NYJC/2021/JC2/Prelim/P1/Q10

One day, Eddie came home from a birthday party and brought back a helium filled balloon. After playing with it, he accidentally released the balloon at the point $(1,2,3)$ and it floated vertically upwards at a speed of 1 unit per second. Shortly after $t$ seconds, a sudden gust of wind caused the balloon to move in the direction of $\mathbf{i}+4 \mathbf{j}+6 \mathbf{k}$.
You may assume that $z=0$ refers to the horizontal ground.
(i) Find the angle in which the balloon has changed in direction after the gust of wind blew it away.



(ii) Find the Cartesian equation of the plane that the balloon is moving along.



(iii) Given that the balloon eventually stayed at the point $(2,6,12)$ on the ceiling, find the time $t$ when the gust of wind blew the balloon away.



Eddie decides to shoot the balloon down with his catapult.
(iv) Assume he was holding his catapult at $(3,2,1)$ initially and he walked along the path parallel to $2 \mathbf{i}+\mathbf{j}$. Find the position vector of the point where he should place his catapult so that the distance between his catapult and the balloon is at its minimum. Hence find this distance.