8. The plane \(\Pi _ { 1 }\) has equation
$$x - 5 y + 3 z = 11$$
The plane \(\Pi _ { 2 }\) has equation
$$3 x - 2 y + 2 z = 7$$
The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).
- 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 \(l\). - Find, in exact simplified form, the shortest distance between \(m\) and \(n\).
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