2022 NYJC J1 CT Q8
A group of campers built a tent in the shape of tetrahedron $OABC$ where points $\left( x,y,z \right)$ are defined relative to the origin ,$O$, where units are metres. The three vertices $A$, $B$ and $C$ of the tetrahedron are securely fastened to three rigid poles represented by the $x-$, $y-$ and $z-$axes respectively with the vertices $O$,$A$ and $B$ resting on horizontal ground as shown in the diagram below.
Measurements taken by the campers yield the following information:
• $OA=3$ metres,
• $AB=5$ metres,
• $\angle OAC=45$ degrees.
(i)
Write down the exact coordinates of $A$ and $C$ and show that the coordinates of $B$ are $\left( 0,4,0 \right)$.
[3]
The camp leader wishes to fasten the side $AB$ of the tent more securely to the ground by hitting a nail along $AB$ which is closest to $O$.
(ii)
Determine the coordinates of the nail.
[4]
One end of a thin straight rod is placed at the point $\left( 1,1,0 \right)$ with the other end just touching the triangular canvas $ABC$ of the tent.
(iii)
Find the exact length of the shortest straight rod required.
[4]
The camp leader intends to construct a horizontal triangular ceiling at a height of $2$ metres above the horizontal base of the tent.
(iv)
Write down the vector equation of the edge $AC$ of the tent and hence find the coordinates of the point where the triangular ceiling meets the edge $AC$ of the tent.
[3]
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