The line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } + \mathbf { j }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(- \mathbf { i } - \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + 2 \mathbf { k }\). The point \(P\) is on \(l _ { 1 }\) and the point \(Q\) is on \(l _ { 2 }\) and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length of \(P Q\).
2. Find the position vector of \(Q\).
3. Show that the perpendicular distance from \(Q\) to the plane containing \(A B\) and the line \(l _ { 1 }\) is \(\sqrt { } 3\).