# normal of plane

The perpendicular or normal line of a plane is a special case of the surface normal, but may be defined separately as follows:

A line $l$ is a normal of a plane $\pi$, if it intersects the plane and is perpendicular to all lines passing through the intersection point in the plane.  Then the plane $\pi$ is a of the line $l$.  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

If a line ($l$) a plane ($\pi$) and is perpendicular to two distinct lines ($m$ and $n$) passing through the cutting point ($L$) in the plane, then the line is a normal of the plane.

Proof.  Let $a$ be an arbitrary line passing through the point $L$ in the plane $\pi$.  We need to show that  $a\perp l$.  Set another line of the plane cutting the lines $m$, $n$ and $a$ at the points $M$, $N$ and $A$, respectively.  Separate from $l$ the equally segments $LP$ and $LQ$.  Then

 $PM\;=\;QM\quad\mbox{and}\quad PN\;=\;QN,$

since any point of the center normal of a line segment ($PQ$) is equidistant from the end points of the segment.  Consequently,

 $\Delta MNP\cong\Delta MNQ\quad(\mbox{SSS}).$

Thus the segments $PA$ and $QA$, being corresponding parts of two congruent triangles, are equally long.  I.e., the point $A$ is equidistant from the end points of the segment $PQ$, and it must be on the perpendicular bisector (http://planetmath.org/CenterNormal) of $PQ$.  Therefore  $AL\bot PQ$, i.e.  $a\bot l$.

All of a plane are parallel.  If a line is parallel to a normal of a plane, then it is a normal of the plane, too.

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 Canonical name NormalOfPlane Date of creation 2013-04-19 15:03:42 Last modified on 2013-04-19 15:03:42 Owner pahio (2872) Last modified by pahio (2872) Numerical id 17 Author pahio (2872) Entry type Theorem Classification msc 51M04 Synonym plane normal Related topic AngleBetweenLineAndPlane Related topic NormalLine Related topic NormalVector Related topic Congruence Related topic ParallelAndPerpendicularPlanes Related topic ParallelismOfTwoPlanes Defines normal plane Defines center normal plane