normal of plane
A line is a normal of a plane , if it intersects the plane and is perpendicular to all lines passing through the intersection point in the plane. Then the plane is a normal plane of the line . The normal plane passing through the midpoint (http://planetmath.org/Midpoint3) of a line segment is the center normal plane of the segment.
There is the
Theorem. If a line () a plane () and is perpendicular to two distinct lines ( and ) passing through the cutting point () in the plane, then the line is a normal of the plane.
Proof. Let be an arbitrary line passing through the point in the plane . We need to show that . Set another line of the plane cutting the lines , and at the points , and , respectively. Separate from the equally segments and . Then
Thus the segments and , being corresponding parts of two congruent triangles, are equally long. I.e., the point is equidistant from the end points of the segment , and it must be on the perpendicular bisector (http://planetmath.org/CenterNormal) of . Therefore , i.e. .
Proposition 2. All normal planes of a line are parallel. If a plane is parallel to a normal plane of a line, then also it is a normal plane of the line.
|Title||normal of plane|
|Date of creation||2013-04-19 15:03:42|
|Last modified on||2013-04-19 15:03:42|
|Last modified by||pahio (2872)|
|Defines||center normal plane|