Another example of a relative motion problem is rain falling on your car. When you look out the front window, it always seems to be falling towards you no matter what direction you drive in. The back window is barely hit, but the wipers are really needed on the front. Why? It's vector velocity addition.
The left part of the figure shows what an observer would see that stands on the side of the road: the car is moving to the right, with its velocity vector shown in red, and the rain is falling straight down, with its velocity vector shown in blue.
The right part of the figure shows what it appears like from
inside the car: If you are moving with 
car
to the right, then the ground, the trees, the rain drops etc. are
moving towards you with velocity vector -car.
The velocity vector of the rain as seen from the moving car is then
the black diagonal arrow in the right figure. This explains why rain
always seems to hit the car from the front while driving.