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 $P = (-1, 2)$