Video: Finding the Planes That Pass through Given Points

Find three planes that pass through both of the points 𝐡 and 𝐢.

03:10

Video Transcript

Find three planes that pass through both of the points 𝐡 and 𝐢.

A plane is a space that extends infinitely in all directions and is a set of all points in three dimensions. Here’s an example of a plane. We will call this plane plane 𝐾𝐿𝑀. We label a plane by three points that are found not on the same line. Let’s consider the shape we were given.

We need a plane that passes through point 𝐡 and 𝐢. Since we know that a plane needs to be named by three points, we can choose a third point like point 𝐷. And we would say that there is a plane that includes point, 𝐡, 𝐢, and 𝐷. This plane would include the face that is the bottom of this rectangular prism. It would include the face 𝐴𝐡𝐢𝐷.

But our goal is to find three different planes. So we need to consider another third point that would form a plane with the points 𝐡 and 𝐢. We could try 𝐡, 𝐢, and 𝐢 prime. They lie on the same plane. We could label this plane as plane 𝐡𝐢𝐢 prime. It’s the plane that includes the front face of this rectangular prism, the face 𝐡𝐢𝐢 prime 𝐡 prime.

What about a third plane? Again, we’ll need the points 𝐡 and 𝐢. This one might not seem as immediately obvious. But 𝐡, 𝐢, and 𝐷 prime also fall in a plane. You can kind of imagine that this plane runs as a diagonal through our rectangular prism. We will call this one 𝐡𝐢𝐷 prime. And so we have a list of three planes that pass through the points 𝐡 and 𝐢.

Now let’s go back to the plane 𝐡𝐢𝐷. That’s the yellow plane that includes the base of the rectangular prism. Remember that we just need three points to name this plane. And so we could also call it 𝐡𝐢𝐴 or 𝐷𝐴𝐡. If we named it 𝐷𝐴𝐡, it would still be a correct answer, as the plane 𝐷𝐴𝐡 includes the points 𝐡 and 𝐢.

In fact, this is true for all of our planes. We could name them in many different ways. We could name the plane that includes the front face 𝐡𝐡 prime 𝐢 prime. We could name the diagonal plane that cuts through our prism 𝐴 prime 𝐷 prime 𝐢. So long as we include three points from that plane.

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