Yes..To show that 2 particles do not collide we can show that when their final position vectors are equated( because they are same) that since times are different they do not collide right?
Yes, but I find that solving the equations represented by the displacement vectors to see if there is a common point in space at a time t is usually an easier way. We typically don't have the final displacement vectors to work with.To show that 2 particles do not collide we can show that when their final position vectors are equated( because they are same) that since times are different they do not collide right?