2021 TJC P1 Q11

Coordinates axes \(Oxyz\) are set up with the origin \(O\) at the base of an airport control tower. The \(x\)-axis is due East, the \(y\)-axis due North and the \(z\)-axis vertical. The units of distances are kilometres. An airplane \(A\) takes off from the point \(X\). For the first \(4\) minutes, the position vector of \(A\) at time \(t\) minutes after take-off, is given by

\(\mathbf{r}=\left( 2+t \right)\mathbf{i}+\left( 1+2t \right)\mathbf{j}+3t\mathbf{k}\),  \(0\le t\le 4\).

1. State the coordinates of \(X\).[1]
2. Find the acute angle the flight path makes with the horizontal.[2]
3. The airplane enters a cloud at a height of \(5\) km. Find the coordinates of the point where it enters the cloud.[2]

A second airplane \(B\) takes off from the point \(\left( -2,-1,0 \right)\) at the same time as the first airplane \(A\) and is traveling at a constant speed in a straight line for the first \(4\) minutes. Two minutes after take-off, \(B\) is at the point \(\left( 1,5,\alpha \right)\).

4. Find in terms of \(\alpha \), the position vector of \(B\) after \(t\) minutes where \(0\le t\le 4\). Explain if it is possible for the two airplanes to collide in the first \(4\) minutes.[4]