Distinguish a geometric flight path from motion in time

Distances are in metres and time in seconds. A radio-controlled airplane takes off from ground level and travels at a steady speed in a straight line. Its position t seconds after takeoff is $\mathbf r=\mathbf a+t\mathbf b$, where a is its departure position and b its velocity vector.

(i) After 6 seconds it reaches P=(−9,4,6) and its velocity is $-2\mathbf i+\mathbf j+\mathbf k$. Find its departure coordinates.1 mark

Paul stands at C=(−7,5,2).

(ii) Find the shortest distance from Paul to the airplane’s flight path.3 marks

A drone takes off at the same time and flies at steady speed on the straight line $(2x-1)/(-6)=(y+7)/4=(z-10)/k$.

(iii) Show its flight path can be written $\mathbf r=\mathbf p+\lambda\mathbf q$, λ≥0, determining p,q.1 mark

(iv) Given the flight paths intersect, find k.3 marks

(v) Do the airplane and drone collide? Justify your answer.1 mark