2010 DHS P2 Q4

The planes \({{\prod }_{1}}\) and \({{\prod }_{2}}\) are defined by

\({{\prod }_{1}}:\mathbf{r}\cdot \left( \begin{matrix} 2 \\ 4 \\ 1 \\ \end{matrix} \right)=10\), \({{\prod }_{2}}:\mathbf{r}\cdot \left( \begin{matrix} 1 \\ 3 \\ 1 \\ \end{matrix} \right)=8\).

1. Find the acute angle between the two planes.[3]
2. Obtain a vector equation of \({{l}_{1}}\), the line of intersection of the two planes.[4]

The Cartesian equation of another line, \({{l}_{2}}\), is given by \(\frac{x-2}{6}=\frac{7-z}{3},y=m\), where \(m\) is a real constant.

3. If the plane \({{\prod }_{1}}\)and line \({{l}_{2}}\)intersect at the point \(\left( 6,m,5 \right)\), find the value of \(m\).[2]
4. Show that the lines \({{l}_{1}}\)and \({{l}_{2}}\)are perpendicular for all values of \(m\).[2]