equation of plane

The position of a plane τ can be fixed by giving the position vector OQ of the projectionPlanetmathPlanetmath point Q of the origin on the plane.

Let the length of the position vector be r and the angles formed by the vector with the positive coordinateMathworldPlanetmathPlanetmath axes α, β, γ. Let  P=(x,y,z)  be an arbitrary point. Then P is in the plane τ iff its projection on the line OQ coincides with Q, i.e. iff (http://planetmath.org/Iff) the projection of the coordinate way of P is r. This may be expressed as the equation  xcosα+ycosβ+zcosγ=r  or

xcosα+ycosβ+zcosγ-r=0, (1)

which thus is the equation of the plane.

Conversely, we may show that a first-degree equation

Ax+By+Cz+D=0 (2)

between the variables x, y, z represents always a plane. In fact, we may without hurting generality suppose that  D0. Now  R:=A2+B2+C2>0.  Thus the length of the radius vector (http://planetmath.org/PositionVector) of the point  (A,B,C)  is R. Let the angles formed by the radius vector with the positive coordinate axes be α, β, γ. Then we can write


(cf. direction cosinesMathworldPlanetmath). Dividing (2) termwise by R gives us


where  DR0. The last equation represents a plane whose distance from the origin is -DR and whose normal line forms the angles α, β, γ with the coordinate axes.

Since the coefficients A,B,C are proportional to the direction cosines of the normal vectorMathworldPlanetmath of this plane, they are direction numbers of the normal line of the plane.

Examples. The equations of the coordinate planes are
x=0 (yz-plane),  y=0 (zx-plane),  z=0 (xy-plane);
the equation of the plane through the points  (1, 0, 0),  (0, 1, 0)  and  (0, 0, 1)  is

The plane can be represented also in a vectoral form, by using the position vector r0 of a point of the plane and two linearly independentMathworldPlanetmath vectors u and v parallelMathworldPlanetmathPlanetmath to the plane:

r=r0+su+tv. (3)

Here, r means the position vector of arbitrary point of the plane, s and t are real parameters. In the coordinate form, (3) may be e.g.

Title equation of plane
Canonical name EquationOfPlane
Date of creation 2013-03-22 17:28:48
Last modified on 2013-03-22 17:28:48
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Topic
Classification msc 51N20
Related topic DirectionCosines
Related topic SurfaceNormal
Related topic RuledSurface
Related topic AnalyticGeometry
Related topic AngleBetweenLineAndPlane
Related topic IntersectionOfQuadraticSurfaceAndPlane