GermanWikipedia - /page/entry.mu

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<title>Lot (Mathematik)</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Lot (Mathematik)</h1>

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Ein Lot ist in der <a href="Geometrie" title="Geometrie">Geometrie</a> eine <a href="Strecke_(Geometrie)" title="Strecke (Geometrie)">Strecke</a> oder <a href="Gerade" title="Gerade">Gerade</a>, die auf einer gegebenen Geraden oder <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> <a href="Orthogonalit%C3%A4t" title="Orthogonalität">senkrecht</a> steht. Je nachdem, ob es sich um eine Gerade oder um eine Strecke handelt, spricht man auch von Lotgerade oder Lotstrecke. Der Schnittpunkt des Lots mit der gegebenen Geraden oder Ebene wird Lotfußpunkt genannt. Das Lot kann auf verschiedene Weisen mit <a href="Konstruktion_mit_Zirkel_und_Lineal" title="Konstruktion mit Zirkel und Lineal">Zirkel und Lineal geometrisch konstruiert</a> werden. Berechnet werden kann es in der <a href="Analytische_Geometrie" title="Analytische Geometrie">analytischen Geometrie</a> mit Hilfe der <a href="Vektorrechnung" class="mw-redirect" title="Vektorrechnung">Vektorrechnung</a> und des <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukts</a>. Die <a href="L%C3%A4nge_(Mathematik)" title="Länge (Mathematik)">Länge</a> der Lotstrecke ist dann gerade der <a href="Abstand" title="Abstand">Abstand</a> (Normalabstand) eines Punkts von der Gerade oder Ebene.

Spezielle Lotgeraden/Lotebenen sind die <a href="Mittelsenkrechte" title="Mittelsenkrechte">Mittelsenkrechte</a> zweier Punkte in der Ebene/Raum.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Eine <a href="Strecke_(Geometrie)" title="Strecke (Geometrie)">Strecke</a> oder <a href="Gerade" title="Gerade">Gerade</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}">

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<annotation encoding="application/x-tex">{\displaystyle l}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"> heißt Lot auf eine Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> oder <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">, wenn

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<annotation encoding="application/x-tex">{\displaystyle l\perp g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bf96fdf3d3c0c94ea85ab98a4663f3f220e3678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.908ex; height:2.509ex;" alt="{\displaystyle l\perp g}" loading="lazy">   bzw.   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\perp E}">

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<annotation encoding="application/x-tex">{\displaystyle l\perp E}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce2de983ce28b8b07a9afe7220e6162e8fe60a8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.567ex; height:2.176ex;" alt="{\displaystyle l\perp E}" loading="lazy"></dd></dl>

gilt, wenn sie also <a href="Orthogonalit%C3%A4t" title="Orthogonalität">senkrecht</a> auf der Geraden oder Ebene steht, also mit ihr einen <a href="Rechter_Winkel" title="Rechter Winkel">rechten Winkel</a> bildet. Der Lotfußpunkt ist dann der Schnittpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\cap g}">

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<annotation encoding="application/x-tex">{\displaystyle l\cap g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2372b09129137c7943b1495659567124ecc6375f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.392ex; height:2.509ex;" alt="{\displaystyle l\cap g}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\cap E}">

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<annotation encoding="application/x-tex">{\displaystyle l\cap E}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b374b2d7aeaf294cb7de7d67c4013248849a04d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.052ex; height:2.176ex;" alt="{\displaystyle l\cap E}" loading="lazy"> des Lots mit der Geraden oder Ebene.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Geometrische_Konstruktionen">Geometrische Konstruktionen</h2></div>

In zwei Dimensionen lässt sich das Lot auf eine Gerade auf einfache Weise <a href="Konstruktion_mit_Zirkel_und_Lineal" title="Konstruktion mit Zirkel und Lineal">mit Zirkel und Lineal konstruieren</a>. Je nachdem, ob ein gegebener Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> auf der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> oder außerhalb liegt, spricht man vom Errichten oder vom Fällen des Lots.

<div class="mw-heading mw-heading3"><h3 id="Errichten_des_Lots">Errichten des Lots</h3></div>

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</style><div class="thumb tright vl-mehrere-bilder" style="width:533px;"><div class="thumbinner"><div class="vl-mehrere-bilder-horizontal" style="width:234px;"><div class="thumbimage"></div><div class="thumbcaption">Errichten eines Lots als <a href="Mittelsenkrechte" title="Mittelsenkrechte">Mittelsenkrechte</a> zweier Punkte.</div></div><div class="vl-mehrere-bilder-horizontal" style="width:285px;"><div class="thumbimage"></div><div class="thumbcaption">Errichten eines Lots mithilfe des <a href="Thaleskreis" class="mw-redirect" title="Thaleskreis">Thaleskreises</a>. Die Position des Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> ist frei wählbar.</div></div><div style="clear:both;"></div>

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Ist ein Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> auf der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> gegeben, dann findet man die Lotgerade durch diesen Punkt wie folgt:

Man sticht den Zirkel in den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> ein und bestimmt durch Ziehen eines <a href="Kreisbogen" title="Kreisbogen">Kreisbogens</a> mit beliebigem Radius zwei Punkte auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> mit gleichem Abstand zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">. Dann vergrößert man den Winkel des Zirkels, sticht ihn jeweils in einen der beiden gefundenen Punkte auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> ein und findet durch Ziehen zweier Kreisbögen einen Punkt (von zwei möglichen) außerhalb der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> mit gleichem Abstand von den beiden Punkten. Die Gerade, die durch diesen Punkt und den gegebenen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> verläuft, ist dann die Lotgerade zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">.

Eine Alternative, auf einer Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> durch den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> mit eingeschränkten Platzverhältnissen ein Lot zu errichten, zeigt das rechte Bild. Man schlägt dazu um einen frei wählbaren Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> einen Kreisbogen mit dem Radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {MP}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {MP}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b951f58617790cde44efffb76a892d359e8a961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.467ex; height:3.009ex;" alt="{\displaystyle {\overline {MP}}}" loading="lazy">, bis er die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> schneidet (bspw. kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> so wählen, dass eine gedachte Linie von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> mit der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> einen Winkel von ca. 45° bildet). Dann zieht man die Gerade von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, bis sie den Kreisbogen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4efa52c8f47f06136f6ebfd1d68c1249aaca39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.506ex; height:2.509ex;" alt="{\displaystyle P'}" loading="lazy"> schneidet. Die abschließende Gerade, die durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4efa52c8f47f06136f6ebfd1d68c1249aaca39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.506ex; height:2.509ex;" alt="{\displaystyle P'}" loading="lazy"> verläuft, ist dann die Lotgerade zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Fällen_des_Lots">Fällen des Lots</h3></div>

<div class="thumb tright vl-mehrere-bilder" style="width:484px;"><div class="thumbinner"><div class="vl-mehrere-bilder-horizontal" style="width:234px;"><div class="thumbimage"></div><div class="thumbcaption">Fällen des Lots</div></div><div class="vl-mehrere-bilder-horizontal" style="width:236px;"><div class="thumbimage"></div><div class="thumbcaption">Alternative Methode zum Fällen des Lots</div></div><div style="clear:both;"></div>

</div></div>

Ist ein Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> außerhalb der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> gegeben, dann findet man das Lot durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> wie folgt: Man sticht den Zirkel in den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> ein und bestimmt durch Ziehen eines Kreisbogens mit hinreichend großem Radius zwei Punkte auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> mit gleichem Abstand von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">. Dann sticht man jeweils in einen der beiden gefundenen Punkte auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> ein und findet durch Ziehen zweier Kreisbögen (mit hinreichend großem Radius) einen weiteren Punkt mit gleichem Abstand von den beiden Punkten. Die Gerade, die durch diesen Punkt und den gegebenen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> verläuft, ist dann die Lotgerade zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> und der Schnittpunkt dieser Lotgeraden mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> ist der Lotfußpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">.

Eine alternative Konstruktion, von einem gegebenen Punkt das Lot auf eine Gerade zu fällen, besteht darin, den Zirkel an zwei beliebigen Punkten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/577d686fc81d1d1eb3ae54e78aeee8957baf6718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{1}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}" loading="lazy"> auf der Geraden einzustechen und jeweils den Kreis, der durch den gegebenen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> verläuft, einzuzeichnen. Diese beiden Kreise schneiden sich dann in einem weiteren Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4efa52c8f47f06136f6ebfd1d68c1249aaca39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.506ex; height:2.509ex;" alt="{\displaystyle P'}" loading="lazy"> außerhalb der Gerade und die Linie die durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef4efa52c8f47f06136f6ebfd1d68c1249aaca39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.506ex; height:2.509ex;" alt="{\displaystyle P'}" loading="lazy"> verläuft, ist dann die Lotgerade durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">. Diese Konstruktion kann auch für <a href="Spiegelung_(Geometrie)" title="Spiegelung (Geometrie)">Spiegelungen</a> benutzt werden.

<div class="mw-heading mw-heading2"><h2 id="Berechnung">Berechnung</h2></div>

<div class="mw-heading mw-heading3"><h3 id="In_der_Ebene">In der Ebene</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Lotgerade,_Fußpunkt">Lotgerade, Fußpunkt</h4></div>

Für einen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\colon {\vec {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\colon {\vec {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dda9233673719f0b83daa22bb834b67dba0db46a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.105ex; height:2.676ex;" alt="{\displaystyle P\colon {\vec {p}}}" loading="lazy"> und eine Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon {\vec {x}}={\vec {a}}+s{\vec {r}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon {\vec {x}}={\vec {a}}+s{\vec {r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35a32ad5349d45159bed4e60af9fe085eecf3f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.962ex; height:2.676ex;" alt="{\displaystyle g\colon {\vec {x}}={\vec {a}}+s{\vec {r}}}" loading="lazy"> in der Ebene hat diejenige Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> (Lotgerade) durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">, die auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> senkrecht steht, die <a href="Normalenform" title="Normalenform">Normalenform</a>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ h\colon ({\vec {x}}-{\vec {p}})\cdot {\vec {r}}=0,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f2887cb641abebbaa1e8dcea1c0831857edf512.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.068ex; height:2.843ex;" alt="{\displaystyle \ h\colon ({\vec {x}}-{\vec {p}})\cdot {\vec {r}}=0,}" loading="lazy"></dd></dl>

denn der Richtungsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"> der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> ist ein <a href="Normalenvektor" title="Normalenvektor">Normalenvektor</a> der Lotgeraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">. Hierbei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"> das <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> zweier Vektoren. Soll der Lotfußpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon {\vec {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\colon {\vec {f}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d07d5b5e7573fb4fbe46c3f5b655fbaef931d9a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.439ex; height:3.343ex;" alt="{\displaystyle F\colon {\vec {f}}}" loading="lazy"> (Schnittpunkt von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g,h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e02d6e92a89cf24b614a0dc77ba984907888a39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.489ex; height:2.509ex;" alt="{\displaystyle g,h}" loading="lazy">) bestimmt werden, setzt man die Parameterdarstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in die Gleichung der Lotgeraden ein, löst nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> auf und setzt das Ergebnis in die Parameterdarstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> ein. Es ergibt sich:

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ F\colon {\vec {f}}={\vec {a}}+{\frac {({\vec {p}}-{\vec {a}})\cdot {\vec {r}}}{{\vec {r}}\cdot {\vec {r}}}}\;{\vec {r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28018f8d9377d78f73b4425676f4322b3bee779e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25ex; height:5.843ex;" alt="{\displaystyle \ F\colon {\vec {f}}={\vec {a}}+{\frac {({\vec {p}}-{\vec {a}})\cdot {\vec {r}}}{{\vec {r}}\cdot {\vec {r}}}}\;{\vec {r}}}" loading="lazy"></dd></dl>

Andere Vorgaben:

a) Falls die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> durch zwei Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/533793cbe3b0ecde32740b1af44ab97ae2c3fe40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.868ex; height:3.176ex;" alt="{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}" loading="lazy"> gegeben ist, kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2a1ccee366cd6620b5217de65db185437abff1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.486ex; height:3.009ex;" alt="{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}" loading="lazy"> setzen.

b) Falls die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> durch die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;y=mx+d\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;y=mx+d\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac7a547d78b7ca3369f3055a83570d383024f939.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.971ex; height:2.509ex;" alt="{\displaystyle \;y=mx+d\;}" loading="lazy"> gegeben ist, hat die Lotgerade durch den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(x_{0},y_{0})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=(x_{0},y_{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a29b6070ac1b7ca23c822ac47b0b1bf4593ab526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.265ex; height:2.843ex;" alt="{\displaystyle P=(x_{0},y_{0})}" loading="lazy"> die <a href="Lineare_Funktion#Orthogonale_Geraden" title="Lineare Funktion">Gleichung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;y=-{\tfrac {1}{m}}(x-x_{0})+y_{0}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;y=-{\tfrac {1}{m}}(x-x_{0})+y_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9958687385decdc31baf00aeab88d315b83a57d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.383ex; height:3.343ex;" alt="{\displaystyle \;y=-{\tfrac {1}{m}}(x-x_{0})+y_{0}}" loading="lazy">. Der Lotfußpunkt ist der Schnittpunkt beider Geraden. Alternativ kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(0,d)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=(0,d)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b56fde14228b250551f4354df736e68fd1b6b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.063ex; height:2.843ex;" alt="{\displaystyle A=(0,d)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}=\textstyle {1 \choose m}}">

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}=\textstyle {1 \choose m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f6ed4f5837a49bf444d1c8426faea561a1c7363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.894ex; height:3.343ex;" alt="{\displaystyle {\vec {r}}=\textstyle {1 \choose m}}" loading="lazy"> setzen und die obige Formel verwenden.

c) Falls die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> durch die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;ax+by=d\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;ax+by=d\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b850fcd26bf59d944888332bc28eacca9e80285.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.158ex; height:2.509ex;" alt="{\displaystyle \;ax+by=d\;}" loading="lazy"> oder in Normalenform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;{\vec {x}}\cdot {\vec {n}}=d\;}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;{\vec {x}}\cdot {\vec {n}}=d\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e7df81d4b8e3c64171101185e5758122b90b30d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.008ex; height:2.343ex;" alt="{\displaystyle \;{\vec {x}}\cdot {\vec {n}}=d\;}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}=\textstyle {a \choose b}}">

<mover>

</mover>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}=\textstyle {a \choose b}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a119b7d19313ec46df4fedcf60aa62e614e12094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.493ex; height:3.176ex;" alt="{\displaystyle {\vec {n}}=\textstyle {a \choose b}}" loading="lazy"> beschrieben wird, kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}=\textstyle {-b \choose a}}">

<mover>

</mover>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}=\textstyle {-b \choose a}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5603ff816d48728417d5e5c534d562fdba01fa97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.435ex; height:3.509ex;" alt="{\displaystyle {\vec {r}}=\textstyle {-b \choose a}}" loading="lazy"> setzen und für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> einen der Achsenschnittpunkte wählen.

<div class="mw-heading mw-heading4"><h4 id="Mittelsenkrechte">Mittelsenkrechte</h4></div>

Die Mittelsenkrechte zweier Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/533793cbe3b0ecde32740b1af44ab97ae2c3fe40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.868ex; height:3.176ex;" alt="{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}" loading="lazy"> ist die Lotgerade durch den Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;P\colon {\vec {p}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}\;}">

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;P\colon {\vec {p}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43dc09e8fcb54da3f1a65c376fed80f6b300c283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.251ex; height:4.343ex;" alt="{\displaystyle \;P\colon {\vec {p}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}\;}" loading="lazy"> der Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AB}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b04153f9681e5b06066357774475c04aaef3a8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}" loading="lazy">. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2a1ccee366cd6620b5217de65db185437abff1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.486ex; height:3.009ex;" alt="{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}" loading="lazy"> erhält man aus der Formel (LG2):

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {r}}=\left({\vec {x}}-{\frac {{\vec {a}}+{\vec {b}}}{2}}\right)\cdot ({\vec {b}}-{\vec {a}})=0\quad \Leftrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8ee5853c59fb10828038a8d5afbf9281a07d4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.362ex; height:7.509ex;" alt="{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {r}}=\left({\vec {x}}-{\frac {{\vec {a}}+{\vec {b}}}{2}}\right)\cdot ({\vec {b}}-{\vec {a}})=0\quad \Leftrightarrow }" loading="lazy"></dd>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\vec {x}}\cdot ({\vec {b}}-{\vec {a}})={\frac {1}{2}}({\vec {b}}^{2}-{\vec {a}}^{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8779a052d59dd19ed6ba6dc2b3e7e5acd944c42c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.742ex; height:5.176ex;" alt="{\displaystyle \ {\vec {x}}\cdot ({\vec {b}}-{\vec {a}})={\frac {1}{2}}({\vec {b}}^{2}-{\vec {a}}^{2})}" loading="lazy"></dd></dl>

In Koordinaten ergibt sich für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(a_{1},a_{2}),\;B=(b_{1},b_{2})\colon }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=(a_{1},a_{2}),\;B=(b_{1},b_{2})\colon }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f768d29740a2a89b44b7ee9129c55c4edad252fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.388ex; height:2.843ex;" alt="{\displaystyle A=(a_{1},a_{2}),\;B=(b_{1},b_{2})\colon }" loading="lazy">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (b_{1}-a_{1})x+(b_{2}-a_{2})y={\frac {1}{2}}(b_{1}^{2}-a_{1}^{2}+b_{2}^{2}-a_{2}^{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/116cf3c2e0abcc6abf133b497940882b14b1d7fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.396ex; height:5.176ex;" alt="{\displaystyle (b_{1}-a_{1})x+(b_{2}-a_{2})y={\frac {1}{2}}(b_{1}^{2}-a_{1}^{2}+b_{2}^{2}-a_{2}^{2})}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Punkt_und_Gerade">Punkt und Gerade</h4></div>

Setzt man in der obigen Formel (LG2) Vektoren aus dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy"> ein, so beschreibt sie diejenige Ebene durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\colon {\vec {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\colon {\vec {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dda9233673719f0b83daa22bb834b67dba0db46a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.105ex; height:2.676ex;" alt="{\displaystyle P\colon {\vec {p}}}" loading="lazy">, die auf der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;g\colon {\vec {x}}={\vec {a}}+s{\vec {r}}\;}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;g\colon {\vec {x}}={\vec {a}}+s{\vec {r}}\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ade32a0c2d2f863d883d1a79a9888ac1a23403b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.252ex; height:2.676ex;" alt="{\displaystyle \;g\colon {\vec {x}}={\vec {a}}+s{\vec {r}}\;}" loading="lazy"> senkrecht steht, also die Lotebene:

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ \lambda \colon ({\vec {x}}-{\vec {p}})\cdot {\vec {r}}=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3943f1d209e68a9a470c4dde81b7dcc383120bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.437ex; height:2.843ex;" alt="{\displaystyle \ \lambda \colon ({\vec {x}}-{\vec {p}})\cdot {\vec {r}}=0}" loading="lazy"></dd></dl>

Der Schnittpunkt der Lotebene mit der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> ergibt sich aus der 3-dimensionalen Form der obigen Formel (LF2):

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ F\colon {\vec {f}}={\vec {a}}+{\frac {({\vec {a}}-{\vec {p}})\cdot {\vec {r}}}{{\vec {r}}\cdot {\vec {r}}}}\;{\vec {r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51ee6fbda90306006645d55938d0b8f798d3f58a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25ex; height:5.843ex;" alt="{\displaystyle \ F\colon {\vec {f}}={\vec {a}}+{\frac {({\vec {a}}-{\vec {p}})\cdot {\vec {r}}}{{\vec {r}}\cdot {\vec {r}}}}\;{\vec {r}}}" loading="lazy"></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> ist der Lotfußpunkt.

Die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {PF}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {PF}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25ed728ba9ae9d5271c82809872bcbf3372c7a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.761ex; height:3.009ex;" alt="{\displaystyle {\overline {PF}}}" loading="lazy"> schneidet die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> senkrecht. Also ist

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ h\colon {\vec {x}}={\vec {p}}+t({\vec {f}}-{\vec {p}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e23215ee3fae5f623257201f0ee5b44b11081962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.026ex; height:3.509ex;" alt="{\displaystyle \ h\colon {\vec {x}}={\vec {p}}+t({\vec {f}}-{\vec {p}})}" loading="lazy"></dd></dl>

die Lotgerade von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Punkt_und_Ebene">Punkt und Ebene</h4></div>

Für den Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\colon {\vec {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\colon {\vec {p}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dda9233673719f0b83daa22bb834b67dba0db46a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.105ex; height:2.676ex;" alt="{\displaystyle P\colon {\vec {p}}}" loading="lazy"> und die Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\varepsilon \colon {\vec {x}}\cdot {\vec {n}}=d\;}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;\varepsilon \colon {\vec {x}}\cdot {\vec {n}}=d\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec606f786e944fa3ca477be222580f073ff759bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.126ex; height:2.343ex;" alt="{\displaystyle \;\varepsilon \colon {\vec {x}}\cdot {\vec {n}}=d\;}" loading="lazy"> ist

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ h\colon {\vec {x}}={\vec {p}}+s{\vec {n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdb90028b9afc294e5f0586953fb97415f21b3d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.032ex; height:2.676ex;" alt="{\displaystyle \ h\colon {\vec {x}}={\vec {p}}+s{\vec {n}}}" loading="lazy"></dd></dl>

die Lotgerade. Für den Lotfußpunkt setzt man die Gleichung der Lotgerade in die Ebenengleichung ein, löst nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> auf und setzt das Ergebnis wiederum in die Lotgleichung ein:

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ F\colon {\vec {f}}={\vec {p}}+{\frac {d-{\vec {p}}\cdot {\vec {n}}}{{\vec {n}}\cdot {\vec {n}}}}{\vec {n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/561158adab724de72be2bba70c35bdd5d0d3b869.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.97ex; height:5.676ex;" alt="{\displaystyle \ F\colon {\vec {f}}={\vec {p}}+{\frac {d-{\vec {p}}\cdot {\vec {n}}}{{\vec {n}}\cdot {\vec {n}}}}{\vec {n}}}" loading="lazy"></dd></dl>

Alternative Vorgabe: Falls die Ebene in der <a href="Normalenform#Aus_der_Parameterform" title="Normalenform">Form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;{\vec {x}}={\vec {a}}+s{\vec {u}}+t{\vec {v}}\;}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;{\vec {x}}={\vec {a}}+s{\vec {u}}+t{\vec {v}}\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a4eb590d80f0a58ae35369bf3fc9f90028bd957.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.064ex; height:2.509ex;" alt="{\displaystyle \;{\vec {x}}={\vec {a}}+s{\vec {u}}+t{\vec {v}}\;}" loading="lazy"> gegeben ist, kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;{\vec {n}}={\vec {u}}\times {\vec {v}}\;}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;{\vec {n}}={\vec {u}}\times {\vec {v}}\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c6345b700459b442988a9b7593d3c36a82ba25f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.129ex; height:2.343ex;" alt="{\displaystyle \;{\vec {n}}={\vec {u}}\times {\vec {v}}\;}" loading="lazy"> setzen.

<div class="mw-heading mw-heading4"><h4 id="Mittellotebene">Mittellotebene</h4></div>

Die Mittellotebene zweier Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/533793cbe3b0ecde32740b1af44ab97ae2c3fe40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.868ex; height:3.176ex;" alt="{\displaystyle \;A\colon {\vec {a}},\;B\colon {\vec {b}}\;}" loading="lazy"> ist die Lotebene durch den Mittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;P\colon {\vec {p}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}\;}">

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;P\colon {\vec {p}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43dc09e8fcb54da3f1a65c376fed80f6b300c283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.251ex; height:4.343ex;" alt="{\displaystyle \;P\colon {\vec {p}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}\;}" loading="lazy"> der Strecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AB}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b04153f9681e5b06066357774475c04aaef3a8bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}" loading="lazy">. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2a1ccee366cd6620b5217de65db185437abff1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.486ex; height:3.009ex;" alt="{\displaystyle {\vec {r}}={\vec {b}}-{\vec {a}}}" loading="lazy"> erhält man, wie im ebenen Fall (Mittelsenkrechte), aus der Formel (PGLE3):

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {r}}=\left({\vec {x}}-{\frac {{\vec {a}}+{\vec {b}}}{2}}\right)\cdot ({\vec {b}}-{\vec {a}})=0\quad \Leftrightarrow }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8ee5853c59fb10828038a8d5afbf9281a07d4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.362ex; height:7.509ex;" alt="{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {r}}=\left({\vec {x}}-{\frac {{\vec {a}}+{\vec {b}}}{2}}\right)\cdot ({\vec {b}}-{\vec {a}})=0\quad \Leftrightarrow }" loading="lazy"></dd>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\vec {x}}\cdot \left({\vec {b}}-{\vec {a}}\right)={\frac {1}{2}}\left({\vec {b}}^{2}-{\vec {a}}^{2}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43bb428764813daa184ce4839530e11a1302094a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.061ex; height:5.176ex;" alt="{\displaystyle \ {\vec {x}}\cdot \left({\vec {b}}-{\vec {a}}\right)={\frac {1}{2}}\left({\vec {b}}^{2}-{\vec {a}}^{2}\right)}" loading="lazy"></dd></dl>

In Koordinaten ergibt sich für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(a_{1},a_{2},a_{3}),\;B=(b_{1},b_{2},b_{3})\colon }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=(a_{1},a_{2},a_{3}),\;B=(b_{1},b_{2},b_{3})\colon }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/656422eebf8ede6e50aa8a950d9588bbb87faf93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.792ex; height:2.843ex;" alt="{\displaystyle A=(a_{1},a_{2},a_{3}),\;B=(b_{1},b_{2},b_{3})\colon }" loading="lazy">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (b_{1}-a_{1})x+(b_{2}-a_{2})y+(b_{3}-a_{3})z={\frac {1}{2}}(b_{1}^{2}-a_{1}^{2}+b_{2}^{2}-a_{2}^{2}+b_{3}^{2}-a_{3}^{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4eb9acfed7886ed3ce91ce69d696e4f29de5fcf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:70.327ex; height:5.176ex;" alt="{\displaystyle (b_{1}-a_{1})x+(b_{2}-a_{2})y+(b_{3}-a_{3})z={\frac {1}{2}}(b_{1}^{2}-a_{1}^{2}+b_{2}^{2}-a_{2}^{2}+b_{3}^{2}-a_{3}^{2})}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Zweitafelprojektion#Lot_auf_eine_Ebene,_Abstand_Punkt-Ebene" title="Zweitafelprojektion">Lot auf eine Ebene, Abstand Punkt-Ebene</a> in der Darstellenden Geometrie</li>

<li><a href="Ophiuride#Als_Lotfußpunktkurve_einer_Parabel" title="Ophiuride">Ophiuride</a>, als Lotfußpunktkurve einer Parabel</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Michael Jung: Ebene Trigonometrie & Analytische Geometrie. Springer Spektrum, Wiesbaden 2024, ISBN 978-3-658-03261-6, S. 571–579.</li>

<li><cite class="lang" lang="en" dir="auto" style="font-style:italic">Perpendicular straight lines</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Perpendicular_straight_lines">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Lot+%28Mathematik%29&rft.atitle=Perpendicular+straight+lines&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2002&rft.genre=book&rft.isbn=1556080107&rft.place=Berlin&rft.pub=Springer-Verlag+und+EMS+Press" style="display:none"> </li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Perpendicular?uselang=de">Commons: Perpendicular</a> – Sammlung von Bildern, Videos und Audiodateien</div>

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Perpendicular.html">Perpendicular.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li>

<li>Warren Buck: <a rel="nofollow" class="external text" href="https://planetmath.org/CompassAndStraightedgeConstructionOfPerpendicular">Compass and straightedge construction of perpendicular.</a> In: <a href="PlanetMath" title="PlanetMath">PlanetMath</a>. (englisch)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Ralf Benölken, Hans-Joachim Gorski, Susanne Müller-Philipp: <cite style="font-style:italic">Leitfaden Geometrie</cite>. 7. Auflage. Springer Spektrum, Wiesbaden 2018, ISBN 978-3-658-23377-8, S. 104.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Lot+%28Mathematik%29&rft.au=Ralf+Ben%C3%B6lken%2C+Hans-Joachim+Gorski%2C+Susanne+M%C3%BCller-Philipp&rft.btitle=Leitfaden+Geometrie&rft.date=2018&rft.edition=7.&rft.genre=book&rft.isbn=9783658233778&rft.pages=104&rft.place=Wiesbaden&rft.pub=Springer+Spektrum" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Harald Scheid, Wolfgang Schwarz: <cite style="font-style:italic">Elemente der Geometrie</cite>. 4. Auflage. Spektrum Akademischer Verlag, Heidelberg 2007, ISBN 978-3-8274-1697-1, S. 9.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Lot+%28Mathematik%29&rft.au=Harald+Scheid%2C+Wolfgang+Schwarz&rft.btitle=Elemente+der+Geometrie&rft.date=2007&rft.edition=4.&rft.genre=book&rft.isbn=9783827416971&rft.pages=9&rft.place=Heidelberg&rft.pub=Spektrum+Akademischer+Verlag" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Philipp Schrenk: <a rel="nofollow" class="external text" href="https://www.mathetreff-online.de/sites/default/files/pdf/matheaufgaben/konstruktionen/grundlagen/konstruktion_eines_rechten_winkels_2.pdf">Konstruktion eines rechten Winkels 2.</a> (PDF) In: mathetreff-online. S. 1–2, abgerufen am 23. Juni 2025.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ALot+%28Mathematik%29&rft.title=Konstruktion+eines+rechten+Winkels+2&rft.description=Konstruktion+eines+rechten+Winkels+2&rft.identifier=https%3A%2F%2Fwww.mathetreff-online.de%2Fsites%2Fdefault%2Ffiles%2Fpdf%2Fmatheaufgaben%2Fkonstruktionen%2Fgrundlagen%2Fkonstruktion_eines_rechten_winkels_2.pdf&rft.creator=Philipp+Schrenk">

</li>

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