You're walking along a road, heading west at 8 km/hr. A car is going by with velocity of 30 km/hr east. There is also a truck passing, traveling at 40 km/hr west.

How fast is the truck traveling relative to you?

How fast is the car traveling relative to you?

How fast is the truck traveling relative to the car?

One way to look at it is this: in an hour, the truck will be 40 km west of where you are now, but you will be 8 km west, so the truck will be 32 km further west than you in an hour. Relative to you, then, the truck has a velocity of 32 km/hr west. Similarly, relative to the truck, you have a velocity of 32 km/hr east.

To find the velocity of other things relative to a particular observer, subtract the observer's velocity from all the other velocities.

Do this to find the velocity of the truck relative to you. Using a subscript Y for you, T for the truck, and G for the ground, we can say:

The velocity of the truck relative to you is v_TY = v_TG - v_YG

This is often written as: v_TY = v_TG + v_GY

If you flip the order of the subscripts the vector is reversed.

The velocity of the truck relative to the ground = v_TG = 40 km/hr west

The velocity of you relative to the ground = v_YG = 8 km/hr west

The velocity of the truck relative to you, v_TY = v_TG - v_YG = +40 -8 = 32 km/hr west

Similarly, the velocity of the car relative to you is 38 km/hr east.

The velocity of the truck relative to the car is 70 km/hr west, and the velocity of the car relative to the truck is 70 km/hr east.