Let $v$ be a vector of $\Bbb{R}^n$ and $c \in \Bbb{R}$. Let $A$ be a point of the affine space $\Bbb{R}^n$.
Show that $\mathcal{E} = \{ B \in \Bbb{R}^n | \langle \overrightarrow{AB},v \rangle = c \}$ is an affine subspace and give its direction and dimension.
When I write down the inner product $\langle \overrightarrow{AB},v \rangle = c$ it gives me : $$ v_1 (b_1 - a_1) + \dots + v_n (b_n - a_n ) = c $$ $$ v_1 b_1 + \dots + v_n b_n = c + v_1 a_1 + \dots v_n a_n $$ This instantaneously show that $\mathcal{E}$ is an affine subspace because of the form of the equation. So the director $\overrightarrow{\mathcal{E}}$ is defined by the homogeneous equation $$ v_1 w_1 + \dots + v_n w_n = 0, ~ w \in \Bbb{R}$$ which is a vector subspace of dimension $n-1$. So the dimension of $\mathcal{E}$ is $n-1$.
Does it seem correct ?
You are correct. To be more precise: the direction is
$$\{v\}^{\perp}.$$