2018 AJC P2 Q2

1. Given that \(\mathbf{a}\) and \(\mathbf{b}\) are non-zero constant vectors, show that the points with position vector \(\mathbf{r}\) and satisfying the equation \(\left( \mathbf{r}-\mathbf{a} \right)\times \mathbf{b}=\mathbf{0}\) lie on a line.[3]

The line \({{L}_{1}}\) has equation \(\left[ \mathbf{r}-\left( \begin{matrix}2\\1\\-1 \end{matrix}\right)\right]\times \left( \begin{matrix} 0 \\ 1 \\1 \end{matrix} \right)=\mathbf{0}\), and the line \({{L}_{2}}\) has equation \(\mathbf{r}=\left( \begin{matrix}-1 \\ 2 \\3 \end{matrix} \right)+\mu \left( \begin{matrix}m \\ 0 \\1 \end{matrix} \right)\), where \(m\) is a constant and \(\mu \) is a real parameter.

2. Find the vector equations of the parallel planes \({{p}_{1}}\) and \({{p}_{2}}\), in scalar product form, such that \({{p}_{1}}\) contains \({{L}_{1}}\) and \({{p}_{2}}\) contains \({{L}_{2}}\), leaving your answers in terms of \(m\).[3]
3. Find the acute angle between the plane \({{p}_{1}}\) and the line which contains both the points \(\left( 2,1,-1 \right)\) and \(\left( -1,2,3 \right)\), leaving your answer in terms of \(m\).[2]
4. It is now given that \({{L}_{1}}\) and \({{L}_{2}}\) is contained in a common plane \({{p}_{3}}\). Using your answer in part (iii), or otherwise, find the cartesian equation of the plane \({{p}_{3}}\).[1]