2026 RI Promo Q9

The lines \(l_1\) and \(l_2\) have equations

\(\mathbf r=\begin{pmatrix}1\\2\\4\end{pmatrix}+\lambda\begin{pmatrix}-2\\1\\a\end{pmatrix}\quad\text{and}\quad x-2=y=\dfrac{3-z}{2}\)

respectively, where \(\lambda\) is a real parameter and \(a\) is a real constant.

1. Given that \(l_1\) and \(l_2\) are skew lines, find the range of values of \(a\).[3]

In the rest of this question, \(a=1\).

The plane \(p\) contains the line \(l_1\) and the plane \(q\) contains the line \(l_2\). Planes \(p\) and \(q\) are parallel and \(\mathbf n\) is a vector perpendicular to both \(p\) and \(q\).

2. Find \(\mathbf n\).[2]

The points \(A\) and \(B\) have coordinates \((1,2,4)\) and \((2,0,3)\) respectively and point \(C\) is the foot of perpendicular from \(A\) to \(q\).

3. By considering the length of projection of a suitable vector onto \(\mathbf n\), find the exact perpendicular distance between \(p\) and \(q\) and hence find \(\overrightarrow{AC}\).[3]

The plane \(m\) contains the point \(A\). It is given that the point of reflection of \(B\) in \(m\) lies on \(q\).

4. State the relationship between \(m\) and \(q\).[1]
5. The line \(l_3\) contains the points \(A\) and \(B\). Given that \(C\) is also the foot of perpendicular from \(B\) to \(m\), find a vector equation of the line of reflection of \(l_3\) in \(m\).[3]