# 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 $a\u27c2l$. 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 |