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How to Plot Points in Three Dimensions

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will look at what a three-dimensional coordinate system looks like. After that, we will learn how to plot points in three dimensions and look at a couple of examples of the process.

Three Dimensions

You may be familiar with the two-dimensional coordinate plane, in which there is an x-axis and a y-axis.


Two Dimensional Coordinate Plane
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However, we don't live in a two-dimensional world! Take a moment to find a corner where two walls and the floor meet. Notice the floor is a two-dimensional plane and we can consider the lines where the floor meets the walls the x-axis and the y-axis.


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Now, consider the line where the walls meet. This line adds in a new axis to the two-dimensional coordinate plane. If we call this line the z-axis, we now have three axes and we are now working in three dimensions!


Three Dimensional Coordinate Plane
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You probably never looked at a corner in this way before, huh?

Plotting Points in Three Dimensions

As we just saw from our corner example, when it comes to working in three dimensions, we simply add an axis to the two-dimensional coordinate plane. We have an x-axis, y-axis, and z-axis, all intersecting at 0.


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In the same way that we can plot a point on the two-dimensional coordinate plane, we can also plot a point in three dimensions. It just takes one extra step!

First of all, let's look at how we plot a point, (x, y), on the two-dimensional coordinate plane.

  1. Locate x on the x-axis.
  2. From that point, moving parallel to the y-axis, move y units. This is your point.

Now, let's take a look at when we are working in three dimensions! To plot the point (x, y, z) in three-dimensions, we simply add the step of moving parallel to the z-axis by z units. That is, to plot a point (x, y, z) in three dimensions, we follow these steps:

  1. Locate x on the x-axis.
  2. From that point, moving parallel to the y-axis, move y units.
  3. From that point, moving parallel to the z-axis, move z units. This is your point.

Well, that doesn't seem so hard! For example, ssume we want to plot the point (1, 2, 3) in three dimensions. First, we locate 1 on the x-axis.


Step One
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Next, from that point, we move 2 units parallel to the y-axis.


Step Two
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Lastly, from that point, we move 3 units parallel to the z-axis.


Step Three
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And there we have it! We've plotted our point!

Another Example

Let's consider another example of this. Suppose you place an imaginary three-dimensional coordinate system in your house. In doing so, you can represent any location in your house with a point.


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