What surface is represented by $\mathbf{r}\cdot\mathbf{a} = \mathrm{const.}$ that is described if $\mathbf{a}$ is a vector of constant magnitude and direction from the origin and $\mathbf{r}$ is the position vector to the point $P(x_1,x_2,x_3)$ on the surface?
My approach was to assume that $\mathbf{a}$ is a constant vector. Then I wrote $$ \mathbf{a} = \begin{pmatrix} \xi_1\\ \xi_2\\ \xi_3 \end{pmatrix}\quad,\quad \mathbf{r} = \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}\quad. $$ Then I proceed to take the dot product: $\mathbf{a}\cdot\mathbf{r} = \xi_1x_1+\xi_2 x_2+\xi_3x_3 = c$ for some constant $c$. This is the equation of a plane.
Is my solution correct? Is there any other way to do this using the alternative definition of the dot product, i.e, $\mathbf{a}\cdot\mathbf{r} = ar\cos\theta$?