2025 JPJC P1 Q12

A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horizontal flat ground, as shown in the diagram below. Assume that the roof and wall are of negligible thickness.

Points \((x, y, z)\) are defined relative to a corner of the wall at \(O\,\mathrm (0, 0, 0)\), where units are metres. Unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are in the directions shown in the diagram. The other corners of the wall are \(A\), \(B\) and \(C\), such that \(AB\) is \(5\) m and \(A\) and \(B\) are \(3\) m vertically above \(O\) and \(C\) respectively. The roof is attached to the wall along \(AB\) and is modelled by a plane parallel to \(4\mathbf{i} + 5\mathbf{j} + \mathbf{k}\).

1. Show that the roof is defined by the plane \(x - 4z = -12\).[3]
2. Find the obtuse angle between the wall and the roof, correct to the nearest degree.[3]

To support the roof, a wooden strut, of negligible thickness, is constructed. It has one end at point \(P\) with coordinates \((4, 3, 0)\) on the horizontal ground and the other end at the roof, such that the length of the wooden strut is a minimum.

3. Find the exact length of the wooden strut.[2]
4. A ray of light is emitted in the direction \(7\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}\) from a light source at \(C\). Show that this light ray meets the wooden strut. Hence, find the coordinates of the point where the light ray meets the wooden strut.[4]