N2009 P1 Q10

The planes \({{\mathrm{p}}_{1}}\) and \({{\mathrm{p}}_{2}}\) have equations \(\mathbf{r}\cdot \left( \begin{matrix}2 \\1 \\3 \end{matrix} \right)=1\) and \(\mathbf{r}\cdot \left( \begin{matrix}-1 \\2 \\1 \end{matrix} \right)=2\) respectively, and meet in a line \(l\).

1. Find the acute angle between \({{\mathrm{p}}_{1}}\) and \({{\mathrm{p}}_{2}}\).[3]
2. Find a vector equation of \(l\).[4]
3. The plane \({{\mathrm{p}}_{3}}\) has equation \(2x+y+3z-1+k\left( -x+2y+z-2 \right)=0\). Explain why \(l\) lies in \({{\mathrm{p}}_{3}}\) for any constant \(k\). Hence, or otherwise, find a cartesian equation of the plane in which both \(l\) and the point \(\left( 2,3,4 \right)\) lie.[5]