Cartesian coordinates

In this topic, we will discuss the Cartesian coordinate system, one way of graphing points in a 2D space. You may have learned about it in school as the $x-y$ plane, so this topic will be more of a reminder.

Cartesian coordinates on the plane

The Cartesian coordinate system is the most common coordinate system. It's based on the idea of orthogonal axes, meaning that you can travel in the direction of one axis without affecting your position on the other. For example, on a map, the north-south axis is orthogonal to the east-west axis.

Let's see how it works on the plane. The Cartesian coordinate system consists of a point $O$, whose coordinates are $(0,0)$, and two orthogonal axes, labeled $x$ and $y$. The axes intersect at $O$ (for origin), making right angles and dividing the plane into four symmetrical parts.

Every point on the plane is defined in relation to these two axes, giving it two coordinates: the $x-$coordinate and the $y-$coordinate. Each of these coordinates is a real number. By convention, we write them in parentheses like $(x,y)$, so the green point below with coordinates $x=2,$ $y=3$ is labeled $(2,3)$.

This figure demonstrates that each point can be considered as an $x$ coordinate (horizontal dotted line) plus a $y$ coordinate (vertical dotted line) – including the origin point $O$, for which each line has $0$ length.

Quadrants

As we mentioned earlier, the two axes divide the infinite plane into four parts. These parts are called Quadrants I through IV, in the following order:

Points in the same quadrant will have matching signs (positive/negative) on their $x$ and $y$ coordinates.

If you forget the order, you can start in the first quadrant – the top-right – and draw a C. If you forget the first quadrant, or the shape of a C, you're out of luck.

Quadrants allow us to talk about the $x-y$ plane in broad strokes. For example, the line $y = x$ includes points in Quadrants I and III, but none in Quadrants II or IV, where $x$ and $y$ have opposite signs. The parabola $y = x^2$ starts in Quadrant II (negative $x$, positive $y$), comes through the origin, and bounces back up into Quadrant I.

Let's look at these points again and sort them into quadrants.

Our green point $(2,3)$ is in Quadrant I. Going counterclockwise we see the red $(-3,1)$ in Quadrant II and finally $(-1.5, -2.5)$, the blue point, in Quadrant III. The origin doesn't belong to any quadrant.

Distance

With two points on the plane, it's natural to wonder about the distance between them. For a simple example, let's look back at the red point above. How far is it from the origin?

If you draw a blue line between the points $(0,0)$ and $(-3,1)$, you see that it divides the red rectangle into two right triangles.

This blue line is equal to the distance between the dot $(-3,1)$ and the origin. How do you find its length? Apply the Pythagorean theorem to one of the right triangles. Since the dot has coordinates $(-3,1)$, the horizontal of any triangle side has the length $3$ and the vertical side of any triangle has the length $1$. So, the blue line has the length $\sqrt{3^2 + 1^2} = \sqrt{10}$.

When comparing any two points $(x_1,y_1)$ and $(x_2,y_2)$, you can find the same kind of triangle, with horizontal and vertical side lengths given by $x_1 - x_2$ and $y_1 - y_2$ respectively. So the whole distance formula is $d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$. This formula is just the result of applying the Pythagorean theorem and nothing more!

Cartesian coordinates in 3D and higher space

Why stop at $x$ and $y$? If we add a third axis, $z$, orthogonal to both, then we will get a three-dimensional space in which the position of a point is defined by three coordinates.

The Cartesian coordinate system can also be expanded for any $n$-dimensional space, where $n$ is a positive whole number. High-dimensional spaces are not easy to visualize, but the gist of it is that in an $n$-dimensional space there will be $n$ orthogonal axes and every point will be defined by $n$ coordinates.

In a database of cities, for example, we could look at their latitude, longitude, average temperature, and population. Each city is represented by a point $(x,y,z,w)$ in a $4D$ space.

Conclusion

The Cartesian coordinate system is the standard way of describing points in a plane, so it's important to get comfortable with how it works. Every point is represented by a unique pair of numbers, so it's trivial to tell when two points are the same. Cartesian coordinates lend themselves to drawing straight lines and right angles; it's difficult to use them to describe a circle or a spiral.