2016 PJC J1 CT Q10

During a camp, a group of scouts builds a structure using wooden poles as shown in the diagram. The structure has a rectangular base \(ABCD\) on the horizontal ground, with \(AB=6\) metres and \(AD=4\) metres. The top of the structure, \(T\), is 2 metres vertically above \(O\), the centre of \(ABCD\). \(T\) is connected to \(P\), \(Q\), \(R\) and \(S\), the mid-points of \(AB\), \(BC\), \(CD\) and \(DA\) respectively.

1. If \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are unit vectors in the directions of \(OQ\), \(OR\) and \(OT\) respectively, write down the position vectors of \(Q\), \(R\) and \(T\). Hence find angle \(QTR\) to the nearest degree.
2. The scouts tie a rope from \(P\) to a point \(E\) on \(ST\) such that \(4\overrightarrow{SE}=\overrightarrow{ST}\). Find the length of the rope between \(E\) and \(P\), assuming that the rope is taut.
3. Another rope is tied from \(P\) to a point \(F\) on \(TQ\). Find the position vector of \(F\) such that the length of the rope between \(F\) and \(P\) is minimised.

4. A flag pole of \(5\) metres is erected vertically at the point \(G\left( \alpha ,\beta ,0 \right)\). Find the values of \(\alpha\) and \(\beta\) if the top of the flag pole is collinear with \(A\) and \(T\).