The plane \(\Pi_1\) has equation \(\mathbf{r} = \mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda (\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}) + \mu (3\mathbf{i} - \mathbf{k})\).

(a) Find an equation for \(\Pi_1\) in the form \(ax + by + cz = d\).

The line \(l\), which does not lie in \(\Pi_1\), has equation \(\mathbf{r} = -3\mathbf{i} + \mathbf{k} + t(\mathbf{i} + \mathbf{j} + \mathbf{k})\).

(b) Show that \(l\) is parallel to \(\Pi_1\).

(c) Find the distance between \(l\) and \(\Pi_1\).

(d) The plane \(\Pi_2\) has equation \(3x + 3y + 2z = 1\).

Find a vector equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\).