Measuring Position 

Consider a point A, in empty space, millions of light years away from
anything else. 



How can its position be stated? 

A few seconds thought will lead to the conclusion that the position of A can
only be stated by making reference to some other point, as shown
here. 



We can now say that A is a distance r away from B in
the direction of... er, let's see now... in the direction of
A! 

That's all you can say about the direction without some extra
points of reference. 



Going a step further: does the idea of a "position for A" even make
any sense in the absence of any other point? 

Anyway, we know where A is in relation to B but what is the position of
B? Same problem... 



We must accept that statements about the positions of points or bodies can only
be relative to some other point or body. 

Things can be made a little easier if we introduce a set of axes
along which the positions can be measured. 

This is equivalent to adding one more point (called the origin,
O) and imagining a couple of rulers fixed there, at 90° to each
other. 



Using this system we can now say that the position of A is (x_{A},
y_{A}) and the position of B is (x_{B}, y_{B}). 

However, we must remember that these are the coordinates of A and B relative
to O. 



Another way of saying this is that these coordinates represent the positions of
A and B in O's frame of reference. 

Summarizing these positions (assuming arbitrary distance units on the axes of
the graph), we have 

Relative to O 

position of A is 
x_{A} = 1 
y_{A} = 2 
position of B is 
x_{B} = 4 
y_{B} = 4 




It is often useful to be able to decide what other observers' measurements will
be for the positions of the same points or bodies. 



We say that, knowing the measurements in O's frame of reference, we want to
transform them into measurements in another frame of reference. 



To do this, we simply move the origin of our axes. 

Let's transform to A's frame of reference as an example. 



Here the origin is on A so we now see things "from A's point of view". 

Relative to A 

position of A is 
x_{A} =
0 
y_{A} = 0 
position of B is 
x_{B} =
3 
y_{B} = 2 
position of O is 
x_{o} =
1 
y_{o} = 2 




Looking at this simple example leads us to the conclusion that
to transform O's measurements of the position of B into A's frame of
reference, we do the following subtraction: 

(coordinates of B relative to O) 
(coordinates of A relative to O) 



Having seen here that all measurements of position must be
relative to some
point or observer or frame of reference, we must also conclude the same for
speeds, velocities, accelerations etc as these quantities are related to
rates of change of positions of bodies. 
