N2011 P1 Q11

The plane \(p\) passes through the points with coordinates \((4, -1, -3)\), \((-2, -5, 2)\) and \((4, -3, -2)\).

1. Find a Cartesian equation of \(p\).[4]

The line \(l_1\) has equation \(\frac{x-1}{2} = \frac{y-2}{-4} = \frac{z+3}{1}\) and the line \(l_2\) has equation \(\frac{x+2}{1} = \frac{y-1}{5} = \frac{z-3}{k}\), where \(k\) is a constant. It is given that \(l_1\) and \(l_2\) intersect.

2. Find the value of \(k\).[4]
3. Show that \(l_1\) lies in \(p\) and find the coordinates of the point at which \(l_2\) intersects \(p\).[4]
4. Find the acute angle between \(l_2\) and \(p\).[3]