The subspace in question: $V=\{ \vec{u} \in \Bbb{R}^n : \vec{n}^T\vec{u}=\vec{0} \}$
I am assuming that $\vec{u} = \begin{bmatrix}x_0 \\ x_1 \\ \vdots \\ x_2\end{bmatrix}$.
The dimension of a vector space/subspace is equal to the number of linearly independent vectors in its basis. So it's either 1 or n. Does $\vec{u}$ count as 1 or as n?