2021 HCI Promo Q11

The points \(A\), \(B\) and \(C\) have coordinates \(\left( -1,-12,4 \right)\), \(\left( 5,0,7 \right)\) and \(\left( 6,1,4 \right)\) respectively.

The line \({{l}_{1}}\) has equations \(\frac{x-1}{2}=\frac{2-y}{3}\), \(z=4\) and the line \({{l}_{2}}\) passes through \(A\) and \(B\).

1. Find the coordinates of the foot of perpendicular from \(C\) to \({{l}_{1}}\).[4]
2. Find the acute angle between \({{l}_{1}}\) and \({{l}_{2}}\).[3]
3. The point \(D\) is on \({{l}_{2}}\) such that the distance from \(D\) to \(A\) is twice the distance from \(D\) to \(B\). Find the possible point(s) \(D\).[4]
4. The line \({{l}_{3}}\) passes through point \(A\) and is perpendicular to both \({{l}_{1}}\) and \({{l}_{2}}\). Find the equation of \({{l}_{3}}\).[2]