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
$\mathbf{r}=\left( 2+t \right)\mathbf{i}+\left( 1+2t \right)\mathbf{j}+3t\mathbf{k}$, $0\le t\le 4$.
(i)
State the coordinates of $X$.
[1]
(ii)
Find the acute angle the flight path makes with the horizontal.
[2]
(iii)
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 $\left( 1,5,\alpha \right)$.
(iv)
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]
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