Cartesian axes and representation of points in the plane

A Cartesian axes are a pair of perpendicular real lines that allow us to identify the different points in the plane. We will identify any point $$P$$ by means of a pair of numbers, $$a$$ and $$b$$, and we will write $$P = (a, b)$$. Before explaining how to find $$a$$ and $$b$$, let's analyze the Cartesian axes more in depth.

This is a graphic representation of the Cartesian axes:


We observe that we have two real lines that cross at point $$0$$ of both.

It is woth mentioning that the above mentioned straight lines divide the plane in four parts called quadrants, which are identified accordingly to the figure:


The different axes have their own names:

  • The horizontal axis is the abscissa axis.
  • The vertical axis is the axis of ordinates.

The point where the two axes are cut is called the origin (sometimes simply $$O$$), and it takes as its coordinates $$O = (0, 0)$$.

Once the notation has been seen, we are already able to locate points.

A rigorous definition of what is considered to be coordinates of a point might be:

Given a Cartesian axes and a point $$P$$ of the plane, if $$a$$ and $$b$$ are the values of the projection of the point $$P$$ on the abscissa and ordinates axes, respectively, then we have $$P = (a, b)$$.

A more constructive definition might be the following one:

The coordinates $$a$$ and $$b$$ of a point $$P$$ of the plane, $$P = (a, b)$$, are the points of intersection of the parallel lines to the axes of coordinates drawn from point $$P$$ to the coordinates axes. The first coordinate $$a$$ is the intersection with the horizontal axis or the abscissa axis, and the second coordinate $$b$$ is the intersection with the vertical axis or the ordinate axis.

A visual example will turn out to be a lot more clearer.

Initially we have the point and the coordinates axes:


If we draw a parallel line from the point $$P$$, we have:


And therefore we can already say that $$P = (2,-3)$$.

The process of representing points is exactly the same one but in the inverse.

Let's suppose that we want to represent point $$P = (-1, 2)$$ in the Cartesian axes, then the procedure to follow is the following :

We mark in the abscissa axis the point $$-1$$ and in the axis of ordinates the point $$2$$:


We draw parallel lines to the axes of ordinates and abscissa from points $$a$$ and $$b$$ respectively:


The intersection of the above mentioned parallel lines is point $$P = (-1, 2)$$