The diagram shows a tetrahedron with vertices $O$, $A$, $B$ and $C$. Points $A$, $B$ and

Question

The diagram shows a tetrahedron with vertices $O$, $A$, $B$ and $C$. Points $A$, $B$ and $C$ have position vectors with respect to the origin $O$ given by $\overrightarrow{OA}=\begin{pmatrix}2\-1\2\end{pmatrix}$, $\overrightarrow{OB}=\begin{pmatrix}0\3\1\end{pmatrix}$ and $\overrightarrow{OC}=\begin{pmatrix}3\0\4\end{pmatrix}$. The point $D$ lies on $BC$, between $B$ and $C$, such that $BD:DC=1:2$. Find the position vector of $D$[1] Find a cartesian equation of face $ABC$ and state its exact distance from $O$.[4] Hence find the acute angle between $OD$ and face $ABC$.[2] Find the coordinates of the point on $OD$ which is closest to $A$.[3]

Accepted Answer

(a) $\overrightarrow{OD}=(1,2,2)$ (b) $3x+y-2z=1$, distance $\dfrac1{\sqrt{14}}$ (c) $5.1^\circ$ (d) $(\dfrac49,\dfrac89,\dfrac89)$

2025 NYJC Promo Q9

2025 NYJC Promo Q9

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