2013 DHS P1 Q10

The lines \({{l}_{1}}\) and \({{l}_{2}}\) have equations \(r=\left( \begin{matrix} 3 \\ 1 \\ 2 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} b \\ 1 \\ -1 \\ \end{matrix} \right)\), where \(b>1\), and \(r=\left( \begin{matrix} 4 \\ 0 \\ 1 \\ \end{matrix} \right)+\mu \left( \begin{matrix} -1 \\ -1 \\ 1 \\ \end{matrix} \right)\) respectively.

1. Given that the acute angle between \({{l}_{1}}\) and \({{l}_{2}}\) is \(30{}^\circ \), find the value of \(b\), giving your answer correct to \(2\) decimal places.[3]

For the rest of the question, use \(b=3\).

2. Find the coordinates of the points \(A\) and \(B\) where \({{l}_{1}}\) and \({{l}_{2}}\) meet the \(xy\)-plane respectively.[3]
3. The point \(C\) has position vector \(2\mathbf{i}+7\mathbf{j}+3\mathbf{k}\). Show that the equation of plane \({{p}_{1}}\) which passes through the points \(A\), \(B\), and \(C\) is given by \(x-2y+5z=3\).[3]
4. Another plane \({{p}_{2}}\) with equation \(x+2y+3z=2\) meets \({{p}_{1}}\) in the line \({{l}_{3}}\). Find the vector equation of \({{l}_{3}}\).[2]
5. Explain whether the lines \({{l}_{2}}\) and \({{l}_{3}}\) are parallel, skew, coincident or intersecting, justifying your answer clearly.[2]