The quadrilateral ABCD is a trapezium in which AB and DC are parallel. With respect to the origin O, the position vectors of A, B, and C are given by \(\overrightarrow{OA} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\), \(\overrightarrow{OB} = \mathbf{i} + 3\mathbf{j} + \mathbf{k}\), and \(\overrightarrow{OC} = 2\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}\).

(a) Given that \(\overrightarrow{DC} = 3\overrightarrow{AB}\), find the position vector of D.

(b) State a vector equation for the line through A and B.

(c) Find the distance between the parallel sides and hence find the area of the trapezium.