The diagram shows a three-dimensional shape OABCDEFG. The base OABC and the upper surface DEFG are identical horizontal rectangles. The parallelograms OAED and CBFG both lie in vertical planes. Points P and Q are the mid-points of OD and GF respectively. Unit vectors i and j are parallel to \(\overrightarrow{OA}\) and \(\overrightarrow{OC}\) respectively and the unit vector k is vertically upwards. The position vectors of A, C and D are given by \(\overrightarrow{OA} = 6\mathbf{i}\), \(\overrightarrow{OC} = 8\mathbf{j}\) and \(\overrightarrow{OD} = 2\mathbf{i} + 10\mathbf{k}\).

(i) Express each of the vectors \(\overrightarrow{PB}\) and \(\overrightarrow{PQ}\) in terms of i, j and k.

(ii) Determine whether P is nearer to Q or to B.

(iii) Use a scalar product to find angle BPQ.