During a camp, a group of scouts builds a structure using wooden poles as shown in the diagram. The

Question

During a camp, a group of scouts builds a structure using wooden poles as shown in the diagram. The structure has a rectangular base $ABCD$ on the horizontal ground, with $AB=6$ metres and $AD=4$ metres. The top of the structure, $T$, is 2 metres vertically above $O$, the centre of $ABCD$. $T$ is connected to $P$, $Q$, $R$ and $S$, the mid-points of $AB$, $BC$, $CD$ and $DA$ respectively. If $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are unit vectors in the directions of $OQ$, $OR$ and $OT$ respectively, write down the position vectors of $Q$, $R$ and $T$. Hence find angle $QTR$ to the nearest degree. The scouts tie a rope from $P$ to a point $E$ on $ST$ such that $4\overrightarrow{SE}=\overrightarrow{ST}$. Find the length of the rope between $E$ and $P$, assuming that the rope is taut. Another rope is tied from $P$ to a point $F$ on $TQ$. Find the position vector of $F$ such that the length of the rope between $F$ and $P$ is minimised. A flag pole of 5 metres is erected vertically at the point $G\left( \alpha ,\beta ,0 \right)$. Find the values of $\alpha$ and $\beta$ if the top of the flag pole is collinear with $A$ and $T$.

Accepted Answer

(i) $66.9°$ (ii) $\frac{\sqrt{149}}{4}$ units (iii) $\overrightarrow{OF} = \frac{1}{13}\begin{pmatrix} 12 \\ 0 \\ 18 \end{pmatrix}$ (iv) $\alpha = \frac{9}{2}$, $\beta = 3$

2016 PJC J1 CT Q10

2016 PJC J1 CT Q10

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