2022 HCI P2 Q5

The diagram below shows a \({3}\)-dimensional structure in which a pentahedron \(DEFGMN\) lies on top of a trapezoidal prism \(OABCDEFG\). Taking \(O\) as the origin, perpendicular vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OA\) and \(OC\) respectively. The base of the structure sits on the horizontal \(x\mathrm{-}y\) plane.

Planes \(OABC\) and \(DEFG\) are parallel to each other. It is given that \(OC\), \(AB\), \(DG\) and \(EF\) are parallel to one another where \(OC=AB=DG=EF=3\) units. It is also given that \(OA\), \(CB\), \(DE\), \(GF\) and \(MN\) are parallel to one another where \(OA=CB=MN=6\) units and \(DE=GF=10\) units. The pentahedron \(DEFGMN\) and the trapezoidal prism \(OABCDEFG\) each has a height of \(12\) units.

1. The point \(D\) has coordinates \(\left( -2,0,s \right)\). State the value of \(s\).[1]
2. The line \(DM\) is parallel to the vector \(\left( \begin{matrix} 1 \\ 1 \\ t \\ \end{matrix} \right)\) and the plane \(ABFE\) has equation\(6x=36+z\). It is given that the line \(DM\) does not intersect with the plane \(ABFE\). Find the value of \(t\).[2]
3. Show that the equation of the plane \(DGM\) is given by \(\mathbf{r}\cdot \left( \begin{matrix} \begin{matrix} 6 \\ 0 \\ -1 \\ \end{matrix} \\ \end{matrix} \right)=k\), where \(k\) is a constant to be determined.[2]
4. Find the acute angle between the planes \(DGM\) and \(DEFG\).[3]
5. Find the coordinates of \(M\) and the exact shortest distance from \(M\) to the plane \(ABFE\).[4]
6. Another plane \(\Pi \) has cartesian equation \(x=c\), where \(c\) is a constant. If the three planes \(OAED\), \(EFN\) and \(\Pi \) all intersect at the point \(E\), find the value of \(c\), showing your working clearly.[2]