In a computer game, a slope can be modelled as a plane $p$ containing three points, \(A(1, 0, -3)\

Question

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 $C(2, 3, -14)$. 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. 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). 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. 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 $q$. Find the cartesian equation of $q$.[3]

Accepted Answer

(b)3m +n=-2 (e)2x+3y+z=35

2025 RI P2 Q4

2025 RI P2 Q4