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Question Video: Finding the Planes That Pass through Given Points Mathematics • 11th Grade

Find three planes that pass through both of the points 𝐴 and 𝐵.

03:14

Video Transcript

Find three planes that pass through both of the points 𝐴 and 𝐵.

Let’s observe on the diagram where the points 𝐴 and 𝐵 lie. We can say that the planes that pass through both points 𝐴 and 𝐵 will be the planes that pass through the line 𝐴𝐵. We should recall that there exists exactly one plane through any three noncollinear points. Notice how we use the word “noncollinear.” If the three points were collinear, they would lie on a line. And there would be an infinite number of planes passing through three collinear points.

So, in order to find a plane passing through 𝐴 and 𝐵, we need to find another point which doesn’t lie on this line, which could define a plane. Let’s take the point 𝐶 and visualize the plane that would pass through 𝐴, 𝐵, and 𝐶. It would look something like this. And so one of the planes is actually one which contains one of the faces of this prism. One way in which we can define a plane is by using three points that we know lie on the plane. So we could call this plane the plane 𝐴𝐵𝐶. However, the plane 𝐴𝐵𝐷 would also work.

Now, let’s see if we can find another plane which passes through the points 𝐴 and 𝐵. We can do this by finding another point. So let’s use the point 𝐵 prime and see if this would create a plane. And since we know that 𝐵 prime is not collinear with 𝐴 and 𝐵, then we can create another plane. We can define this plane as the plane 𝐴𝐵𝐵 prime.

We know that we are looking for three planes passing through 𝐴 and 𝐵. So let’s see if we can find the third plane. It might be very tempting to think that, well, we’ve found a plane on the base of this prism. We’ve found a plane on the side of this prism. So maybe there might be a plane at the back of this prism. If we visualized a plane such as this, it might contain the points 𝐴, 𝐷, 𝐷 prime, and 𝐴 prime, but it doesn’t contain the point 𝐵. Even if we visualize this plane at the front of the prism, it would have the points 𝐵, 𝐶, 𝐶 prime, and 𝐵 prime but not the point 𝐴. This plane would not contain both of the points 𝐴 and 𝐵. In fact, the third plane containing the points 𝐴 and 𝐵 will look something like this. As it contains the point 𝐶 prime, we can define it as the plane 𝐴𝐵𝐶 prime.

And so we have the answer that three planes passing through the points 𝐴 and 𝐵 are the planes 𝐴𝐵𝐶, 𝐴𝐵𝐵 prime, and 𝐴𝐵𝐶 prime.

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