Derivation of the Riemannian metric tensor

Firstly, I would like to say that I didn't do a course in Riemannian geometry, so my doubt is probably something that is not clear for me from the basis of the Riemannian geometry, which I'm self-studying.

I'm reading Interior estimates for hypersurfaces moving by mean curvature and I'm trying understand the statement:

"Now note that $$\nabla u = \omega - \langle \nu , \omega \rangle$$ and $$\nabla (|\textbf{x}|^2 - u^2) = 2(\textbf{x} - \langle \textbf{x} , \nu \rangle \nu - u \nabla u)$$",

where $$M$$ is an hypersurface of dimension $$n$$ which evolves under Mean Curvature Flow by the immersion $$F: M \times [0,T) \longrightarrow \mathbb{R}^{n+1}$$, $$\textbf{x} = F(p,t)$$, $$u := \langle \textbf{x} , \omega \rangle$$, $$\nu$$ is the normal unit vector and $$\omega$$ is a fixed vector on $$\mathbb{R}^{n+1}$$ such that $$\langle \omega, \nu \rangle > 0$$.

First, I think the authors wanted to write $$\nabla u = \omega - \langle \nu, \omega \rangle \nu$$, i.e., $$\nabla u$$ it's the projection of $$\omega$$ on tangent space of $$M$$.

I tried compute $$\nabla u$$ and I found $$\nabla u = \langle \nabla \textbf{x}, \omega \rangle$$, which is the length of the projection of $$\omega$$ on tangent space of $$M$$. I did this computation by the fact that $$u$$ is a function defined on $$M$$, by the compatibility of the tensor metric, but I found some problems because my computation of $$\nabla u$$ doesn't match with the computation of $$\nabla u$$ did by the authors and the other problem is that the compatibility of the tensor metric it's relationed with the derivation of a function along to a vector field by relation

$$\nabla_X \langle Y,Z \rangle = X \langle Y,Z \rangle = \langle \nabla_X Y,Z \rangle + \langle Y, \nabla_X Z \rangle,$$

but I don't have explicitly along what vector field $$u$$ is being differentiated. I think this vector field defined on $$M$$ is arbitrary and this explain why the direction was omitted in the differentiation of $$u$$. I had these same problems when I tried compute $$\nabla |\textbf{x}|^2$$ by an analagous argument.

I would like to know how compute $$\nabla u$$ and $$\nabla |\textbf{x}|^2$$. Thanks in advance!