In the diagram, $O$ is the centre of the rectangular base $ABCD$ of a right pyramid with vertex

Question

In the diagram, $O$ is the centre of the rectangular base $ABCD$ of a right pyramid with vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $BC$ and $OV$ respectively. The length of $AB$, $BC$ and $OV$ are $12\text{ cm}$, $6\text{ cm}$ and $6\text{ cm}$ respectively. A line $l$ has Cartesian equation $\frac{-x-4}{10} = y+2 = \frac{t-z}{-2}.$ Find a vector equation of $AV$. If the line $l$ intersects $AV$ at $M$, find the position vector of $M$ and the value of $t$. Find the acute angle between $AV$ and the line $l$. Hence find the perpendicular distance from $A$ to the line $l$.

Accepted Answer

(i) $\mathbf r=\begin{pmatrix}0\0\6\end{pmatrix}+\lambda\begin{pmatrix}2\1\2\end{pmatrix}$ (ii) $\overrightarrow{OM}=\begin{pmatrix}-4\-2\2\end{pmatrix}$, $t=2$ (iii) $60.8^\circ$, $2.62$ cm

2006 HCI P1 Q14

2006 HCI P1 Q14

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