm{RootedVector}\left(\mathrm{root}=\mathrm{vs}\,\left[1\,0\,1\right]\right)$

$\mathrm{v2}≔\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]$

$\mathrm{v1}\&x\mathrm{v2}$

$\left[\begin{array}{c}1\\ 0\\ \mathrm{-1}\end{array}\right]$

$\mathrm{GetRootPoint}\left(\mathrm{v1}\&x\mathrm{v2}\right)$

$\mathrm{v1}≔\mathrm{RootedVector}\left(\mathrm{root}=\left[1\,\frac{\mathrm{\pi }}{4}\,1\right]\,\left[1\,1\,1\right]\,\mathrm{cylindrical}\left[r,p,h\right]\right)$

$\mathrm{v1}≔\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

$\mathrm{v2}≔\mathrm{Vector}\left(\left[0\,0\,1\right]\,\mathrm{coordinates}=\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{v2}≔\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$

$\mathrm{v2}\&x\mathrm{v1}$

$\left[\begin{array}{c}\mathrm{-1}\\ 1\\ 0\end{array}\right]$

$\mathrm{GetRootPoint}\left(\mathrm{v2}\&x\mathrm{v1}\right)$

$\mathrm{GetCoordinates}\left(\mathrm{v2}\&x\mathrm{v1}\right)$

${\mathrm{cylindrical}}_{r,p,h}$

$\mathrm{vf1}≔\mathrm{VectorField}\left(⟨r\,\mathrm{\phi }\,\mathrm{\theta }⟩\,\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)$

$\mathrm{vf2}≔\mathrm{VectorField}\left(⟨r^2\,\mathrm{\phi }\,\mathrm{\phi }⟩\,\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)$

$\mathrm{CrossProduct}\left(\mathrm{vf1}\,\mathrm{vf2}\right)$

$F≔\mathrm{VectorField}\left(⟨1\,-x\,y⟩\,\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{Del}\&xF$

$G≔\mathrm{VectorField}\left(⟨r\,t\,p⟩\,\mathrm{spherical}\left[r,p,t\right]\right)$

$\mathrm{Del}\&xG$

$\mathrm{pv1}≔\mathrm{PositionVector}\left(\left[1\,p\,q\right]\,\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)$

$\mathrm{pv1}≔\left[\begin{array}{c}\sin \left(p\right)\cos \left(q\right)\\ \sin \left(q\right)\sin \left(p\right)\\ \cos \left(p\right)\end{array}\right]$

$\mathrm{pv2}≔\mathrm{PositionVector}\left(\left[1\,p\,p\right]\,\mathrm{cylindrical}\left[r,\mathrm{\phi },h\right]\right)$

$\mathrm{pv2}≔\left[\begin{array}{c}\cos \left(p\right)\\ \sin \left(p\right)\\ p\end{array}\right]$

$\mathrm{CrossProduct}\left(\mathrm{pv1}\,\mathrm{pv2}\right)$

$\left[\begin{array}{c}\sin \left(q\right)\sin \left(p\right)p-\cos \left(p\right)\sin \left(p\right)\\ -\sin \left(p\right)\cos \left(q\right)p+{\cos \left(p\right)}^2\\ {\sin \left(p\right)}^2\cos \left(q\right)-\sin \left(q\right)\sin \left(p\right)\cos \left(p\right)\end{array}\right]$

$\mathrm{About}\left(\mathrm{CrossProduct}\left(\mathrm{pv1}\,\mathrm{pv2}\right)\right)$

$\left[\begin{array}{cc}\mathrm{Type:}& \mathrm{Position\; Vector}\\ \mathrm{Components:}& \left[\sin \left(q\right)\sin \left(p\right)p-\cos \left(p\right)\sin \left(p\right)\,-\sin \left(p\right)\cos \left(q\right)p+{\cos \left(p\right)}^2\,{\sin \left(p\right)}^2\cos \left(q\right)-\sin \left(q\right)\sin \left(p\right)\cos \left(p\right)\right]\\ \mathrm{Coordinates:}& \mathrm{cartesian}\\ \mathrm{Root\; Point:}& \left[0\,0\,0\right]\end{array}\right]$