With reference to the origin $O$, the coordinates of $A$, $B$ and $C$ are $(2,8,-a)$, \((1

Question

With reference to the origin $O$, the coordinates of $A$, $B$ and $C$ are $(2,8,-a)$, $(1,-10,3)$ and $(5,6,a)$ respectively. The plane $\pi$ contains points $O$, $A$, $B$ and $C$. Show that $a=1$.[4] Find the acute angle between $\pi$ and the $yz$-plane.[2] $\pi$ and the $yz$-plane meet in the line $l$. By evaluating an appropriate cross product, find a vector equation for $l$.[2] Write down the distance from $A$ to the $yz$-plane. Hence or otherwise, find the distance from $A$ to $l$.[3]

Accepted Answer

(a) $a=1$ (b) $64.1^\circ$ (c) $\mathbf r=\lambda\begin{pmatrix}0\-4\1\end{pmatrix},\ \lambda\in\mathbb R$ (d) $2$; $2\sqrt{\frac{21}{17}}\approx2.22$

2026 RI P1 Q11

2026 RI P1 Q11

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