With \(O\) as the origin, the points \(A, B, C\) have position vectors
\(\mathbf{i}-\mathbf{j}, \quad 2 \mathbf{i}+\mathbf{j}+7 \mathbf{k}, \quad \mathbf{i}-\mathbf{j}+\mathbf{k}\)
respectively.
(i) Find the shortest distance between the lines \(O C\) and \(A B\).

(ii) Find the cartesian equation of the plane containing the line \(O C\) and the common perpendicular of the lines \(O C\) and \(A B\).