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Determine intersections of two planes

The intersection of two planes in three-dimensional space is determined by solving their simultaneous linear equations: it is a line when the planes are distinct and nonparallel, empty when they are parallel and distinct, or the entire plane when their equations describe the same plane. The line may be represented parametrically, with a direction related to the planes’ normal vectors; this scope is limited to Cartesian three-dimensional geometry and does not include higher-dimensional or projective generalizations.

Detailed Explanation: Determine intersections of two planes

To find the intersection of two planes, solve their equations simultaneously. The solutions are all the points that lie on both planes.

  • If there is one free variable, the intersection is a line.
  • If the equations are inconsistent, the planes are parallel and distinct, so there is no intersection.
  • If the equations are equivalent, the planes are the same plane.

Example

Find the intersection of

Π1:x+y+z=6\Pi_1: x+y+z=6

and

Π2:2x−y+z=3.\Pi_2: 2x-y+z=3.

We have two equations with three variables, so we expect one free variable. Let

z=t.z=t.

Substitute z=tz=t into both equations:

x+y=6−tx+y=6-t

and

2x−y=3−t.2x-y=3-t.

Add the equations to eliminate yy:

3x=9−2t,3x=9-2t,

so

x=3−23t.x=3-\frac{2}{3}t.

Now substitute this into x+y=6−tx+y=6-t:

(3−23t)+y=6−t.\left(3-\frac{2}{3}t\right)+y=6-t.

Therefore,

y=3−13t.y=3-\frac{1}{3}t.

Thus the intersection consists of all points

(3−23t,  3−13t,  t).\left(3-\frac{2}{3}t,\;3-\frac{1}{3}t,\;t\right).

To avoid fractions, let t=3st=3s. Then

(x,y,z)=(3,3,0)+s(−2,−1,3).\boxed{(x,y,z)=(3,3,0)+s(-2,-1,3)}.

So the planes intersect in the line

x=3−2s,y=3−s,z=3s,s∈R.\boxed{ \begin{aligned} x&=3-2s,\\ y&=3-s,\\ z&=3s, \end{aligned} \qquad s\in\mathbb R.}

The direction vector of the line is ⟨−2,−1,3⟩\langle -2,-1,3\rangle. It is perpendicular to both planes’ normal vectors, ⟨1,1,1⟩\langle1,1,1\rangle and ⟨2,−1,1⟩\langle2,-1,1\rangle, as expected for a line lying in both planes.

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    3D Space - Plane Relationships - Two Scalar Equations to Intersection Line Vector Equation

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