2022 EJC J2 MCT Q8

Planes \(p\) and \(q\) have equations \(x+y+z=5\) and \(3x+y-z=9\) respectively, and meet in the line \(l\).

1. Find a vector equation for \(l\).[2]
2. Verify that the point \(A\) with position vector \(\mathbf{i}+\mathbf{j}+3\mathbf{k}\) lies on \(p\).[1]
3. Find the position vector of \(F\), the foot of perpendicular from \(A\) to \(q\). Hence find the coordinates of \(A'\), the reflection of the point \(A\) about \(q\).[5]
4. The point \(B\) lies on the line segment \(AA'\). The plane \(r\) contains the line \(l\) and the point \(B\). Find the range of values for \(\theta \), the acute angle between \(r\) and \(q\).[3]