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

Find three planes that pass through both of the points ๐ด and ๐ต.

03:14

### Video Transcript

Find three planes that pass through both of the points ๐ด and ๐ต.

Letโs observe on the diagram where the points ๐ด and ๐ต lie. We can say that the planes that pass through both points ๐ด and ๐ต will be the planes that pass through the line ๐ด๐ต. We should recall that there exists exactly one plane through any three noncollinear points. Notice how we use the word โnoncollinear.โ 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 ๐ด and ๐ต, we need to find another point which doesnโt lie on this line, which could define a plane. Letโs take the point ๐ถ and visualize the plane that would pass through ๐ด, ๐ต, and ๐ถ. 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 ๐ด๐ต๐ถ. However, the plane ๐ด๐ต๐ท would also work.

Now, letโs see if we can find another plane which passes through the points ๐ด and ๐ต. We can do this by finding another point. So letโs use the point ๐ต prime and see if this would create a plane. And since we know that ๐ต prime is not collinear with ๐ด and ๐ต, then we can create another plane. We can define this plane as the plane ๐ด๐ต๐ต prime.

We know that we are looking for three planes passing through ๐ด and ๐ต. So letโs see if we can find the third plane. It might be very tempting to think that, well, weโve found a plane on the base of this prism. Weโ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 ๐ด, ๐ท, ๐ท prime, and ๐ด prime, but it doesnโt contain the point ๐ต. Even if we visualize this plane at the front of the prism, it would have the points ๐ต, ๐ถ, ๐ถ prime, and ๐ต prime but not the point ๐ด. This plane would not contain both of the points ๐ด and ๐ต. In fact, the third plane containing the points ๐ด and ๐ต will look something like this. As it contains the point ๐ถ prime, we can define it as the plane ๐ด๐ต๐ถ prime.

And so we have the answer that three planes passing through the points ๐ด and ๐ต are the planes ๐ด๐ต๐ถ, ๐ด๐ต๐ต prime, and ๐ด๐ต๐ถ prime.