2010 PJC P2 Q3

The diagram above shows a partial design of the roof of a gallery of an amphitheatre. \(ABCD\) is an inclined rectangular roof, where \(AB=10\)m and \(BC=5\)m. \(EFGH\) is a rectangle on a horizontal ground, where \(EF=10\)m and \(EH=4\)m.

The points \(A\) and \(B\) are \(3\)m directly above \(E\) and \(F\) respectively. The points \(C\) and \(D\) are \(6\)m directly above \(G\) and \(H\) respectively.

Point \(O\) is the centre of \(EFGH\) and is taken as the origin. Perpendicular unit vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are in the direction of \(\overrightarrow{EF}\), \(\overrightarrow{EH}\) and \(\overrightarrow{EA}\) respectively.

1. Find a vector equation of the line passing through \(A\) and \(C\).[2]
2. A spot light is fixed at the top of a vertical pillar erected at a point \(Q\) with coordinates \((15,\,\,6,\,\,0)\). Find the height of the pillar if the top of the pillar is collinear with \(A\) and \(C\).[3]
3. The main electrical supply to the gallery runs along the diagonal \(AC\). Find the position vector of point \(X\) on \(AC\) such that the length of the electric cable that is required to provide electricity from the main electrical supply to an amplifier mounted at \(D\), will be the least.[3]