ACJC Tutorial 15 Q9

The diagram shows a cuboid \(OABCA'B'C'D'\) in which the lengths of \(OA\),\(OC\) and \(OD'\) are \(3a\),\(2a\) and \(a\) respectively, where \(a\ne 0\). The point \(O\) is taken as origin, with unit vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) in the direction of \(\overrightarrow{OA}\), \(\overrightarrow{OC}\), \(\overrightarrow{OD'}\) respectively. The points \(P\) on \(C'B'\) and \(Q\) on \(OD'\) are such that \(C'P=\frac{1}{3}C'B'\) and \(OQ=\frac{1}{3}OD'\) respectively.

1. Find the position vectors of \(P\) and \(Q\).
2. Hence find the vector equation of line \(PQ\).
3. Determine whether the lines \(PQ\) and \(AC'\) are skew lines.
4. Find the shortest distance from \(M\), the midpoint of \(AB'\) to the line \(PQ\).
5. Use vector product to find the area of triangle \(PQM\).