2025 RI P2 Q4

In a computer game, a slope can be modelled as a plane \(p\) containing three points, \(A(1, 0, -3)\), \(B(1, 4, -15)\) and \(C(2, 3, -14)\).

1. Show that \(p\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} = -1\).[2]

In Stage \(1\) of the game, Griffles travels on the slope from \(A\) along a path with equation \(\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ m \\ n \end{pmatrix}\) where \(m\) and \(n\) are constants and \(\lambda\) is a real parameter.

2. Find an equation relating \(m\) and \(n\).[1]

In the rest of the question, it is given that \(m = \frac{1}{3}\), \(n = -3\) and Griffles is modelled as a point travelling on the slope. An obstacle in the form of a rectangular virtual screen stands on the slope with its base modelled by the line segment \(BC\) (see diagram).

3. By considering the line segment \(BC\), show that Griffles is able to successfully navigate this obstacle without colliding into it.

   [3]

A laser gun with a sensor is mounted at the point \(E(5, 2, 1)\) above the slope. The sensor is activated when an object is within a range of \(5\) units.

4. Show that Griffles does not activate the sensor.[3]

After completing Stage \(1\), Griffles is teleported to another slope. This slope can be modelled as a plane \(q\) parallel to \(p\) such that \(E\) is equidistant from \(p\) and \(q\).

5. Find the cartesian equation of \(q\).[3]