With respect to the origin O, the vertices of a triangle ABC have position vectors \(\overrightarrow{OA} = 2\mathbf{i} + 5\mathbf{k}\), \(\overrightarrow{OB} = 3\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}\) and \(\overrightarrow{OC} = \mathbf{i} + \mathbf{j} + \mathbf{k}\).

(a) Using a scalar product, show that angle ABC is a right angle. [3]

(b) Show that triangle ABC is isosceles. [2]

(c) Find the exact length of the perpendicular from O to the line through B and C. [4]