Amy has a master's degree in secondary education and has taught math at a public charter high school.
After watching this video lesson, you will know how read the three coordinates given to you and graph any given point in the three-dimensional coordinate space. Then test your knowledge with a quiz.
Plotting points and graphing lines in two-dimensions is easy because you can simply draw it on a flat piece of paper. Whatever you draw on that sheet of paper will be two-dimensional. It only has length and width. For points on the coordinate plane, they only have an x component and a y component. But, what happens when you add in a third component, a z component? What then? Welcome to the coordinate space, a place where you can plot three-dimensional points and graph three-dimensional lines.
We define three-dimensional space as that space where you have a width, a length, and a height. In coordinate space, this translates to having an x component, a y component, and a z component. Why do you need to know how to plot points and graph in three-dimensions? You need to know how to do this because you will be tested on this subject in your standardized tests that you take for college. Also, you need to know this information if you want to get involved with computer aided design (or CAD) as the objects in CAD images are sometimes drawn in the three-dimensional space.
Graphing a Point
So first, let's consider plotting a single point in three-dimensions
Imagine that you are beginning to design something used a computer aided design program such as SketchUp. You are looking at the three-dimensional space. The point that you want to plot is (4, 2, 1). Breaking this point down to its components, this point is telling you that its x component is 4, its y component is 2, and its z component is 1. To plot points in the three-dimensional coordinate space, you do it pretty much like you would in the two-dimensional plane, except you add another dimension. The third dimension tells you how high your point is. If your third dimension happens to be negative, then you will be going lower instead of higher. Your point is no longer always at ground level. It can now float! To graph this point, you first find your x component on the x-axis. Then you follow this line to your y component on the y-axis. Then you go up until you've reached your z component. So, your point (4, 2, 1) is the point (4, 2) at a height of 1.
Looking at this graph, notice that your axes are in different locations. Your axis that goes up and down is now the z axis. The x axis here is now going diagonal. We need to draw it like this because we are trying to fit three dimensions into two.
Graphing a Line/Plane
Next, let's see about graphing a linear equation in three dimensions. When you graph a liner equation in three dimensions, you won't get a straight line like you would when graphing in two dimensions. Instead, you will get a plane, an endless flat surface. Let's take a look.
Over 79,000 lessons in all major subjects
Get access risk-free for 30 days,
just create an account.
After plotting your point in SketchUp, you now need to graph the equation 3x - 2y + 4z = 12 to create the surface on which all your objects will stay. To do this, you will need to find all the intercepts: the x-intercept, the y-intercept, and the z-intercept. Once you've found all the intercepts, then your plane is the flat surface that goes through all three points. So, for the equation 3x - 2y + 4z = 12, to find the x intercept, you set the y and z variables to 0 and solve for x. You get 3x - 0 + 0 = 12. Then, solving for x, you get x = 4. Finding the y intercept, you set the x and z variables to 0. You get 0 - 2y + 0 = 12. Solving for y, you get y = -6. To find the z intercept, you set the x and y variables to 0. You get 0 - 0 + 4z = 12. Solving for z, you get z = 3. Now that you've found all your intercepts, you can plot all of them on the three-dimensional coordinate plane.
This equation graphs into the plane that passes through all these points. This particular plane slants downward from the front.
Let's try another example: Plot the point (3, 0, 3).
To plot this point, you first locate the 3 on the x-axis. Then you follow this line and see where it intersects with the 0 on the y-axis. Then you take it up to a height of 3 on the z-axis.
And you're done.
Let's review what you've learned. The three-dimensional space is the space where you have a width, a length, and a height. Your points in the coordinate space will have three parts to them. They have an x-component, a y-component, and a z-component. To graph them, you first find the x-axis number. Then you follow this line to where it intersects with the y-axis number. Then you take it up to where it meets the z-axis number. If the z number is negative, then you take it down, of course. When graphing a linear equation in three dimensions, you first find all your intercepts. After you've found your intercepts, then you plot these. The plane that passes through all three points is your graph of the equation.
Did you know… We have over 200 college
courses that prepare you to earn
credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the
first two years of college and save thousands off your degree. Anyone can earn
credit-by-exam regardless of age or education level.