(i) Describe geometrically the following surfaces in three-dimensional space:

(a) $\mathbf{r} \cdot \mathbf{u}=\alpha|\mathbf{r}|$, where $0<|\alpha|<1$

(b) $|\mathbf{r}-(\mathbf{r} \cdot \mathbf{u}) \mathbf{u}|=\beta$, where $\beta>0$.

Here $\alpha$ and $\beta$ are fixed scalars and $\mathbf{u}$ is a fixed unit vector. You should identify the meaning of $\alpha, \beta$ and $\mathbf{u}$ for these surfaces.

(ii) The plane $\mathbf{n} \cdot \mathbf{r}=p$, where $\mathbf{n}$ is a fixed unit vector, and the sphere with centre $\mathbf{c}$ and radius $a$ intersect in a circle with centre $\mathbf{b}$ and radius $\rho$.

(a) Show that $\mathbf{b}-\mathbf{c}=\lambda \mathbf{n}$, where you should give $\lambda$ in terms of $a$ and $\rho$.

(b) Find $\rho$ in terms of $\mathbf{c}, \mathbf{n}, a$ and $p$.