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.

1. 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 \(O\).

2. 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.

3. 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.

4. 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]