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# Planes

It's easy to feel a little confused when we first hear about "planes" in a math classroom. No -- planes in math have nothing to do with flying through the air. But they can be interesting -- even fun! One use of a plane is to allow us to see mathematical equations in a visual manner. Using a plane, we can translate algebra into geometric shapes and vice versa. But what exactly is a plane? How do planes work? What can they teach us about math? Let's find out:

## Planes defined

A plane is essentially just a flat surface. For example, when you put cards down on a table, you are arranging them on a "plane" of sorts. But a real plane has no beginning or end, and it extends indefinitely in all directions.

We can plot any ordered pair on a plane. We can even plot multiple points to form shapes -- such as the three vertices of a triangle. These points may also be linear, which means that they form a straight line.

## The history of the plane

French mathematician René Descartes observed a fly zipping around from one part of the ceiling to the next during the 17th century. He began plotting the fly's position using one corner of the ceiling as a reference point.

He then realized that he had found a way to translate algebra into geometry. By giving his reference point (also known as the "origin") a value of 0,0, he could plot any point on the ceiling. He could also translate these coordinates into algebraic expressions and vice versa.

We now call this concept the "Cartesian plane" in honor of Descartes. However, we often simply say that we're graphing coordinates on a "plane" for short.

## Visualizing a plane

We know that there is exactly one plane for any three non-collinear points. Consider the following diagram:

Note that P in this situation stands for "plane." We can either call this "plane ABC" or simply "plane P."

## More complex planes

We can also create more complex diagrams by combining two planes together. Take a look at this diagram:

Note that planes A and C are parallel, while plane B transverses or intersects both of them.

## The dimensions of planes

Planes have two "dimensions." These are generally represented by values x and y.

The x value represents the point's horizontal displacement in relation to the origin.

On the other hand, the y value represents the point's vertical displacement from the origin.

An easy way to remember the difference between the x and y values is to tell ourselves that "x" is a cross, so it must represent the values that move across the plane.

We should also note that a plane can be three-dimensional. Figures and points that have three dimensions have an x, y, and a "z" value. Theoretically, we could even add a fourth value for the fourth dimension -- which is time. But this would be very difficult to visualize!

As we begin our journey into the world of planes, we will mostly be dealing with two-dimensional coordinates and figures. For example, you might see the Cartesian coordinates (0,1).

The first value is always the x value, while the second value is always the y value. In other words, we write Cartesian coordinates in the format (x, y).

We know that if the x coordinate is 0, the point must lie directly on the x-axis. If the y coordinate is 0, we know that the point must lie directly on the y-axis.

Cartesian Plane

Collinear Points

Coplanar

## Flashcards covering the Planes

Common Core: High School - Geometry Flashcards

## Practice tests covering the Planes

Common Core: High School - Geometry Diagnostic Tests