The DotProduct(v1, v2) command (scalar product) computes the dot product of v1 and v2, where v1 and v2 can be free Vectors, rooted Vectors, position Vectors, vector fields, Del or Nabla.

The function can be accessed through  . or DotProduct exports.

If v2 is a VectorField, the divergence of v2 can be computed as DotProduct(Del, v2), or $\nabla .\mathrm{v2}$.

If v1 is a VectorField, an operator representing the directional derivative in the direction of v1 is obtained as DotProduct(v1, Del), or $\mathrm{v1}.\nabla$.

The behavior of the dot product of two Vectors is described by the following table:

$\mathrm{restart}$

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

$⟨1\,1\,1⟩·⟨-1\,-1\,1⟩$

$\mathrm{-1}$

$\mathrm{v1}≔\mathrm{Vector}\left(\left[1\,-1\,2\right]\,\mathrm{coordinates}=\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{v1}≔\left[\begin{array}{c}1\\ \mathrm{-1}\\ 2\end{array}\right]$

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

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

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

$1$

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

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

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

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

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

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

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

$0$

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

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

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

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

$\mathrm{v2}·\mathrm{v1}$

$1$

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

$\mathrm{vf2}≔\mathrm{VectorField}\left(⟨r^2\,\mathrm{\phi }+\mathrm{\theta }\,0⟩\,\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)$

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

$r^3+\mathrm{\phi }\left(\mathrm{\phi }+\mathrm{\theta }\right)$

$\mathrm{vf1}≔\mathrm{VectorField}\left(⟨x\,-yz\,z⟩\,\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{Del}·\mathrm{vf1}$

$2-z$

$\mathrm{vf2}≔\mathrm{VectorField}\left(⟨rt\,\mathrm{\phi }\,t⟩\,\mathrm{cylindrical}\left[r,\mathrm{\phi },t\right]\right)$

$\mathrm{Del}·\mathrm{vf2}$

$\frac{2rt+r+1}{r}$

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

$W≔\mathrm{VectorField}\left(⟨yz\,xz\,xy⟩\,\mathrm{cartesian}\left[x,y,z\right]\right)$

$\left(V·\mathrm{Del}\right)\left(W\right)$

$\mathrm{pv1}≔\mathrm{PositionVector}\left(\left[p\,p\right]\,\mathrm{polar}\left[r,t\right]\right)$

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

$\mathrm{pv2}≔\mathrm{PositionVector}\left(\left[1\,p\right]\,\mathrm{parabolic}\left[u,v\right]\right)$

$\mathrm{pv2}≔\left[\begin{array}{c}\frac{1}{2}-\frac{p^2}{2}\\ p\end{array}\right]$

$\mathrm{pv1}·\mathrm{pv2}$

$p\cos \left(p\right)\left(\frac{1}{2}-\frac{p^2}{2}\right)+p^2\sin \left(p\right)$

$\mathrm{vf3}≔\mathrm{VectorField}\left(⟨yz\,xz\,xy⟩\,\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{v3}≔\mathrm{Vector}\left(\left[1\,2\,1\right]\,\mathrm{coordinates}=\mathrm{cartesian}\left[x,y,z\right]\right)$

$\mathrm{v3}≔\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right]$

$\mathrm{vf3}·\mathrm{v3}$

$xy+2xz+yz$