When are two particles moving in the same or opposite direction?

Question

Given that particles $P$ and $Q$ have velocities $a$i+$b$j and $x$i+$y$j, what is the condition that they are going in the same direction, and what is the condition that they are going in the opposite direction?

Answer 1

If $P$ has velocity $ai + bj$ then $Q$ is moving in the same (opposite) direction as $P$ if $xi + yj$ is a positive (negative) scalar multiple of $ai + bj$.