Answer: velocity \(\binom{12}{16}\), position of \(P\) \(\binom{1+12t}{2+16t}\), collision at \(t=4\) with position vector \(\binom{49}{66}\).

The magnitude of the direction vector \(\binom34\) is

\(\sqrt{3^2+4^2}=5.\)

So the unit vector in this direction is \(\binom{3/5}{4/5}\). Since the speed is \(20\), the velocity vector of \(P\) is

\(20\binom{3/5}{4/5}=\binom{12}{16}.\)

The position vector of \(P\) after \(t\) seconds is

\(\binom12+t\binom{12}{16}=\binom{1+12t}{2+16t}.\)

The position vector of \(Q\) after \(t\) seconds is

\(\binom{17}{18}+t\binom8{12}=\binom{17+8t}{18+12t}.\)

At collision, the position vectors are equal. Comparing the first components,

\(1+12t=17+8t.\)

So

\(4t=16,\qquad t=4.\)

The position vector of the collision point is

\(\binom{1+12(4)}{2+16(4)}=\binom{49}{66}.\)