Is this vectorial identity between operators true?

Question

$$\nabla^2(\mathbf{t} u(\mathbf{r}) ) =\mathbf{t} div(\nabla u(\mathbf{r}))$$ Where $\mathbf{t}$ is constant vector and $\nabla^2$ is the vector laplacian (defined here: https://en.wikipedia.org/wiki/Vector_Laplacian). Thanks for your help.

Answer 1

Yes: $\nabla^2 (tu) = \partial_i^2 (t_i u_i) = t_i (\partial_i^2 u_i) = t . \operatorname{div} (\nabla u)$.