N2011 P1 Q7

Referred to the origin \(O\), the points \(A\) and \(B\) are such that \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\). The point \(P\) on \(OA\) is such that \(OP:PA=1:2\), and the point \(Q\) on \(OB\) is such that \(OQ:QB=3:2\). The mid-point of \(PQ\) is \(M\) (see diagram).

1. Find \(\overrightarrow{OM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) and show that the area of triangle \(OPM\) can be written as \(k\left| \mathbf{a}\times \mathbf{b} \right|\), where \(k\) is a constant to be found.[6]
2. The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are now given by
   \(\mathbf{a}=2p\mathbf{i}-6p\mathbf{j}+3p\mathbf{k}\) and \(\mathbf{b}=\mathbf{i}+\mathbf{j}-2\mathbf{k}\),
   where \(p\) is a positive constant. Given that \(\mathbf{a}\) is a unit vector,
   1. find the exact value of \(p\),[2]
   2. give a geometrical interpretation of \(\left| \mathbf{a}\cdot \mathbf{b} \right|\),[1]
   3. evaluate \(\mathbf{a}\times \mathbf{b}\).[2]