Coordinate System

 

Motivation:

 

                       How do you usually describe the route to a new place so that people can go there by themselves?  You need to describe a place that people are familiar with. And from that place, you describe the route to the new place so that people can understand how to reach there. Similarly, we would like formulate the problem mathematically. We start with two labeled axes that are perpendicular to each other and choose the intersection of the two axes as the origin.  One axis is known as x-axis and the other is y-axis.   With the distance measured with unit length specified in each axis,  we can identify each point on the same plane.

 

                                  

 

                As shown in the diagram above, the arrow sign stands for the positive direction on each axis.

 

                           

              And how do you describe the position of point A ?   It could be described as: starting from O, move along with the positive direction of x-axis two units. And then, move along with the positive direction of y-axis for 2 units.  For simplicity, we might as well use  (x,y) to represent it.  In this case, the position of  A is (2,2) .

 

              There exists other convention (left-handed rule ) to specify the positions of points in a plane:

 

                               

               But we just go with the previous one ( right-handed rule ).

 

Property:

 

                    The following diagram shows the property of point in each area.

                      

 

                     And we also list the name for each area .

 

 

                    This diagram originally is used to show label the position in each point. But later on, people found out that it also can be used to graph the relationship between some variables.  For example,   y=2x .  We know that y varies as x varies.  For each value of x, there exists a corresponding for y.  If we can draw all the possible points for (x,y) on the coordinate system above,  it would be easier to figure out the property between those two variables.   And that plot is so-called XY plot.

 

 

Example: Plot y=2x .

 

               At first, we list the relationship between x and y by setting x=1,2,3,4:                    

 

x	1	2	3	4
y	2	4	6	8

 

              And then, draw those points on X-Y plane:

 

              

 

               We find out that  y=2x is a line on the graph if we sample more points to draw.