2023 ACJC P1 Q12

An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. These flight paths may be assumed to be straight lines, and the aircraft may be assumed to be point objects.

In the diagram below, the labelled points are defined relative to an origin \(O\) with coordinates \((0,0,0)\) on the airfield. The \(x\)-, \(y\)- and \(z\)-axes are in the directions east, north and vertically upwards respectively, with units in metres. The direction vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) correspond to the east, north and vertically upwards directions respectively. The point \(A\) has coordinates \((2000,500,0)\) and the point \(B\) has coordinates \((1400,2900,0)\).

Aircraft Alpha takes off from Point \(A\) in the direction \(4\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) as shown.

1. Find the initial take-off angle \(\theta\), the angle that Aircraft Alpha's flight path makes with the horizontal ground.[2]

The peak of a majestic mountain has coordinates \((18000,4500,2000)\).

2. Find the coordinates of the closest point on the flight path of Aircraft Alpha to the peak. Hence find the closest distance between the flight path and the peak, giving your answer to the nearest metre.[5]
3. Aircraft Bravo takes off from Point \(B\). For aviation safety reasons, the flight paths of Aircraft Alpha and Aircraft Bravo should never cross each other. Given that Aircraft Bravo will take off in the direction \(4\mathbf{i} + 2\mathbf{j} + h\mathbf{k}\), find the value of \(h\) which will result in the flight paths crossing each other, giving your answer to \(3\) decimal places.[2]

The value of \(h\) is now given to be \(1\).

4. Find the cartesian equation of the plane containing point \(B\) and the flight path of Aircraft Alpha. Using this answer, provide another mathematical explanation as to why the flight path of Aircraft Bravo will not cross the flight path of Aircraft Alpha.[4]