2012 ACJC P1 Q11

\(OABC\) is a rhombus. The points \(A\) and \(C\) are such that \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OC}=\mathbf{c}\). The point \(M\) is the centre of the rhombus and angle \(AOC\) is \({{60}^{\circ }}\).

1. By using scalar products, show that the length of projection of \(\overrightarrow{OM}\) on \(\overrightarrow{OC}\) is \(\frac{3}{4}\left| \mathbf{c} \right|\).[4]
2. By using a vector product, show that the area of triangle \(OMC\) can be written as \(k{{\left| c \right|}^{2}}\) where \(k\) is a constant to be found.[3]
3. The point \(D\) lies on \(OC\) produced such that \(OC:CD=2:3\). Show that the shortest distance from \(D\) to the line \(OA\) can be written as \(t\left| \mathbf{c} \right|\) where \(t\) is a constant to be found.[3]