Consider point A, in empty space, millions of light-years away from anything else. How could its position be stated?

The position of point A can only be stated with reference to another point.

We can now say that A is r metres away from B. However, what is the position of B?

Measurement of position is easier if we imagine a set of axes, as in the next diagram. This is equivalent to introducing a third point, O, (the origin of the axes) and two rulers at 90° to each other.

We can now say that the position of A is (x_A, y_A) and the position of B is (x_B, y_B).

These co-ordinates are the positions of A and B relative to O. Another way of saying the same thing is that they are positions measured in O’s frame of reference.

If we have measurements made in O’s frame of reference and we want to transform them to A’s frame of reference, we simply move the origin of the axes to coincide with A, as shown in the next diagram (in which we finally forget all that "empty space"!).

Now, with the origin moved to A, it is clear that the position of O relative to A is

and the position of B relative to A is

To find the position of B relative to A we have simply done the following subtraction:

(co-ordinates of B relative to O) - (co-ordinates of A relative to O)

All measurements of position must be relative to a certain point or object and, since speed is a measure of rate of change of position, it should not be surprising to find that all speeds (or velocities) are also relative measurements.