The Vector Equation of a Plane in Space
Simple Explanation
A plane through a known point (position vector r₀), with normal vector n, is described by the vector equation n·(r−r₀) = 0 — every point r in the plane forms a vector (r−r₀) that is perpendicular to n.
Why Do We Need It?
This is the vector-notation version of the point-normal plane equation from the previous chapter — the same idea, expressed compactly using the dot product.
Formula
The Vector Equation of a Plane
n · (r − r₀) = 0
A plane through the point with position vector r₀, with normal vector n, is described by this equation — it says every point r in the plane forms a vector (r−r₀) perpendicular to n.
- n
- — the normal vector to the plane
- r₀
- — the position vector of a known point on the plane
- r
- — the position vector of a general point on the plane
When to use it: Whenever a plane needs to be described using vector notation.
Worked Example
Write a plane's vector equation, then convert to component form
Write the vector equation of the plane through r₀=(1,0,2) with normal vector n=(3,1,−2), then convert to component (ax+by+cz=d) form.
Why Does This Work?
n·(r−r₀)=0 literally states that (r−r₀), the vector from the known point to any other point on the plane, is perpendicular to the normal — expanding this dot product using components gives exactly the same a(x−x₀)+b(y−y₀)+c(z−z₀)=0 equation as before, just derived using vector notation this time.
Real-Life Example
Programming a flat virtual surface in a 3D engine
A game engine needs a compact, efficient mathematical test for whether a point lies on a flat surface, given the surface's normal and one known point.
The vector equation n·(r−r₀)=0 is exactly the compact test used — it evaluates to zero precisely for points on the plane.
Practice
For a plane through r₀=(2,1,0) with normal n=(1,1,1), find d in the equation x+y+z=d.
MediumCommon mistake
Treating this as a completely new, unrelated technique — it is really the same point-normal plane equation from before, just written compactly with the dot product.
Quick Review
- n·(r−r₀) = 0.
- Expands directly into the familiar a(x−x₀)+b(y−y₀)+c(z−z₀)=0 form.
- n is the normal vector; r₀ is a known point on the plane.