2020 CJC Promo Q11

A hanging ornament is constructed by connecting six metal rods to form a tetrahedron \(VABC\) as shown in the diagram.

The points \(\left( x,y,z \right)\) are defined relative to a reference point \(O\left( 0,0,0 \right)\), where units are in centimetres. The coordinates of the points \(A\), \(B\), \(C\) and \(V\) are \(\left( 1,2,0 \right)\), \(\left( 6,3,0 \right)\), \(\left( 2,7,0 \right)\) and \(\left( 3,4,5 \right)\) respectively.

1. Find the equation of the plane \(VBC\) in scalar-product form.[3]
2. Find the coordinates of the point \(P\) on the plane \(VBC\) which is nearest to the point \(A\).[3]
3. Hence or otherwise, find the shortest distance from the point \(A\) to the point \(VBC\).[2]
4. Find the acute angle between the line \(VA\) and the plane \(VBC\).[3]