In the diagram, \(OABCD\) is a pyramid with vertex \(D\). The horizontal base \(OABC\) is a square of side 4 units. The edge \(OD\) is vertical and \(OD = 4\) units. The unit vectors \(\mathbf{i}, \mathbf{j}\) and \(\mathbf{k}\) are parallel to \(OA, OC\) and \(OD\) respectively.

The midpoint of \(AB\) is \(M\) and the point \(N\) on \(CD\) is such that \(DN = 3NC\).

(a) Find a vector equation for the line through \(M\) and \(N\).

(b) Show that the length of the perpendicular from \(O\) to \(MN\) is \(\frac{1}{3}\sqrt{82}\).