2025 TJC Promo Q12

An adventure park is designing a new zipline that will span between two hilltops. The diagram below shows a vertical cross-section of the zipline setup. Points are defined relative to a main control centre at \((0, 0, 0)\). The ground is horizontal and is modelled as the \(x\)-\(y\) plane.

The adventure park owner proposes to place the upper launch station at point \(A(0, 75, 50)\), and the lower landing station at point \(B(600, 25, 20)\). Points \(C\) and \(D\) are taken as the foot of perpendicular from points \(B\) and \(A\) onto the \(x\)-\(y\) plane respectively.

1. Find a vector equation of line \(AB\).[2]
2. Write down the coordinates of \(C\) and \(D\). Hence or otherwise, find the slope angle of descent, \(\beta\), for the proposed zipline.[3]

A viewing platform is to be constructed at a scenic point in the valley, located at point \(P(105, 30, 30)\).

3. Show that a vector equation of the plane \(ABCD\) is \(\mathbf{r}\cdot \begin{pmatrix} 1 \\ 12 \\ 0 \end{pmatrix}=900\).[3]
4. Find the exact coordinates of the foot of perpendicular from \(P\) to the plane \(ABCD\).[4]