Lines \(l_1\) and \(l_2\) have equations
respectively, where \(\lambda\) is a parameter and \(a, b\) are constants.
It is given that lines \(l_1\) and \(l_2\) intersect each other at a right angle at point \(A\). Find the values of \(a\) and \(b\), and show that point \(A\) has coordinates \((4,1,3)\)[5]
Plane \(p_1\) has equation \(2x+z=-4\).
Find the shortest distance from point \(A\) to plane \(p_1\).[2]
Plane \(p_2\) contains both lines \(l_1\) and \(l_2\), and has equation \(-11x+8y-5z=-51\).
Find the acute angle between planes \(p_1\) and \(p_2\).[2]
Hence, or otherwise, find the shortest distance of point \(A\) to line \(l_3\), the line of intersection between \(p_1\) and \(p_2\). (Give your answer to three significant figures.)[2]