Course: Calculus III
Topic: Vectors and Geometry of Space
Subtopic: Planes


A plane in space is the simplest of 3D surfaces, infinitely large and flat, easy to describe based on where it intersects the three coordinate axes. Our analysis will include the intersection of a plane and a line, or that of two non-parallel planes.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: plane in R3, planes spanned by axes (ie. xy-plane, xz-plane, yz-plane)

Formulae to have in your notes: equation of a plane (scalar and vector forms) and distance between a point and a line

Equation of a Plane
through point `P_o (x_o,y_o,z_o)` normal to vector `vecn=<<a,b,c>>`

Scalar form
(a.k.a. Component form)



`ax+by+cz=d` where `d=ax_o + by_o + cz_o`

Vector form

`vecn` `vec(P_o P) = 0`

Distance Between a Point and a Line

Distance d between the point Q and the line `vecr = vec(r_o) + t vecv` is

`d = || vecv xx vec(PQ) || / || vecv ||`

where P is any point on the line and `vecv` is a vector parallel to the line.

Supplemental Resources (optional)

Paul's OL Notes - Calc III: Equations of Planes

Patrick JMT Just Math Tutorials:
Finding the Scalar Equation of a Plane
Finding the Point where a Line Intersects a Plane

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down right column to "Vectors in Space". There are lots of related titles in the second half of that list starting with The Equation of a Plane in 3D Using Vectors.