equation of plane
The position of a plane can be fixed by giving the position vector of the projection point of the origin on the plane.
Let the length of the position vector be and the angles formed by the vector with the positive coordinate axes , , . Let be an arbitrary point. Then is in the plane iff its projection on the line coincides with , i.e. iff (http://planetmath.org/Iff) the projection of the coordinate way of is . This may be expressed as the equation or
(1) |
which thus is the equation of the plane.
Conversely, we may show that a first-degree equation
(2) |
between the variables , , represents always a plane. In fact, we may without hurting generality suppose that . Now . Thus the length of the radius vector (http://planetmath.org/PositionVector) of the point is . Let the angles formed by the radius vector with the positive coordinate axes be , , . Then we can write
(cf. direction cosines). Dividing (2) termwise by gives us
where . The last equation represents a plane whose distance from the origin is and whose normal line forms the angles , , with the coordinate axes.
Since the coefficients are proportional to the direction cosines of the normal vector of this plane, they are direction numbers of the normal line of the plane.
Examples. The equations of the coordinate planes are
(-plane), (-plane), (-plane);
the equation of the plane through the points , and is
.
The plane can be represented also in a vectoral form, by using the position vector of a point of the plane and two linearly independent vectors and parallel to the plane:
(3) |
Here, means the position vector of arbitrary point of the plane, and 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 |