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Vector Equation of a Plane

A plane is determined by a point and a normal vector. All points r on the plane satisfy (r − r₀)·n = 0. Point-normal, general, and intercept forms.

Concept Fundamentals
a(x−x₀)+b(y−y₀)+c(z−z₀)=0
Point-normal
Ax+By+Cz+D=0
General

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Plane through origin has D=0 in general form. Distance from point to plane uses the normal vector. Two non-parallel planes intersect in a line.

Key quantities
a(x−x₀)+b(y−y₀)+c(z−z₀)=0
Point-normal
Key relation
Ax+By+Cz+D=0
General
Key relation

Ready to run the numbers?

Why: Plane equations are used in collision detection, clipping, and 3D graphics.

How: Given point r₀ and normal n=(a,b,c), the plane is a(x−x₀)+b(y−y₀)+c(z−z₀)=0. Expand to get general form Ax+By+Cz+D=0.

Plane through origin has D=0 in general form.Distance from point to plane uses the normal vector.
Sources:Wikipedia - Plane (geometry)

Run the calculator when you are ready.

Start CalculatingEnter a point on the plane and the normal vector.

Sample Examples

Input

Point on Plane

Normal Vector

Results

Point-normal: 1(x-1) + 1(y-2) + 1(z-3) = 0
General: 1x + 1y + 1z + -6 = 0
Intercept: x/6 + y/6 + z/6 = 1

Step-by-Step

Point r₀ = (1, 2, 3)

Normal n = (1, 1, 1)

Point-normal: a(x-x₀) + b(y-y₀) + c(z-z₀) = 0

General: Ax + By + Cz + D = 0, D = -d = -6

Intercept form: x/6 + y/6 + z/6 = 1

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

Planes are used in collision detection and clipping.

— Graphics

🔬

SVM uses hyperplanes (n-dimensional planes).

— ML

Key Takeaways

  • Point-normal form: a(x−x0)+b(y−y0)+c(z−z0)=0a(x-x_0) + b(y-y_0) + c(z-z_0) = 0.
  • General form: Ax+By+Cz+D=0Ax + By + Cz + D = 0 with D=−(Ax0+By0+Cz0)D = -(Ax_0+By_0+Cz_0).
  • Intercept form: fracxp+fracyq+fraczr=1\\frac{x}{p} + \\frac{y}{q} + \\frac{z}{r} = 1 when plane has non-zero intercepts.
  • Normal vector must be non-zero.
  • Parallel planes have proportional normals.

Did You Know?

Planes are used in collision detection and clipping.
Support vector machines use hyperplanes (n-dimensional planes).
Plane through origin has D=0 in general form.
Distance from point to plane uses the normal vector.
Two non-parallel planes intersect in a line.
CAD uses plane equations for surface modeling.

Understanding

A plane is determined by a point and a normal vector. All points r on the plane satisfy (vecr−vecr0)cdotvecn=0(\\vec{r} - \\vec{r}_0) \\cdot \\vec{n} = 0.

(vecr−vecr0)cdotvecn=0(\\vec{r} - \\vec{r}_0) \\cdot \\vec{n} = 0
ax+by+cz=d,quadd=vecncdotvecr0ax + by + cz = d,\\quad d = \\vec{n} \\cdot \\vec{r}_0
Ax+By+Cz+D=0,quadD=−(Ax0+By0+Cz0)Ax + By + Cz + D = 0,\\quad D = -(Ax_0+By_0+Cz_0)

Expert Tips

Normalize n for unit normal; simplifies distance formulas.
Plane through origin: use point (0,0,0) or D=0.
Check n ≠ 0 before computing.
Dot product of normals gives angle between planes.

FAQ

Q: What if normal is zero?
A: Not a valid plane.
Q: How to find plane through 3 points?
A: n = (P₂-P₁)×(P₃-P₁), use one point as r₀.
Q: Distance from point to plane?
A: d=frac∣Ax+By+Cz+D∣sqrtA2+B2+C2d = \\frac{|Ax+By+Cz+D|}{\\sqrt{A^2+B^2+C^2}}.
Q: When is plane parallel to axis?
A: When corresponding normal component is zero.
Q: Relation to line-plane intersection?
A: Substitute line parametric into plane equation, solve for t.
Q: Applications?
A: Graphics, physics, robotics, machine learning.
Q: Can D be zero?
A: Yes, when plane passes through origin.

How to Use

  1. Enter point on plane (x₀, y₀, z₀).
  2. Enter normal vector (a, b, c).
  3. View point-normal, general, and intercept forms.

Disclaimer

Normal vector must be non-zero. Intercept form is only valid when plane does not pass through origin and has non-zero intercepts on all axes.

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