2021 EJC J2 MYE P1 Q10

A flag pole is erected on a football field. Points \(\left( x,y,z \right)\) are defined relative to the centre of the field at \(\left( 0,0,0 \right)\). The football field is part of the \(x\)-\(y\) plane and the girth of the pole can be neglected. The top of the pole is at the point \(T\left( 2,13,9 \right)\). A light source at point \(A\left( 10,15,12 \right)\) casts a shadow of the pole onto the field (see diagram).

1. One end of the shadow of the pole is at the foot of the pole. \(R\), and the other end is at point \(P\). State the coordinates of \(R\), and show that the point \(P\) is at \(\left( -22,7,0 \right)\).[5]
2. Hence, or otherwise, find the length of the shadow.[2]

Another light source at point \(B\) casts another shadow of the pole onto the field that is perpendicular to the shadow cast by the light source at point \(A\).

3. Find a Cartesian equation of the plane that \(B\) must lie in.[3]

After a storm, the pole tilted such that the top of the pole is now instead at point \(M\left( -2,9,7 \right)\) while bottom of the pole remained at \(R\).

4. What is the acute angle between the tilted pole and the football field?[3]