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 normal plane 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

Theorem.  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  $a⟂l$.  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

$$PM=QM\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}PN=QN,$$

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

$$\mathrm{\Delta }MNP\cong \mathrm{\Delta }MNQ\mathit{\hspace{1em}}(\text{SSS}).$$

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  $AL\perp PQ$, i.e.  $a\perp l$.

Proposition 1.  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