Cylinder and cone ornament in three dimensions: axis line, centre and radius of the join, surface area, turning angle and plane

A glass ornament is made by fixing a solid right circular cone to one end of a solid cylinder of the same radius. The base of the cone and the end of the cylinder meet in the circle \(C\). The cone has apex \(V\), and \(P\) is a point on the circumference of \(C\), as shown in the diagram.

In this question, all lengths are in centimetres.

The ornament is placed so that the circle \(C\) lies in the plane \(2x-y+2z=3\). The apex \(V\) has coordinates \((5,1,6)\) and \(P\) has coordinates \((4,5,0)\). The cylinder is \(4\text{ cm}\) long.

1. Find a vector equation of the axis of the ornament, which passes through \(V\) and the centre of \(C\).[3]
2. The centre of \(C\) is \(M\). Find its coordinates.[3]
3. 1. Show that \(C\) has radius \(\sqrt{17}\text{ cm}\).
   2. Find the ornament's total surface area.[6]
4. The ornament is now turned about the fixed point \(V\), and the point \(P\) moves to \(P'(11,2,2)\).
   Find the size, in degrees, of angle \(P\hat{V}P'\).[3]
5. Find a Cartesian equation for the plane that contains \(P\), \(V\) and \(P'\), in the form \(ax+by+cz=d\), where \(a,b,c,d\in\mathbb{Z}\).[4]