equation of 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
which thus is the equation of the plane.
Conversely, we may show that a first-degree equation
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
Examples. The equations of the coordinate planes are
(-plane), (-plane), (-plane);
the equation of the plane through the points , and is
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|
|Date of creation||2013-03-22 17:28:48|
|Last modified on||2013-03-22 17:28:48|
|Last modified by||pahio (2872)|