The definition of average velocity is:

According to this definition, the vector average velocity is obtained by dividing a vector by a scalar. Since the scalar () is always positive, this definition indicates that the two vectors ( and ) have the same direction. We would like to apply this definition to find the velocity at the turning point. Then and in the definition correspond to an interval that spans the turning point, i.e., with an initial point just before the turning point and with a final point just after the turning point. The diagrams below illustrate this idea graphically for a ball that is turning around.

Suppose a ball rolls uphill past point 1, turns around at point 2 and then rolls downhill past point 3. Since the ball has turned around, points 1 and 3 correspond to the same location. The position of the ball is a vector that points from the origin to the location of the ball. The location of the origin in the diagram is arbitrary. Its location will not affect the velocity we are trying to compute. Since points 1 and 3 represent the same point, the position vectors at these two instants are also the same. If this is true, what is ? What is ?