Vector calculus identities navigation

[Dafe]

Homework Statement

I'm reading in a fluid dynamics book and in it the author shortens an equation using identities my rusty vector calculus brain cannot reproduce.

Homework Equations

$$\vec{e} \cdot \frac{\partial}{\partial t}(\rho \vec{u}) = -\nabla\cdot (\rho\vec{u})\cdot\vec{e} - \rho(\vec{u}\cdot\nabla)\vec{u}\cdot\vec{e} - (\nabla p)\cdot\vec{e} + \rho\vec{b}\cdot\vec{e}$$

The author turns the left side of the equation into:

$$-\nabla\cdot(p\vec{e} + \rho\vec{u}(\vec{u}\cdot\vec{e})) + \rho\vec{b}\cdot\vec{e}$$

Just to be clear;
$$\vec{u}$$ is a vector valued function,
$$\vec{e}$$ is a fixed vector,
$$\rho, p$$ are scalar valued functions.

The Attempt at a Solution

The first part is fine:
$$\nabla\cdot (p\vec{e}) = \nabla(p)\cdot\vec{e} + p(\nabla\cdot\vec{e})$$
The divergence of a fixed vector is zero and so,
$$\nabla\cdot (p\vec{e}) = \nabla(p)\cdot\vec{e}$$

Next I need to find
$$\nabla\cdot (\rho\vec{u}(\vec{u}\cdot\vec{e}))$$

I am not sure what to do with is. $$\rho$$ is a scalar valued function, but so is I think $$\vec{u}\cdot\vec{e}$$. I know of the product rule between a scalar and a vector valued function, but what happens when there are two scalar valued functions?

Any suggestions are welcome, thanks.