Introduction to Cartesian Coordinates

What is The Cartesian Coordinate System

The cartesian coordinate system is a system used to define the position of points, bodies, and objects by how far they are along a line from a reference point. The most basic form of cartesian coordinates is the number line, which can be used to locate the position of any point in one dimension.

In this system, an arbitrary point on the line is chosen as the reference / origin point, and the location of any point P is defined as the distance from the origin to point P. The distance can be either positive or negative depending on which side of the line P lies. Each point is given a unique coordinate, and each real number is the coordinate of a unique point.

By convention, the right side of the line is considered positive, and the left side is negative.

For example, the point located 5 units right from the origin can be referenced as point (5), and the point located 4 units left from the origin can be referenced as point (-4).

Locating points in one dimension is quite a straightforward task. However, it does not have many use cases and applications. So let's expand our world into two dimensions by introducing the Cartesian plane or Cartesian coordinates in 2d, which we will discuss in the next section.

Two-Dimensional Cartesian Coordinate System

A two-dimensional cartesian coordinate system or cartesian plane can be constructed by drawing two perpendicular lines called axes, in which the origin or reference point is defined as the point where the two lines intersect. The position of any point P can be determined by how far it's from the origin point along each axis.

The two lines are often denoted by the letters x (for the horizontal line) and y (for the vertical line), and they are referred to as the x-axis and y-axis. By convention, the right side of the x-axis is considered positive, and the left side is negative. Similarly, the top side of the y-axis is considered positive, and the bottom side is negative.

Any point on a two-dimensional cartesian system is referenced by an ordered pair of numbers (x,y).

Plotting Points on a Cartesian Plane Example 1.1

Locate the point (3,2) on the cartesian plane.

Solution

Step 1) We identify the x and y coordinates in the ordered pair (3,2) as:

x-coordinate = 3 y-coordinate = 2

Step 2) We locate the x value (3) on the x-axis. Then we sketch a vertical line through x=3, as shown in the diagram below.

Step 3) Now, we locate the y value (2) on the y-axis and sketch a horizontal line through y=2. The point where these two lines meet is the point with coordinates (3,2). We plot the point there, as shown in the diagram below.

Identifying Points on a Cartesian Plane Example 1.2

The ability to identify the coordinates of a point shown on a graph is just as important as being able to plot points. So try to identify the coordinate of point Q shown in the graph below.

Solution

Step 1) We identify the x-coordinate of the point by sketching a vertical line that goes through point Q and intersects the x-axis, as shown in the diagram below. The number where the line intersects the x-axis is the x-coordinate of point Q.

Point Q x-coordinate = -4

Step 2) We identify the y-coordinate of the point by sketching a horizontal line that goes through point Q and intersects the y-axis, as shown in the diagram below. The number where the line intersects the y-axis is the y-coordinate of point Q.

Point Q y-coordinate = -3

We now have identified both coordinates of point Q and can refer to it using the ordered pair (-4,-3).

The Four Quadrants

The two axes in 2d cartesian coordinates divide the plane into four regions we call quadrants. The four quadrants are numbered in Roman numerals, as I, II, III, and IV, in a counterclockwise direction.

Following the sign convention for each axis, in which the right side of the x-axis is positive, and the left side is negative. The top side of the y-axis is positive, and the bottom side is negative. We can determine in which quadrant a point lies by looking at the signs of its x and y coordinates, where:

• In quadrant I both x and y are positive (+,+)
• In quadrant II x is negative and y is positive (-,+)
• In quadrant III both x and y are negative (-,-)
• In quadrant IV x is positive and y is negative (+,-)

Plane Quadrants Example 1.3

Identify the cartesian coordinates of points P and Q, shown in the diagram below, and find in which quadrant each point is located.

Solution

Point P, shown in the graph below, has the coordinates (2, 1) and is located in the first quadrant.

Point Q has the coordinates (-3, 2) and is located in the second quadrant.

Hopefully, by now you have a good grasp on how to identify the position of points in a two-dimensional plane using the cartesian coordinates, and with this tool under your belt, you can explore a multitude of ideas in mathematics, physics, engineering, and many other fields. But we live in a three-dimensional world, and in order to be able to locate the position of points, bodies, and objects around us we need to use the 3D Cartesian coordinates.

Three-Dimensional Cartesian Coordinate System

We can create a three-dimensional cartesian coordinate system by adding another line / axis perpendicular to the two-dimensional cartesian plane that also goes through the origin point. Using this system, we'll be able to locate the position of any point in 3d space by how far it's from the origin along the three lines / axes.

Any point on a three-dimensional cartesian system is referenced by an ordered triplet of numbers (x,y,z).

Plotting Points in Three Dimensions Example 2.1

Plotting points in three dimensions using the cartesian coordinates includes the same steps we used in two dimensions plus an extra axis/dimension. Try to locate point (6,5,4) on a 3d cartesian coordinate system

Solution

Step 1) We identify the x, y, and z coordinates in the ordered triplet (6,5,4) as:

x-coordinate = 6 y-coordinate = 5 z-coordinate = 4

Step 2) We locate the x value (6) on the x-axis. Then we sketch a line parallel to the y-axis that goes through x=6, as shown in the diagram below.

Step 3) We locate the y value (5) on the y-axis. Then we sketch a line parallel to the x-axis that goes through y=5, as shown in the diagram below.

Step 4) Now, we sketch a line of length z=4 parallel to the z-axis that goes through the point where the previous two lines intersect on the x-y plane. The point where the line ends is the point with coordinates (6,5,4). We plot the point there, as shown in the diagram below.

The Eight Cartesian Octants

Similar to the plane quadrants, 3d cartesian coordinates divide the space into eight regions called octants. By convention, each octant is named based on its signs. For instance, the first octant where the x, y, and z axes are all positive is named (+, +, +).

The same methodology can be used to create a four-dimensional cartesian coordinate system where any point can be referenced by an ordered quadruple of numbers (w,x,y,z). Similarly, we can create a 5-dimensions system or even more!

Frequently Asked Questions

When was the cartesian coordinate system invented?

The coordinate system was invented by René Descartes (Latinized name: Cartesius) and was first published in 1637.

What are the applications of the cartesian coordinate system?

The coordinate system is the foundation of analytic geometry and has many applications in mathematics, as well as other fields that deal with geometry, such as astronomy, physics, engineering, and many more.

How many axes are there in a cartesian plane?

There are two perpendicular axes in a cartesian plane.

What is the origin of a cartesian plane?

The origin or reference point is the point that has the coordinates (0,0).