The "Laplacian", \nabla, is the "vector operator", in Cartesian coordinates,
\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z}\vec{j}
which, applied to a numerical function, f, gives the "gradient", \nabla f.
We can think of that as the "scalar product" of f with \nabla.
The most common products of vector with vector are the dot product and the cross product which, with \nabla give the "divergence", \nabla\cdot \vec{f}, and the "curl", \nabla\times \vec{f}, respectively.
A less used product of two vectors is the "outer product" or "tensor product" which gives a tensor or matrix:
<a_1, a_2, a_3> tensor <b_1, b_2, b_3>= \begin{bmatrix}a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & a_3b_3\end{bmatrix}.
The "outer product" of \nabla with \vec{F}= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k} would be
\begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} & \frac{\partial g}{\partial z} \\ \frac{\partial h}{\partial x} & \frac{\partial h}{\partial y} & \frac{\partial h}{\partial z}\end{bmatrix}
The "dot product" of that with <x, y, z> is the matrix product
\begin{bmatrix}x & y & z\end{bmatrix}\begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} & \frac{\partial g}{\partial z} \\ \frac{\partial h}{\partial x} & \frac{\partial h}{\partial y} & \frac{\partial h}{\partial z}\end{bmatrix}= \begin{bmatrix}x\frac{\partial f}{\partial x}+ y\frac{\partial g}{\partial x}+ z\frac{\partial h}{\partial x} \\ x\frac{\partial g}{\partial y}+ y\frac{\partial g}{\partial y}+ z\frac{\partial h}{\partial y} \\ x\frac{\partial f}{\partial z}+ y\frac{\partial g}{\partial z}+ z\frac{\partial y}{\partial z}\end{bmatrix}
