dot product of surface and gradient

Question

I've been asked to answer if the following dot product make sense: $F \cdot \nabla f$, when $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is $C^1(\mathbb{R}^3)$ and $F: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ is $C^1(\mathbb{R}^3, \mathbb{R}^3)$. Following the solutions the answer is yes. It has been a while since my linear algebra class. What are the requirements for dot products? Only the length of the vectors? So we ignore further information like if it is a row or column vector? Furthermore, can we treat a gradient just as a normal row vector?