Planes (in SPAAAAACE!)

Links and useful resources

• gr10's Geometry >>>START HERE<<<
• AoPS Online Textbook
• AoPS Alcumus
• Big Ideas Geometry textbook
• GeoGebra Online Geometry Constuction Tool
• Two-column math templates
• Proofs unit slides from mathgiraffe.com
• Proofs unit printables for two-column proofs

Concept summary and connections

• 3-dimensional geometry
• parallel planes
• parallel lines
• skew lines
• 3-point plane
• intersection of planes
• equation of a plane

Lesson and worked examples

What is a plane? A plane is a flat two-dimensional entity that continues forever. Note: that's not a definition, it's just intuition! We'll get to an actual definition later...

This lesson is basically just going to exercise your intuition about planes, so here are some things to think about:

• If you have a plane, and a point "floating" above it so that it never touches the plane, how would you measure the distance between them?
• If you have a plane, and a line "floating" above it so that it never touches the plane, how would you measure the distance between them? It needs to be a unique number or it has no meaning...
• Is it possible to have two planes that have a "distance" between them? If so, how would you measure that?
• What happens when two planes intersect each other? What kind of a geometric thing does the intersection make?

Definitions of a plane

Remember the equation of a line: $Ax+By=C$. We have order-one variables, which are summed and add up to a constant. In 2d, this is a sort of definition of "flatness." The plane in 3d is similar:

Now we have three linear terms adding to a constant. In 3d, the plane is what we think of as "flat." There are a lot of ways to define a plane, which are useful in different contexts:

• The plane is then the set of all $(x,y,z)$ points that satisfy the plane equation for a particular assignment of $\{A,B,C,D\}$ coefficients.
• If you have three points that aren't collinear, they always form a triangle. That triangle is part of a plane. The plane can be defined to be the set of all points that make the same plane when you replace any of the first three with the new one.
• A plane is the set of all points that you can get by adding linear combinations of two independent vectors to a starting point.
• Given a point and a line through the point, a plane is the set of all points that, when used to make another line through the point, make a line that is perpendicular to the starting line.

Media resources

• Youtube search for "3-dimensional geometry"
• Youtube search for "parallel planes"
• Youtube search for "parallel lines"
• Youtube search for "skew lines"
• Youtube search for "3-point plane"
• Youtube search for "intersection of planes"

Guided practice

• Given two points, how many different planes pass through the two points?
• Given three points, is it always possible to find a plane that passes through all three points?
• Given any four points, is it always possible to find a plane that passes through all four points?
• Given any two intersecting lines, is there always a plane that contains both lines?

Homework

• #hw (geom-gr10) 14.1 Planes (p. 356) (gr10) 2025-03-03