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Equation of a Plane: Concepts, Formulas and Examples

MathematicsTry the interactive 3D model →

A plane in 3D is fixed by one point on it and a vector perpendicular to it, called the normal vector. If the point is (x1, y1, z1) and the normal is n = (a, b, c), the plane equation is a(x − x1) + b(y − y1) + c(z − z1) = 0.

Expanding gives the general form ax + by + cz + d = 0, where (a, b, c) is still the normal vector. This one equation is the starting point for everything about planes: intersections, angles and distances.

Key Concepts

  • Normal vector: n = (a, b, c) is perpendicular to every direction lying in the plane. The coefficients of x, y, z in the plane equation ARE the normal's components.
  • Point–normal form: a(x − x1) + b(y − y1) + c(z − z1) = 0, which comes from the condition n · (r − r1) = 0.
  • General form: ax + by + cz + d = 0, where d = −(ax1 + by1 + cz1).
  • Three-point form: a plane through three non-collinear points P, Q, R has normal n = PQ × PR.
  • Scaling: multiplying the whole equation by any non-zero constant gives the same plane.
  • Special cases: if d = 0 the plane passes through the origin; if a = 0 the plane is parallel to the x-axis.

Formulas

  • Vector form: (r − a) · n = 0, or r · n = d.
  • Cartesian: ax + by + cz + d = 0.
  • Normal from three points P, Q, R: n = (Q − P) × (R − P).
  • Two planes are parallel when their normals are proportional, and perpendicular when n1 · n2 = 0.

Worked Example

Find the equation of the plane through (1, 0, 2) with normal (3, −1, 2).

  1. Point–normal form: 3(x − 1) − (y − 0) + 2(z − 2) = 0.
  2. Expand: 3x − 3 − y + 2z − 4 = 0, which simplifies to 3x − y + 2z = 7.
  3. Verify the point: 3(1) − 0 + 2(2) = 3 + 4 = 7. Correct.
  4. Check the normal: pick another point on the plane, for example x = 0, y = 0 gives z = 7/2, so Q = (0, 0, 3.5). The in-plane direction PQ = (1, 0, −1.5).
  5. Dot with the normal: 3(1) + (−1)(0) + 2(−1.5) = 3 − 3 = 0, confirming the normal is perpendicular to the plane. Correct.

Common Misconceptions

  • (a, b, c) is a point on the plane. No — (a, b, c) is the normal vector. Points on the plane are the (x, y, z) triples satisfying the equation.
  • The equation of a plane is unique. Multiplying by any non-zero constant gives an equivalent equation of the same plane.
  • ax + by + cz + d = 0 always describes a plane. Only when (a, b, c) is not (0, 0, 0); if all three coefficients are zero it is not a plane equation.

Real-World Applications

  • 3D graphics: clipping planes and polygon surfaces in rendering.
  • Construction: floors, walls and roof planes in CAD models.
  • Physics: equipotential surfaces and wavefronts are often planar.
  • Surveying: fitting a plane to terrain measurements.

A plane is one point plus a normal vector: a(x − x1) + b(y − y1) + c(z − z1) = 0, or ax + by + cz + d = 0. Read the normal straight off the coefficients, and remember that scaling the equation changes nothing.

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