2018 RVHS P2 Q3

The diagram below shows the structure of an indoor farm to be built. Edges \(OA\), \(DC\), \(IH\) and \(EF\) are vertical, and edges \(OA\), \(AB\), \(BC\) and \(CD\) are equal to \(EF\), \(FG\), \(GH\) and \(HI\) respectively. Diagram is not drawn to scale.

Taking \(O\) as the origin and the vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) as unit vectors along the \(OD\), \(OE\) and \(OA\) respectively, the coordinates of \(A\) and \(C\) are \(\left( 0,0,20 \right)\) and \(\left( 50,0,25 \right)\) respectively. All lengths are measured in metres.

1. Given that \(-\mathbf{i}+4\mathbf{k}\) is a vector perpendicular to the flat roof \(ABGF\), and \(\mathbf{i}+2\mathbf{k}\) is perpendicular to the flat roof \(BCHG\), find the obtuse angle between the two roofs.[2]
2. A humidity sensor is to be installed along the line \(BG\), \(50\) metres away from \(B\). Find the coordinates of the location of the sensor.[4]
3. A metal cable is anchored on the ground outside the farm in front of \(OABCD\), at a point \(25\) metres away from \(O\) along \(OD\) and \(10\) metres in front of \(OD\). The cable is attached to a point on \(AB\) to secure the roof \(ABGF\). Find the length of the shortest cable needed.[3]
4. State an assumption for the above calculations to be valid.[1]