2025 NJC P1 Q12

A model of a triangular canopy that provides shade outdoors is shown in Figure \(1\) below.

Figure 1

With point \(O\) taken as the origin, the canopy \(ABC\) is supported by three vertical columns given by \(OA\), \(DB\) and \(EC\). The unit vectors \(\mathbf{i}\) and \(\mathbf{k}\) are such that \(\mathbf{i}\) is along \(OD\), \(\mathbf{k}\) along \(OA\), and unit vector \(\mathbf{j}\) is perpendicular to both. The bases of the vertical columns are anchored to the horizontal ground \(ODE\), which is perpendicular to \(OA\).

1. State the cartesian equation of plane \(OAB\).[1]

Points \(A\) and \(B\) have position vectors given by \(\overrightarrow{OA}=3\mathbf{k}\) and \(\overrightarrow{OB}=10\mathbf{i}+4\mathbf{k}\). A marking, given by point \(M\), is to be placed on the line segment \(AB\).

2. Find \(\overrightarrow{OM}\) in terms of a parameter \(\lambda\), stating the range of \(\lambda\).[2]

Point \(C\) has position vector given by \(\overrightarrow{OC}=6\mathbf{i}+4\mathbf{j}+2\mathbf{k}\). The plan is to lay cables to connect \(M\) to \(C\) and then \(C\) to \(E\). All cables are laid in straight lines and have negligible thickness.

3. Explain why it is not possible for angle \(MCE\) to be \(90{}^\circ\).[1]

With reference to Figure \(2\) below, points \(F\) and \(G\) lie on lines \(AB\) and \(OD\) respectively. Party streamers, with negligible thickness, are laid in straight lines to connect \(E\), \(C\), \(F\) and \(G\). The quadrilateral formed lies on a plane with cartesian equation \(x-y=2\).

Figure 2

4. Show that quadrilateral \(ECFG\) is a trapezium.[3]
5. Find the shortest distance between \(F\) and the line \(CE\). Hence or otherwise, find the area enclosed by the streamers.[5]