The diagram shows a cuboid with rectangular base $OABC$ and top $EFGH$, where $OA=4$ unit

Question

The diagram shows a cuboid with rectangular base $OABC$ and top $EFGH$, where $OA=4$ units, $OC=3$ units and $OE=2$ units. The point $O$ is taken as the origin and unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$, are taken along $OA$, $OC$ and $OE$ respectively.

Find the cartesian equation of the plane $p$ which contains the points $A$, $C$ and $E$.[3]
Find the acute angle between $p$ and the base $OABC$.[2]

The line $l$, passing through $O$, is perpendicular to $p$ and intersects the plane containing $B$, $C$, $G$ and $H$ at the point $T$.

Find the position vector of the point $T$ and deduce the perpendicular distance from $T$ to $p$.[5]
A point $Q$ lies on the line passing through $C$ and $T$ such that its distance from $p$ is twice that of the distance from $T$ to $p$. Find the possible position vectors of the point $Q$.[3]

Accepted Answer

(i) $3x+4y+6z=12$ (ii) $39.8^\circ$ (iii) $\overrightarrow{OT}=\begin{pmatrix}\frac94\3\\frac92\end{pmatrix}$, distance $=\frac{135\sqrt{61}}{244}$ units (iv) $\overrightarrow{OQ}=\begin{pmatrix}\frac92\3\9\end{pmatrix}$ or $\begin{pmatrix}-\frac92\3\-9\end{pmatrix}$

2013 DHS P1 Q12

2013 DHS P1 Q12

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