2017 AJC P1 Q9

The position vectors of \(A\), \(B\) and \(C\) with respect to the origin \(O\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. It is given that \(\overrightarrow{AC}=4\,\overrightarrow{CB}\) and \({{\left| \mathbf{a}+\mathbf{b} \right|}^{2}}={{\left| \mathbf{a} \right|}^{2}}+{{\left| \mathbf{b} \right|}^{2}}\).

1. By considering \(\left( \mathbf{a}+\mathbf{b} \right)\cdot \left( \mathbf{a}+\mathbf{b} \right)\), show that \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular.[2]
2. Find the length of the projection of \(\mathbf{c}\) on \(\mathbf{a}\) in terms of \(\left| \mathbf{a} \right|\).[3]
3. Given that \(F\) is the foot of the perpendicular from \(C\) to \(OA\) and \(\mathbf{f}\) denotes the position vector \(\overrightarrow{OF}\), state the geometrical meaning of \(\left| \mathbf{c}\times \mathbf{f} \right|\).[1]
4. Two points \(X\) and \(Y\) move along line segments \(OA\) and \(AB\) respectively such that
   \(\overrightarrow{OX}=\left( \cos 3t \right)\,\mathbf{i}+\left( \sin 3t \right)\,\mathbf{j}+\frac{1}{2}\,\mathbf{k}\),
   \(\overrightarrow{OY}=\left( \sin t \right)\,\mathbf{i}+\left( \cos t \right)\,\mathbf{j}-2\mathbf{k}\),
   where \(t\) is a real parameter, \(0\le t\le 2\pi \). By expressing the scalar product of \(\overrightarrow{OX}\) and \(\overrightarrow{OY}\) in the form of \(p\sin \left( qt \right)+r\) where \(p\), \(q\) and \(r\) are real values to be determined, find the greatest value of the angle \(XOY\).[5]