**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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