The points A, B, C have position vectors

\(2\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}, \quad 2\mathbf{i} + 4\mathbf{j} - \mathbf{k}, \quad -3\mathbf{i} - 3\mathbf{j} + 4\mathbf{k},\)

respectively, relative to the origin O.

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

The point D has position vector \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).

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

(c) Find the shortest distance between the lines AB and CD.