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normal of plane
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(Theorem)
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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 of a line segment is the center normal plane
of the segment.
There is the
Theorem. If a line ($l$ ) cuts 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 long line 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 of $PQ$ . Therefore $AL \perp PQ$ , i.e.
$a \perp l$ .
Proposition 1. All normals 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.
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"normal of plane" is owned by pahio.
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Cross-references: parallel, proposition, triangles, congruent, end points, center normal, proof, theorem, segment, line segment, point, passing through, intersects, line, surface normal, plane, normal line, perpendicular
There are 4 references to this entry.
This is version 13 of normal of plane, born on 2009-02-02, modified 2009-02-15.
Object id is 11595, canonical name is NormalOfPlane.
Accessed 2090 times total.
Classification:
| AMS MSC: | 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries) |
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Pending Errata and Addenda
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