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Lines & Planes in 3D

Learn the core relationships of 3D geometry: parallelism, perpendicularity, spatial angles, and distances—powered by the vector method.

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Quick Facts
Criteria and formulas you should be able to quote instantly.

Line ∥ Plane

Line outside plane and ∥ a line in the plane

Line ⟂ Plane

Perpendicular to two intersecting lines in the plane

Plane ⟂ Plane

Normals perpendicular: n1·n2=0

Line–Plane angle

sin θ = |u·n|/(|u||n|)

Plane–Plane angle

cos θ = |n1·n2|/(|n1||n2|) (acute)

Spatial Angle Calculator (Interactive)
Compute vector–vector, line–plane, and plane–plane angles using dot products.

Vector u

Vector v

Choose a mode, enter vectors, then click Calculate.

3D Vector Calculator (Interactive)
Compute dot/cross products, magnitudes, and angles between 3D vectors with step-by-step work.

Vector a

Vector b

Enter vectors a and b, then click Calculate.

Core Knowledge

1) Parallel Relationships
Line–line, line–plane, and plane–plane parallelism.

Line ∥ Plane (criterion)

If a line outside a plane is parallel to some line contained in the plane, then it is parallel to the plane.

l∥α ⟸ ∃ s⊂α such that l∥sl\parallel \alpha\ \Longleftarrow\ \exists\ s\subset\alpha\text{ such that }l\parallel s

Plane ∥ Plane (criterion)

If a plane contains two intersecting lines that are each parallel to another plane, the two planes are parallel.

If a⊂α, b⊂α, a∩b≠∅, a∥β, b∥β, then α∥β.\text{If }a\subset\alpha,\ b\subset\alpha,\ a\cap b\neq\varnothing,\ a\parallel\beta,\ b\parallel\beta,\ \text{then }\alpha\parallel\beta.

Parallelism (concept diagram)

xyplane αlm

If a line outside a plane is parallel to a line inside the plane, then it is parallel to the plane.

Proof template

Try to produce a line in the plane that is parallel to your target line. Usually you do this by finding a pair of similar triangles or using midpoint/parallel-line theorems.

2) Perpendicular Relationships
Line–plane and plane–plane perpendicularity criteria.

Line ⟂ Plane (criterion)

If a line is perpendicular to two intersecting lines in a plane, then it is perpendicular to the plane.

(l⊥a ∧ l⊥b ∧ a∩b≠∅)⇒l⊥α\left(l\perp a\ \wedge\ l\perp b\ \wedge\ a\cap b\neq\varnothing\right)\Rightarrow l\perp \alpha

Plane ⟂ Plane (normals)

In the vector method, use normals: planes are perpendicular if their normals are perpendicular.

α⊥β ⟺ n⃗α⋅n⃗β=0\alpha\perp\beta\ \Longleftrightarrow\ \vec n_\alpha\cdot \vec n_\beta=0

Line ⟂ Plane (concept diagram)

xyplane αPH

If a line is perpendicular to two intersecting lines in a plane, then it is perpendicular to the plane.

Common “bridge” move

If the goal is line ⟂ plane, you often prove the line is perpendicular to two carefully chosen intersecting lines in the plane (typically edges/diagonals).

3) Spatial Angles
Line–plane angles and dihedral angles via dot products.

Angle between a line and a plane

If a line has direction u and the plane has normal n, then:

sin⁡θ=∣u⃗⋅n⃗∣∣u⃗∣∣n⃗∣\sin\theta=\frac{|\vec u\cdot \vec n|}{|\vec u||\vec n|}

Dihedral angle between planes

Use normals n1 and n2:

cos⁡θ=∣n⃗1⋅n⃗2∣∣n⃗1∣∣n⃗2∣\cos\theta=\frac{|\vec n_1\cdot \vec n_2|}{|\vec n_1||\vec n_2|}

Absolute value gives the acute angle (common convention).

Dihedral angle (concept diagram)

xyhinge lineplane αplane β

A dihedral angle is the angle between two planes along their intersection line (often taken as the acute angle).

Fast setup

If the problem gives a clear “right” structure (perpendicular edges/faces), choose axes along those edges. Normals become simple coordinate vectors, and dot products become one-line computations.

4) Distances in 3D
Point-to-plane distance and why normals matter.

Point-to-plane distance

For plane Ax + By + Cz + D = 0 and point (x0,y0,z0):

d=∣Ax0+By0+Cz0+D∣A2+B2+C2d=\frac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}

Geometric meaning

The denominator is the norm of the normal vector (A,B,C). The formula is “signed distance divided by normal length.”

Point-to-plane distance (concept diagram)

xyplane αPHd

The distance from a point to a plane is measured along the perpendicular segment.

5) Vector Method (A Practical Checklist)
How to build a coordinate system and not get lost.

Checklist

  1. Choose an origin at a “corner” point with many known perpendicular edges.
  2. Align axes with perpendicular edges/lines when possible.
  3. Write direction vectors for lines, and normal vectors for planes.
  4. Use dot products for angles; use distance formulas for point-to-plane.
  5. Keep algebra symbolic as long as possible; plug numbers near the end.

Normal vectors in practice

For plane through points A,B,C, a normal is:

n⃗=AB→×AC→\vec n=\overrightarrow{AB}\times \overrightarrow{AC}

Then plane–plane angles and line–plane angles become dot products.

6) Common Pitfalls
Where solutions most often break.
  • Mixing up line–plane angle with line–normal angle. (They are complementary.)
  • Using a non-normal vector as a “normal” (always verify orthogonality to two independent directions in the plane).
  • Forgetting absolute values in plane-angle formulas when the convention uses the acute angle.
  • Coordinate choices that do not satisfy given lengths/angles (always check quickly).
Practice Quiz (10 Questions)
Reinforce theorems and vector formulas with fast recall questions.

Practice Quiz

10 questions
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