When is this equality true?

Question

Given two vectors $\vec{u}$ and $\vec{v}$, I noticed that $$\vec{u}\cdot\nabla\vec{v}+\vec{v}\cdot\nabla\vec{u}=\vec v(\nabla\cdot \vec u)+\vec u(\nabla\cdot \vec v)$$ when $\vec u$ and $\vec v$ only have at most one non-zero component along the same standard basis. Is this true any other time?