The points A, B, C have position vectors \(\mathbf{i} + \mathbf{j}, \ -\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}, \ -2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\), respectively, relative to the origin O.

(a) Find the equation of the plane ABC, giving your answer in the form \(ax + by + cz = d\).

(b) Find the perpendicular distance from O to the plane ABC.

(c) Find a vector equation of the common perpendicular to the lines OC and AB.