Specifying planes in three dimensions | Introduction to Euclidean geometry | Geometry | Khan Academy

TL;DR

Planes in three dimensions are flat surfaces that extend infinitely in all directions and are defined by three non-collinear points.

Transcript

We've already been exposed to points and lines. Now let's think about planes. And you can view planes as really a flat surface that exists in three dimensions, that goes off in every direction. So for example, if I have a flat surface like this, and it's not curved, and it just keeps going on and on and on in every direction. Now the question is, h... Read More

Key Insights

• 🫓 A plane is a flat surface that extends infinitely in all directions in three dimensions.
• 😥 One point is not enough to specify a plane, as there are infinitely many planes that can go through that point.
• 🫥 Two points define a line, but both the points and the line exist on multiple planes, so they cannot uniquely define a plane.
• 😥 Three non-collinear points are needed to uniquely define a plane.
• 😥 A plane can be specified using different sets of three non-collinear points.
• 🫥 Points on the same line do not specify a unique plane, as the line can be rotated to intersect with multiple planes.
• 🫥 The three non-collinear points used to define a plane must not be on the same line.

Questions & Answers

Q: What is a plane in three dimensions?

A plane in three dimensions is a flat surface that extends infinitely in all directions.

Q: Can a plane be specified with only one point?

No, one point is not enough to specify a plane. There are an infinite number of planes that can go through a single point.

Q: How are two points related to a plane?

Two points define a line, and both the points and the line exist on multiple planes. Two points alone are not enough to uniquely define a plane.

Q: How many non-collinear points are needed to specify a plane?

Three non-collinear points are needed to uniquely define a plane, as they do not all lie on the same line.

Summary & Key Takeaways

• A plane is a flat surface in three dimensions that extends infinitely in all directions.

• One point is not enough to specify a plane, as there are infinitely many planes that can go through that point.

• Two points define a line, but both points and the line exist on multiple planes. Three non-collinear points are needed to uniquely define a plane.