2024 MI PRU2 P1 Q11

A company has plans to assemble multiple solar cells together to form large rectangular shaped solar panels which could be installed as a solar panel system. It will be installed at an angle from a horizontal ground to improve efficiency and reduce maintenance cost.

Points \((x, y, z)\) are defined relative to an origin \((0, 0, 0)\) on the horizontal ground represented by the \(xy\)-plane, where units are in metres. The surface of the solar panel system is modelled as a plane and the three points \((-2, 3, 2)\), \((-5, 2, 2)\) and \((3, -2, 1)\) lie on this plane \(p\). It can be assumed that the solar panel system has negligible thickness.

1. Show that the cartesian equation of the plane \(p\) is \(x - 3y + 20z = 29\).[4]
2. Find the acute angle that plane \(p\) makes with the horizontal ground.[2]

There are several support beams used in the mounting structure to keep the solar panel system in place. These support beams are laid in straight lines and their thickness can be neglected. One of the support beams is placed vertically upwards from a specific point \((2, 3, 0)\).

3. Find the coordinates of point of intersection where this vertical support beam meets the solar panel system.[3]
4. The company wishes to improve the stability of the structure by building another support beam to connect the solar panel system to the point \((2, 3, 0)\). Find the shortest length of the support beam required.[3]