To construct a special tentage, $3$ steel poles $OR$, $AQ$ and $BP$ are secured to the horiz

Question

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). Find $\vec{PR}$ and $\vec{PQ}$.[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 $\mathbf{r}\cdot \left( \begin{matrix} \begin{matrix} 3 \ 8 \ 20 \ \end{matrix} \ \end{matrix} \right)=240$. [2] Find the acute angle between the canvas roof and the plane $OAB$.[2] 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] 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]

Accepted Answer

(i) $\overrightarrow{PR}=\left( \begin{matrix} \begin{matrix} 0 \ -15 \ 6 \ \end{matrix} \ \end{matrix} ight)$, $\overrightarrow{PQ}=\left( \begin{matrix} \begin{matrix} 20 \ -15 \ 3 \ \end{matrix} \ \end{matrix} ight)$ (iii) ${{23.1}^{\circ }}$ (v) $s=\frac{9640}{1419}$ and $t=\frac{4184}{473}$

2024 MI PRU1 P1 Q9

2024 MI PRU1 P1 Q9