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SAT Subject Test Mathematics Level 2: Practice and Study Guide25 chapters | 240 lessons | 15 flashcard sets

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.

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

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.

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!

You probably never looked at a corner in this way before, huh?

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.

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.

- Locate
*x*on the*x*-axis. - 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:

- Locate
*x*on the*x*-axis. - From that point, moving parallel to the
*y*-axis, move*y*units. - 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.

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

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

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

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.

One day, you are in your room and you realize you need something from the living room on the second floor of your house. You wonder if your roommate is near that living room so they can grab it for you and bring it to you. You call out to your roommate, asking them where they are located in the house, and they say they are at the point (2, 4, 11). You pull out your grid and figure out where they are.

First, you locate 2 on the *x*-axis.

Now, from that point, you move 4 units parallel to the *y*-axis.

Lastly, from that point, you move 11 units parallel to the *z*-axis.

Ah-ha! It looks like your roommate is in the living room! You call out and ask them to grab your needed item, and they gladly oblige!

A three-dimensional coordinate system has an *x*-axis, a *y*-axis and a *z*-axis. To plot a point (*x*, *y*, *z*) in three dimensions, we use the following steps:

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

Because we live in a three-dimensional world, this process is very handy! Although it may be unlikely that you would place a three-dimensional coordinate system in your house specifically as our last example suggests, this concept is used very much in locating different points in the world, in longitudinal and latitudinal coordinates, or in locating points in any large space. Thank goodness the process itself is fairly simple!

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SAT Subject Test Mathematics Level 2: Practice and Study Guide25 chapters | 240 lessons | 15 flashcard sets

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