The figure shows a right pyramid $VABCD$ with a square base $ABCD$, standing horizontally on a c

Question

The figure shows a right pyramid $VABCD$ with a square base $ABCD$, standing horizontally on a cuboid $ABCDEFGH$. It is given that $VA=VB=VC=VD=5$ cm, $EF=FG=4$ cm and $AE=2$ cm, as shown in the diagram. $O$ is the centre of the square base $EFGH$. Perpendicular unit vectors $\mathbf{i},\mathbf{j},\mathbf{k}$ are parallel to $EF, FG, EA$ respectively.$C$ lies on $OA$ produced with $OC:AC=3:1$ and $D$ divides $OB$ in a ratio of $2:3$. $X$ is the point of intersection of $AB$ and $CD$. Show that the height of the figure, $OV$ is $2+\sqrt{17}$.[2] State the geometrical meaning of $\frac{\left| \overrightarrow{VA}\times \overrightarrow{OV} \right|}{2+\sqrt{17}}$.[1] Find the equation of the line passing through $B$ and $V$.[2]

Accepted Answer

(i) $OV = 2 + \sqrt{17}$; (iii) $\mathbf{r} = \begin{pmatrix} 0 \ 0 \ 2+\sqrt{17} \end{pmatrix} + \lambda \begin{pmatrix} -2 \ 2 \ \sqrt{17} \end{pmatrix}$

2013 SAJC P1 Q5

2013 SAJC P1 Q5

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