RI/2021/JC1/Promo/P1/Q09

Distances in this question are in metres.
Harry and Tom’s model airplanes are taking off from the horizontal ground, which is the $x-y$ plane. Tom’s airplane takes off after Harry’s. The position of Harry’s airplane $t$ seconds after it takes off is given by $\mathbf{r}=(5 \mathbf{i}+6 \mathbf{j})+t(-4 \mathbf{i}+2 \mathbf{j}+4 \mathbf{k})$. The position of Tom’s airplane $s$ seconds after it takes off is given by $\mathbf{r}=(-39 \mathbf{i}+44 \mathbf{j})+s(4 \mathbf{i}-6 \mathbf{j}+7 \mathbf{k})$.
(i) State the height of Harry’s airplane two seconds after it takes off and find its distance travelled in the two seconds.



(ii) Find the acute angle between the path of Harry’s airplane and the ground.



(iii) Show that the paths of the airplanes are perpendicular.



(iv) Given that the two airplanes collide, find the coordinates of the point of collision. How long after Harry’s airplane takes off does Tom’s airplane take off?



(v) Find the cartesian equation of the plane in which both paths of the airplanes lie.