2022 ACJC J2 MYE Q5

The diagram shows a prism with triangular ends \(ABF\) and \(DCE\). \(AD\), \(BC\) and \(FE\) are parallel edges of the prism. The rectangular plane \(ADEF\) has equation \(x-2y=0\) and the rectangular base \(ABCD\) has equation \(3x-y+z=4\). The coordinates of \(C\) and \(F\) are \(\left( 1,0,1 \right)\) and \(\left( 4,2,0 \right)\) respectively.

1. Find the acute angle between the plane \(ADEF\) and the line \(CF\).[2]
2. Find the length of projection of vector \(\overrightarrow{CF}\) on the base \(ABCD\).[2]
3. Find the position vector of the point \(P\) on the line passing through \(A\) and \(D\), which is closest to the mid-point of line segment \(CF\).[4]
4. Find the Cartesian equation of the third rectangular plane \(BCEF\).[2]
5. Find the distance between the two triangular ends of the prism.[2]