2019 MI P1 Q11

In this question, the distance is measured in metres and time is in seconds.

A radio-controlled airplane takes off from ground level and is assumed to be travelling at a steady speed in a straight line. The position vector of the airplane \(t\) seconds after it takes off is given by \(\mathbf{r}=\mathbf{a}+t\mathbf{b}\), where \(\mathbf{a}\) refers to the position vector of the point where it departs and \(\mathbf{b}\) is known as its velocity vector.

1. Given that the airplane reaches the point \(P\) with coordinates \(\left( -9,\,\,4,\,\,6 \right)\) after \(6\) seconds and its velocity vector is \(-2\mathbf{i}+\mathbf{j}+\mathbf{k}\), find the coordinates of the point where it departs.[1]

Paul stands at the point \(C\) with coordinates \(\left( -7,\,\,5,\,\,2 \right)\) to observe the airplane.

2. Find the shortest distance from Paul’s position to the flight path of the airplane.[3]

While the airplane is in the air, a drone is seen flying at a steady speed in a straight line with equation

\(\frac{2x-1}{-6}=\frac{y+7}{4}=\frac{z-10}{k}\).

3. Show that the equation of the flight path of the drone can be written as \(\mathbf{r}=\mathbf{p}+\lambda \mathbf{q}\), where \(\lambda \) is a real parameter and \(\mathbf{p}\) and \(\mathbf{q}\) are vectors to be determined, leaving your answer in terms of \(k\).[1]
4. Given that the flight paths of the radio-controlled airplane and the drone intersect, find \(k\).[3]

At \(P\), the airplane suddenly changes its speed and direction. The position vector of the airplane \(s\) seconds after it leaves \(P\) is given by

\[\mathbf{r} = \begin{pmatrix} -9 \\ 4 \\ 6 \end{pmatrix} + s \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix}, \text{ where}\hspace{0.5em} s \in \mathbb{R}, s \geq 0.\]

It travels at a steady speed in a straight line towards an inclined slope, which is assumed to be a plane with equation

\(x-y+7z=2\).

5. Determine if the new flight path is perpendicular to the inclined slope.[2]
6. The airplane eventually collides with the slope. Find the coordinates of the point of collision.[3]

TGM correction to part (iii): the symmetric equation describes a full line, so use \(\lambda\in\mathbb R\). The source paper says “non-negative constant”, but gives no starting point or direction that would restrict the drone to a ray. The parameter is not assumed to be elapsed time.