Problem

The planes $\Pi_1$ and $\Pi_2$ have equations

$$\Pi_1 : x - 5y + 3z = 11,$$ $$\Pi_2 : 3x - 2y + 2z = 7.$$

The planes $\Pi_1$ and $\Pi_2$ intersect in the line $l$.

(a) Find a vector equation for $l$, giving your answer in the form $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$ where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors and $\lambda$ is a scalar parameter.

The point $P(2, 0, 3)$ lies on $\Pi_1$.

The line $m$, which passes through $P$, is parallel to $l$.

The point $Q(3, 2, 1)$ lies on $\Pi_2$.

The line $n$, which passes through $Q$, is also parallel to $l$.

(b) Find, in exact simplified form, the shortest distance between $m$ and $n$.