2024 MI PRU1 P1 Q9

To construct a special tentage, \(3\) steel poles \(OR\), \(AQ\) and \(BP\) are secured to the horizontal ground such that each of the poles are perpendicular to the ground. The point \(O\) on the ground is taken as the origin and the unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are parallel to \(OA\), \(OB\) and \(OR\) respectively, with units in metres. It is given that \(OR=12\), \(AQ=9\), \(BP=6\), \(OA=20\) and \(OB=15\) (see diagram).

1. Find \(\vec{PR}\) and \(\vec{PQ}\).[2]

2. The canvas roof \(PQR\), which is of negligible thickness, is part of the plane \(p\). Using your answers in part (i), show that the equation of \(p\) is given by \({\mathbf{r} \cdot \begin{pmatrix} 3 \\ 8 \\ 20 \end{pmatrix} = 240.}\)

   [2]
3. Find the acute angle between the canvas roof and the plane \(OAB\).[2]
4. A cable is to be installed close to the tentage. This cable is part of the line \(L\) with vector equation \(\mathbf{r}=\left( \begin{matrix} \begin{matrix} -2 \\ 8 \\ 12 \\ \end{matrix} \\ \end{matrix} \right)+\lambda \left( \begin{matrix} \begin{matrix} 20 \\ 13 \\ -4 \\ \end{matrix} \\ \end{matrix} \right)\), where \(\lambda \in \mathbb{R}\). Show that this cable is not parallel to the edge \(PR\) of the roof and does not meet the edge \(PR\). [4]
5. A lamp is hung vertically downwards from the canvas roof using a cable such that it is at the point \(M\) with coordinates \(\left( \frac{20}{3},5,8 \right)\). To secure the lamp, a second cable is connected from the lamp to the point \(N\) on the roof. It is assumed that the cables are laid in straight lines and have negligible thickness. For the shortest length to be used for the second cable, show that \(\overrightarrow{ON}=\left( \begin{matrix} \begin{matrix} s \\ \frac{2525}{473} \\ t \\ \end{matrix} \\ \end{matrix} \right)\), where \(s\) and \(t\) are constants to be found.
   [3]