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

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


Video Transcript

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

A plane is a space that extends infinitely in all directions and is a set of all points in three dimensions. Hereโ€™s an example of a plane. We will call this plane plane ๐พ๐ฟ๐‘€. We label a plane by three points that are found not on the same line. Letโ€™s consider the shape we were given.

We need a plane that passes through point ๐ต and ๐ถ. Since we know that a plane needs to be named by three points, we can choose a third point like point ๐ท. And we would say that there is a plane that includes point, ๐ต, ๐ถ, and ๐ท. This plane would include the face that is the bottom of this rectangular prism. It would include the face ๐ด๐ต๐ถ๐ท.

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 ๐ต and ๐ถ. We could try ๐ต, ๐ถ, and ๐ถ prime. They lie on the same plane. We could label this plane as plane ๐ต๐ถ๐ถ prime. Itโ€™s the plane that includes the front face of this rectangular prism, the face ๐ต๐ถ๐ถ prime ๐ต prime.

What about a third plane? Again, weโ€™ll need the points ๐ต and ๐ถ. This one might not seem as immediately obvious. But ๐ต, ๐ถ, and ๐ท 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 ๐ต๐ถ๐ท prime. And so we have a list of three planes that pass through the points ๐ต and ๐ถ.

Now letโ€™s go back to the plane ๐ต๐ถ๐ท. Thatโ€™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 ๐ต๐ถ๐ด or ๐ท๐ด๐ต. If we named it ๐ท๐ด๐ต, it would still be a correct answer, as the plane ๐ท๐ด๐ต includes the points ๐ต and ๐ถ.

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 ๐ต๐ต prime ๐ถ prime. We could name the diagonal plane that cuts through our prism ๐ด prime ๐ท prime ๐ถ. So long as we include three points from that plane.

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