Problem with dimensions in divergence of vector fields

Question

I'm looking at the following problem: Let $u: \mathbb{T}^3 \times [0,T] \rightarrow \mathbb{S}^1$ and the smooth vector field $\xi : \mathbb{T}^3 \times [0,T] \rightarrow \mathbb{R}^3$, then the equality $\xi \cdot \nabla u = \nabla \cdot (\xi u)-(\nabla \cdot \xi)u$ holds. Since $u \in \mathbb{S}^1$ we have that $\nabla u \in \mathbb{R}^{2 \times 3}$ but then the dimensions of the two components on the lefthand side $\xi \cdot \nabla u$ don't match to use the inner product. And on the righthand side I also don't understand what is meant by the product $\xi u$ because here the dimensions also don't match. I don't know any mathematical operations which combine a $\mathbb{R}^2$ and a $\mathbb{R}^3$ term the way it is stated here. Is there maybe a mistake in the source or how should I interpret these terms?