Abstand – Wikipedia

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border-bottom-width: 1px; font-size:95%; margin-bottom:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="Vorlage_Begriffsklärungshinweis"><div class="noviewer noresize bksicon" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/25px-Disambig-dark.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/38px-Disambig-dark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/50px-Disambig-dark.svg.png 2x" data-file-width="444" data-file-height="340" /></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div role="navigation"> Der Titel dieses Artikels ist mehrdeutig. Weitere Bedeutungen sind unter <a href="/wiki/Abstand_(Begriffskl%C3%A4rung)" class="mw-disambig" title="Abstand (Begriffsklärung)">Abstand (Begriffsklärung)</a> aufgeführt.</div> </div></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Datei:01_Abstand_zweier_Punkte.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/01_Abstand_zweier_Punkte.svg/300px-01_Abstand_zweier_Punkte.svg.png" decoding="async" width="300" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/01_Abstand_zweier_Punkte.svg/450px-01_Abstand_zweier_Punkte.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/01_Abstand_zweier_Punkte.svg/600px-01_Abstand_zweier_Punkte.svg.png 2x" data-file-width="341" data-file-height="119" /></a><figcaption></figcaption></figure> Der Abstand (auch Entfernung oder Distanz) zweier <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkte</a> ist die <a href="/wiki/L%C3%A4nge_(Mathematik)" title="Länge (Mathematik)">Länge</a> der kürzesten Verbindung dieser Punkte. Im <a href="/wiki/Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a> ist dies die Länge der <a href="/wiki/Strecke_(Geometrie)" title="Strecke (Geometrie)">Strecke</a> zwischen den beiden Punkten. Der Abstand zweier <a href="/wiki/Geometrische_Figur" title="Geometrische Figur">geometrischer Objekte</a> ist die Länge der kürzesten Verbindungslinie der beiden Objekte, also der Abstand der beiden einander nächstliegenden Punkte. Werden nicht die einander nächstliegenden Punkte zweier Objekte betrachtet, so wird dies explizit angegeben oder ergibt sich aus dem Zusammenhang, wie beispielsweise der Abstand der geometrischen Mittelpunkte oder der <a href="/wiki/Geometrischer_Schwerpunkt" title="Geometrischer Schwerpunkt">geometrischen Schwerpunkte</a>. Die <a href="/wiki/Metrischer_Raum" title="Metrischer Raum">Metrik</a> ist der Teil der Mathematik, der sich mit der Abstandsmessung beschäftigt. Der Abstand, die Entfernung, die Distanz zwischen zwei Werten einer Größe oder zwischen zwei <a href="/wiki/Zeitpunkt" title="Zeitpunkt">Zeitpunkten</a> wird bestimmt, indem man den Absolutbetrag ihrer <a href="/wiki/Subtraktion" title="Subtraktion">Differenz</a> bildet, das heißt, indem sie voneinander abgezogen werden und vom Ergebnis der Absolutbetrag gebildet wird. Der gemessene Abstand ist unabhängig vom gewählten <a href="/wiki/Referenzpunkt" class="mw-redirect" title="Referenzpunkt">Referenzpunkt</a> des <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinatensystems</a>, nicht aber von dessen Skalierung (siehe auch <a href="/wiki/Ma%C3%9Fstabsfaktor" title="Maßstabsfaktor">Maßstabsfaktor</a>). In der <a href="/wiki/Beobachtende_Astronomie" title="Beobachtende Astronomie">beobachtenden Astronomie</a> wird der scheinbare Abstand am Himmel zwischen zwei Himmelsobjekten als <a href="/wiki/Winkelabstand" class="mw-redirect" title="Winkelabstand">Winkelabstand</a> angegeben. Der Abstand zweier <a href="/wiki/Mengentheorie" class="mw-redirect" title="Mengentheorie">Mengen</a> im euklidischen Raum (oder allgemeiner in einem <a href="/wiki/Metrischer_Raum" title="Metrischer Raum">metrischen Raum</a>) kann über die <a href="/wiki/Hausdorff-Metrik" title="Hausdorff-Metrik">Hausdorff-Metrik</a> definiert werden. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Euklidischer_Abstand">1 Euklidischer Abstand</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Abstand_in_der_Ebene">1.1 Abstand in der Ebene</a> <ul> <li class="toclevel-3 tocsection-3"><a href="#Abstand_zwischen_Punkt_und_Gerade">1.1.1 Abstand zwischen Punkt und Gerade</a></li> </ul> </li> <li class="toclevel-2 tocsection-4"><a href="#Abstand_im_dreidimensionalen_Raum">1.2 Abstand im dreidimensionalen Raum</a> <ul> <li class="toclevel-3 tocsection-5"><a href="#Abstand_zwischen_Punkt_und_Gerade_2">1.2.1 Abstand zwischen Punkt und Gerade</a></li> <li class="toclevel-3 tocsection-6"><a href="#Abstand_zwischen_zwei_windschiefen_Geraden">1.2.2 Abstand zwischen zwei windschiefen Geraden</a></li> <li class="toclevel-3 tocsection-7"><a href="#Abstand_zwischen_Punkt_und_Ebene">1.2.3 Abstand zwischen Punkt und Ebene</a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#Andere_Definitionen">2 Andere Definitionen</a> <ul> <li class="toclevel-2 tocsection-9"><a href="#Manhattan-Metrik">2.1 Manhattan-Metrik</a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#Abstandsmessung_auf_gekrümmten_Flächen">3 Abstandsmessung auf gekrümmten Flächen</a></li> <li class="toclevel-1 tocsection-11"><a href="#Dichtestes_Punktpaar">4 Dichtestes Punktpaar</a></li> <li class="toclevel-1 tocsection-12"><a href="#Siehe_auch">5 Siehe auch</a></li> <li class="toclevel-1 tocsection-13"><a href="#Weblinks">6 Weblinks</a></li> <li class="toclevel-1 tocsection-14"><a href="#Anmerkungen">7 Anmerkungen</a></li> <li class="toclevel-1 tocsection-15"><a href="#Einzelnachweise">8 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Euklidischer_Abstand">Euklidischer Abstand</h2>[<a href="/w/index.php?title=Abstand&veaction=edit&section=1" title="Abschnitt bearbeiten: Euklidischer Abstand" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Euklidischer Abstand">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Euklidischer_Abstand" title="Euklidischer Abstand">Euklidischer Abstand</a></div> Im <a href="/wiki/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinatensystem</a> berechnet man den Abstand (euklidischer Abstand) zweier Punkte mit Hilfe des <a href="/wiki/Satz_von_Pythagoras" class="mw-redirect" title="Satz von Pythagoras">Satzes von Pythagoras</a>: <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:01-Abstand_zweier_Punkte.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/01-Abstand_zweier_Punkte.svg/290px-01-Abstand_zweier_Punkte.svg.png" decoding="async" width="290" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/01-Abstand_zweier_Punkte.svg/435px-01-Abstand_zweier_Punkte.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/01-Abstand_zweier_Punkte.svg/580px-01-Abstand_zweier_Punkte.svg.png 2x" data-file-width="423" data-file-height="363" /></a><figcaption>Der Abstand zweier Punkte in der Ebene</figcaption></figure> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}{\text{, wobei }}A=(a_{1},\dotsc ,a_{n})\in \mathbb {R} ^{n}{\text{ und }}B=(b_{1},\dotsc ,b_{n})\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>, wobei </mtext> </mrow> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> und </mtext> </mrow> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}{\text{, wobei }}A=(a_{1},\dotsc ,a_{n})\in \mathbb {R} ^{n}{\text{ und }}B=(b_{1},\dotsc ,b_{n})\in \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e1f9362f8dae1089f0e59199342d1b045ff386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:82.687ex; height:7.509ex;" alt="{\displaystyle d(A,B)={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}{\text{, wobei }}A=(a_{1},\dotsc ,a_{n})\in \mathbb {R} ^{n}{\text{ und }}B=(b_{1},\dotsc ,b_{n})\in \mathbb {R} ^{n}}"><a href="#cite_note-1">[1]</a></dd></dl> Für die Ebene (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\in \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8c2e1c58da45c6b01844054c652be124463a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.114ex; height:3.009ex;" alt="{\displaystyle A,B\in \mathbb {R} ^{2}}">): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e7d18552ecd457573fed2b6ff560b61d0290f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:35.908ex; height:4.843ex;" alt="{\displaystyle d(A,B)={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}}"></dd></dl> Für den dreidimensionalen Raum (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B\in \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\in \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf89b6cac097d017bfc3c7db30c4af046483c639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.114ex; height:3.009ex;" alt="{\displaystyle A,B\in \mathbb {R} ^{3}}">): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+(a_{3}-b_{3})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+(a_{3}-b_{3})^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50c2718252d6fca09a701672635215c3fcbd441a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:48.789ex; height:4.843ex;" alt="{\displaystyle d(A,B)={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+(a_{3}-b_{3})^{2}}}}"><a href="#cite_note-2">[2]</a></dd></dl> Der Abstand eines Punkts von einer <a href="/wiki/Gerade" title="Gerade">Geraden</a> oder einer ebenen Fläche ist der Abstand vom <a href="/wiki/Fu%C3%9Fpunkt" title="Fußpunkt">Fußpunkt</a> des darauf gefällten <a href="/wiki/Lot_(Mathematik)" title="Lot (Mathematik)">Lots</a>, der von einer gekrümmten Linie ist stets ein Abstand von einer ihrer <a href="/wiki/Tangente" title="Tangente">Tangenten</a>. Berechnungsmöglichkeiten für die Abstände von Punkten zu Geraden oder Ebenen sind in der <a href="/wiki/Formelsammlung_analytische_Geometrie" title="Formelsammlung analytische Geometrie">Formelsammlung analytische Geometrie</a> aufgeführt. <div class="mw-heading mw-heading3"><h3 id="Abstand_in_der_Ebene">Abstand in der Ebene</h3>[<a href="/w/index.php?title=Abstand&veaction=edit&section=2" title="Abschnitt bearbeiten: Abstand in der Ebene" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Abstand in der Ebene">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Abstand_zwischen_Punkt_und_Gerade">Abstand zwischen Punkt und Gerade</h4>[<a href="/w/index.php?title=Abstand&veaction=edit&section=3" title="Abschnitt bearbeiten: Abstand zwischen Punkt und Gerade" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Abstand zwischen Punkt und Gerade">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:01_Abstand_Ebene_Punkt-Gerade.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/01_Abstand_Ebene_Punkt-Gerade.svg/290px-01_Abstand_Ebene_Punkt-Gerade.svg.png" decoding="async" width="290" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/01_Abstand_Ebene_Punkt-Gerade.svg/435px-01_Abstand_Ebene_Punkt-Gerade.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/01_Abstand_Ebene_Punkt-Gerade.svg/580px-01_Abstand_Ebene_Punkt-Gerade.svg.png 2x" data-file-width="596" data-file-height="478" /></a><figcaption>Beispiel: Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6f0f204c2851d263548a59a2a02f694973ea8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.921ex; height:2.843ex;" alt="{\displaystyle d(P,g)}"> zwischen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" 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}"> und Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"> in der Ebene.</figcaption></figure> Der Abstand zwischen dem <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x_{0},y_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c7b4383bfe8c664f9feae5c2c132db4bafce3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.166ex; height:2.843ex;" alt="{\displaystyle P(x_{0},y_{0})}"> und der <a href="/wiki/Gerade" title="Gerade">Geraden</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"> mit der <a href="/wiki/Koordinatenform" title="Koordinatenform">Koordinatenform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+by+c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+by+c=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ffa3f9c01bec425db7c1acc330497b6831697b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.661ex; height:2.509ex;" alt="{\displaystyle ax+by+c=0}"> beträgt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,g)={\frac {|ax_{0}+by_{0}+c|}{\sqrt {a^{2}+b^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,g)={\frac {|ax_{0}+by_{0}+c|}{\sqrt {a^{2}+b^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce40c3655fe72a4829b2f5c52d9845cef81b832b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.641ex; height:7.009ex;" alt="{\displaystyle d(P,g)={\frac {|ax_{0}+by_{0}+c|}{\sqrt {a^{2}+b^{2}}}}}"></dd></dl> Der Punkt auf der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}">, der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}"> am nächsten liegt, hat die <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinaten</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x={\frac {b(bx_{0}-ay_{0})-ac}{a^{2}+b^{2}}},\;y={\frac {a(-bx_{0}+ay_{0})-bc}{a^{2}+b^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo stretchy="false">(</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>a</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mi>c</mi> </mrow> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>b</mi> <mi>c</mi> </mrow> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x={\frac {b(bx_{0}-ay_{0})-ac}{a^{2}+b^{2}}},\;y={\frac {a(-bx_{0}+ay_{0})-bc}{a^{2}+b^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516c92405d1a378c38a36d449dbb1c9d36817788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.321ex; height:6.343ex;" alt="{\displaystyle \left(x={\frac {b(bx_{0}-ay_{0})-ac}{a^{2}+b^{2}}},\;y={\frac {a(-bx_{0}+ay_{0})-bc}{a^{2}+b^{2}}}\right)}"></dd></dl> Wenn die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"> durch die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2},y_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}"> verläuft, ist <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=y_{2}-y_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=y_{2}-y_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e9139c2034ab09e003e077564fdfedf0b924d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.556ex; height:2.343ex;" alt="{\displaystyle a=y_{2}-y_{1}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=x_{1}-x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=x_{1}-x_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1202427da4aa2a25c39f284194f879459d36ffd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.704ex; height:2.509ex;" alt="{\displaystyle b=x_{1}-x_{2}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=x_{2}y_{1}-x_{1}y_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=x_{2}y_{1}-x_{1}y_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e37a1363a89cdd892a82bcc643e3c54f5b63059b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.1ex; height:2.343ex;" alt="{\displaystyle c=x_{2}y_{1}-x_{1}y_{2}}"></dd></dl> Diese Werte können in die <a href="/wiki/Formel" title="Formel">Formeln</a> eingesetzt werden.<a href="#cite_note-3">[3]</a> Beispiel Eingesetzte Werte für Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-3,\;b=4,\;c=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>b</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>c</mi> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=-3,\;b=4,\;c=10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee9d0eead5ba41ce58f6e4fd3e1e6c6c9f1d67e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.346ex; height:2.509ex;" alt="{\displaystyle a=-3,\;b=4,\;c=10}"> und für Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P:\;x_{0}=4,\;y_{0}=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>:</mo> <mspace width="thickmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P:\;x_{0}=4,\;y_{0}=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33e005ad7a54ebcfb8d8f847013fbbfde98332bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.106ex; height:2.509ex;" alt="{\displaystyle P:\;x_{0}=4,\;y_{0}=6}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,g)={\frac {(-3)\cdot 4+4\cdot 6+10}{\sqrt {(-3)^{2}+4^{2}}}}={\frac {22}{5}}=4{,}4\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> <mo>+</mo> <mn>10</mn> </mrow> <msqrt> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>22</mn> <mn>5</mn> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>4</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,g)={\frac {(-3)\cdot 4+4\cdot 6+10}{\sqrt {(-3)^{2}+4^{2}}}}={\frac {22}{5}}=4{,}4\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb0a9e53955cfee5b9ab6f24c2164d3964d480b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:43.461ex; height:8.509ex;" alt="{\displaystyle d(P,g)={\frac {(-3)\cdot 4+4\cdot 6+10}{\sqrt {(-3)^{2}+4^{2}}}}={\frac {22}{5}}=4{,}4\;}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Abstand_im_dreidimensionalen_Raum">Abstand im dreidimensionalen Raum</h3>[<a href="/w/index.php?title=Abstand&veaction=edit&section=4" title="Abschnitt bearbeiten: Abstand im dreidimensionalen Raum" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Abstand im dreidimensionalen Raum">Quelltext bearbeiten</a>]</div> Für die Konstruktion des Abstandes bedarf es als zusätzliches Hilfsmittel einer <a href="/wiki/Dynamische_Geometrie" title="Dynamische Geometrie">Dynamischen-Geometrie-Software (DGS).</a> <div class="mw-heading mw-heading4"><h4 id="Abstand_zwischen_Punkt_und_Gerade_2">Abstand zwischen Punkt und Gerade</h4>[<a href="/w/index.php?title=Abstand&veaction=edit&section=5" title="Abschnitt bearbeiten: Abstand zwischen Punkt und Gerade" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Abstand zwischen Punkt und Gerade">Quelltext bearbeiten</a>]</div> Der Abstand zwischen dem <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2728b2a274122fbaf50539fa2dd9c885afca413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}"> und der <a href="/wiki/Gerade" title="Gerade">Geraden</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}">, die durch die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a846e6741a44ffc62819d8842ef8677e888bf0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec3b67f42fe95e4cf78de50e3a82fdcd66aca95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}"> verläuft, beträgt mit den Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p_{0}}},\;{\vec {p_{1}}},\;{\vec {p_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p_{0}}},\;{\vec {p_{1}}},\;{\vec {p_{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1cca2fa99c6a65cbb57814fff87296352124c20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.039ex; width:10.369ex; height:3.343ex;" alt="{\displaystyle {\vec {p_{0}}},\;{\vec {p_{1}}},\;{\vec {p_{2}}}}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P_{0},g)={\frac {\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{1}}}-{\vec {p_{0}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}={\frac {\left|({\vec {p_{0}}}-{\vec {p_{1}}})\times ({\vec {p_{0}}}-{\vec {p_{2}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P_{0},g)={\frac {\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{1}}}-{\vec {p_{0}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}={\frac {\left|({\vec {p_{0}}}-{\vec {p_{1}}})\times ({\vec {p_{0}}}-{\vec {p_{2}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f38cdac3a782608bb9bc1509257c7198e5e523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:61.048ex; height:9.509ex;" alt="{\displaystyle d(P_{0},g)={\frac {\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{1}}}-{\vec {p_{0}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}={\frac {\left|({\vec {p_{0}}}-{\vec {p_{1}}})\times ({\vec {p_{0}}}-{\vec {p_{2}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}}"><a href="#cite_note-4">[4]</a></dd></dl> Beispiel <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:01_Abstand_Raum_Punkte-Gerade.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/01_Abstand_Raum_Punkte-Gerade.png/330px-01_Abstand_Raum_Punkte-Gerade.png" decoding="async" width="330" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/01_Abstand_Raum_Punkte-Gerade.png/495px-01_Abstand_Raum_Punkte-Gerade.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b4/01_Abstand_Raum_Punkte-Gerade.png 2x" data-file-width="586" data-file-height="418" /></a><figcaption>Beispiel: Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P_{0},g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P_{0},g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b9e752e4716f35ca63a991c2306dd322399a10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.722ex; height:2.843ex;" alt="{\displaystyle d(P_{0},g)}"> zwischen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"> und Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"> im Raum.</figcaption></figure> Konstruktion des Abstandes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P_{0},g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P_{0},g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b9e752e4716f35ca63a991c2306dd322399a10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.722ex; height:2.843ex;" alt="{\displaystyle d(P_{0},g)}">. Gegeben sind die Koordinaten der Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}=\left(3{,}5\left|2{,}5\right|0\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mrow> <mo>|</mo> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mrow> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}=\left(3{,}5\left|2{,}5\right|0\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea50a63c67e4fc8643a3a6d03a363b448791678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.628ex; height:2.843ex;" alt="{\displaystyle P_{1}=\left(3{,}5\left|2{,}5\right|0\right)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}=\left(-1\left|7\right|0\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mrow> <mo>|</mo> <mn>7</mn> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}=\left(-1\left|7\right|0\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd05f8a0f2c54f2523170ec7d2af12a9602ce92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.818ex; height:2.843ex;" alt="{\displaystyle P_{2}=\left(-1\left|7\right|0\right)}">, durch die die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"> verläuft, und der Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}=\left(5\left|6\right|3{,}5\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow> <mo>|</mo> <mn>6</mn> <mo>|</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}=\left(5\left|6\right|3{,}5\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/479a1823c90ce1e6007b33209f8eeab93dd915be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.819ex; height:2.843ex;" alt="{\displaystyle P_{0}=\left(5\left|6\right|3{,}5\right)}">. Nach dem Einzeichnen der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"> und dem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"> werden die <a href="/wiki/Vektor#Definition" title="Vektor">Verbindungsvektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c77ed661878e6e1cdd2a207b689f7d4ccad31ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.039ex; width:6.366ex; height:3.343ex;" alt="{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p_{0}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2728382cfe9fd2e278bd7cd7692243d2002864d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.039ex; width:2.363ex; height:3.343ex;" alt="{\displaystyle {\vec {p_{0}}}}"> eingezeichnet. Eine abschließend errichtete <a href="/wiki/Orthogonalit%C3%A4t" title="Orthogonalität">Senkrechte</a> auf die Gerade <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" 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}"> durch Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"> liefert den Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P_{0},g)=4{,}974\ldots \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4,974</mn> <mo>…</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P_{0},g)=4{,}974\ldots \;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8148d57a7e00237cd9589c91162b9db914c3a41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.872ex; height:2.843ex;" alt="{\displaystyle d(P_{0},g)=4{,}974\ldots \;}">[LE]. Nachrechnung Diese Werte in die Formel eingesetzt, ergeben <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P_{0},g)={\frac {\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{1}}}-{\vec {p_{0}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}={\frac {\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-1{,}5\\-3{,}5\\-3{,}5\end{pmatrix}}\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\right|}}={\frac {\left|{\begin{pmatrix}-15{,}75\\-15{,}75\\22{,}5\end{pmatrix}}\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\right|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>75</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>75</mn> </mtd> </mtr> <mtr> <mtd> <mn>22</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P_{0},g)={\frac {\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{1}}}-{\vec {p_{0}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}={\frac {\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-1{,}5\\-3{,}5\\-3{,}5\end{pmatrix}}\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\right|}}={\frac {\left|{\begin{pmatrix}-15{,}75\\-15{,}75\\22{,}5\end{pmatrix}}\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\right|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8076987fde06c8e4b05532937b1ee1fe234b9679" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:78.801ex; height:19.509ex;" alt="{\displaystyle d(P_{0},g)={\frac {\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{1}}}-{\vec {p_{0}}})\right|}{\left|{\vec {p_{2}}}-{\vec {p_{1}}}\right|}}={\frac {\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-1{,}5\\-3{,}5\\-3{,}5\end{pmatrix}}\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\right|}}={\frac {\left|{\begin{pmatrix}-15{,}75\\-15{,}75\\22{,}5\end{pmatrix}}\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\right|}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {\left|{\sqrt {(-15{,}75)^{2}+(-15{,}75)^{2}+22{,}5^{2}}}\right|}{\left|{\sqrt {(-4{,}5)^{2}+4{,}5^{2}+0^{2}}}\right|}}=4{,}974\ldots \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mo>−</mo> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>75</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>75</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>22</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4,974</mn> <mo>…</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {\left|{\sqrt {(-15{,}75)^{2}+(-15{,}75)^{2}+22{,}5^{2}}}\right|}{\left|{\sqrt {(-4{,}5)^{2}+4{,}5^{2}+0^{2}}}\right|}}=4{,}974\ldots \;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64a4b897b023f0470c9fd97647cb92b9801a4a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:49.863ex; height:11.176ex;" alt="{\displaystyle ={\frac {\left|{\sqrt {(-15{,}75)^{2}+(-15{,}75)^{2}+22{,}5^{2}}}\right|}{\left|{\sqrt {(-4{,}5)^{2}+4{,}5^{2}+0^{2}}}\right|}}=4{,}974\ldots \;}">[LE].</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Abstand_zwischen_zwei_windschiefen_Geraden">Abstand zwischen zwei windschiefen Geraden</h4>[<a href="/w/index.php?title=Abstand&veaction=edit&section=6" title="Abschnitt bearbeiten: Abstand zwischen zwei windschiefen Geraden" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Abstand zwischen zwei windschiefen Geraden">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Windschiefe" title="Windschiefe">Windschiefe</a></div> Zwei windschiefe Geraden (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1},\;g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1},\;g_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f193f7305cab5f98e031936136d159090abbb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.006ex; height:2.009ex;" alt="{\displaystyle g_{1},\;g_{2}}">), wobei die eine durch die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a846e6741a44ffc62819d8842ef8677e888bf0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec3b67f42fe95e4cf78de50e3a82fdcd66aca95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}"> und die andere durch die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}=(x_{3},y_{3},z_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}=(x_{3},y_{3},z_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2842a5d94e8010623e5112ee2548de52a996af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{3}=(x_{3},y_{3},z_{3})}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}=(x_{4},y_{4},z_{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}=(x_{4},y_{4},z_{4})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa74771d370169adaca9aad1c6ee186d807bb9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{4}=(x_{4},y_{4},z_{4})}"> verläuft, haben mit den Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}},\;{\vec {p_{3}}},\;{\vec {p_{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}},\;{\vec {p_{3}}},\;{\vec {p_{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa937243e64800eeea2d3b993a20a414643da61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.039ex; width:14.372ex; height:3.343ex;" alt="{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}},\;{\vec {p_{3}}},\;{\vec {p_{4}}}}"> folgenden Abstand: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(g_{1},g_{2})={\frac {\left|({\vec {p_{3}}}-{\vec {p_{1}}})\cdot (({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}}))\right|}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}})\right|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(g_{1},g_{2})={\frac {\left|({\vec {p_{3}}}-{\vec {p_{1}}})\cdot (({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}}))\right|}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}})\right|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce52770a61592c4e7b625262ffd898817ca573e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:47.834ex; height:9.509ex;" alt="{\displaystyle d(g_{1},g_{2})={\frac {\left|({\vec {p_{3}}}-{\vec {p_{1}}})\cdot (({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}}))\right|}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}})\right|}}}"><a href="#cite_note-5">[5]</a></dd></dl> Beispiel <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:01_Abstand_Raum_Gerade-Gerade.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/01_Abstand_Raum_Gerade-Gerade.png/330px-01_Abstand_Raum_Gerade-Gerade.png" decoding="async" width="330" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/01_Abstand_Raum_Gerade-Gerade.png/495px-01_Abstand_Raum_Gerade-Gerade.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/01_Abstand_Raum_Gerade-Gerade.png/660px-01_Abstand_Raum_Gerade-Gerade.png 2x" data-file-width="889" data-file-height="708" /></a><figcaption>Beispiel: Konstruktion des Abstandes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(g_{1},g_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(g_{1},g_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59e8ecc88ca5655b7f67b25020903785eb5b3187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle d(g_{1},g_{2})}"> zwischen zwei windschiefen Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3755e3e04ec295992b2b5331655ef83a500a05c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}"> im Raum.</figcaption></figure> Konstruktion des Abstandes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(g_{1},g_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(g_{1},g_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59e8ecc88ca5655b7f67b25020903785eb5b3187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.386ex; height:2.843ex;" alt="{\displaystyle d(g_{1},g_{2})}"> mithilfe einer Hilfsebene. Gegeben seien die Koordinaten der vier Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}=\left(3{,}5\left|2{,}5\right|0\right),\;P_{2}=\left(-1\left|7\right|0\right),\;P_{3}=\left(5\left|6\right|3{,}5\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> <mrow> <mo>|</mo> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mrow> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mrow> <mo>|</mo> <mn>7</mn> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow> <mo>|</mo> <mn>6</mn> <mo>|</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}=\left(3{,}5\left|2{,}5\right|0\right),\;P_{2}=\left(-1\left|7\right|0\right),\;P_{3}=\left(5\left|6\right|3{,}5\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae426b5eb214a8face084bcd39b9191c8c2ddbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.397ex; height:2.843ex;" alt="{\displaystyle P_{1}=\left(3{,}5\left|2{,}5\right|0\right),\;P_{2}=\left(-1\left|7\right|0\right),\;P_{3}=\left(5\left|6\right|3{,}5\right)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}=\left(0{,}2\left|2{,}5\right|6\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>2</mn> <mrow> <mo>|</mo> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mrow> <mo>|</mo> </mrow> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}=\left(0{,}2\left|2{,}5\right|6\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed881f99337e5b350b300b8709374eb99d5b8fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.662ex; height:2.843ex;" alt="{\displaystyle P_{4}=\left(0{,}2\left|2{,}5\right|6\right).}"> Nach dem Einzeichnen der Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3755e3e04ec295992b2b5331655ef83a500a05c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"> werden zunächst die <a href="/wiki/Vektor#Definition" title="Vektor">Verbindungsvektoren</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}}\;{\vec {p_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}}\;{\vec {p_{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d188d469d5d378e85b2887c0c2e0a9ffd75c9917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.039ex; width:9.335ex; height:3.343ex;" alt="{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}}\;{\vec {p_{3}}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p_{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p_{4}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da8f845ab5b9a0221e3c234cfb6f6e59e3677cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.039ex; width:2.363ex; height:3.343ex;" alt="{\displaystyle {\vec {p_{4}}}}"> eingezeichnet. Für das Bestimmen der Hilfsebene wird eine Parallele zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"> gezogen und anschließend der Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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}"> beliebig auf der Parallele markiert. Mithilfe der somit gegebenen drei Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,P_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759a2b27d0a3c7a7aac00854437891337e04b93e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.324ex; height:2.509ex;" alt="{\displaystyle A,P_{1}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"> wird die Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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}"> generiert. Es folgt das Fällen des Lots vom Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"> auf die Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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}"> mit Fußpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> und eine Parallele zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee518ac53e84056ba9ff09b34fef75f627992470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.81ex; height:2.009ex;" alt="{\displaystyle g_{2},}"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3755e3e04ec295992b2b5331655ef83a500a05c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> (rot) schneidet. Abschließend liefert die Parallele zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P_{3}B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P_{3}B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c79b8fb65fa37c4ee1289a4ca94a1c20ef78b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.425ex; height:3.343ex;" alt="{\displaystyle {\overline {P_{3}B}}}"> ab dem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> bis zur Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}"> den Abstand: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(g_{1},g_{2})=4{,}605\ldots \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4,605</mn> <mo>…</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(g_{1},g_{2})=4{,}605\ldots \;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d6df812973096117ac5ca6458f5cda89415ac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.536ex; height:2.843ex;" alt="{\displaystyle d(g_{1},g_{2})=4{,}605\ldots \;}">[LE]. Nachrechnung Diese Werte eingesetzt in die Formel ergeben <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(g_{1},g_{2})={\frac {\left|({\vec {p_{3}}}-{\vec {p_{1}}})\cdot (({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}}))\right|}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}})\right|}}={\frac {\left|{\begin{pmatrix}1{,}5\\3{,}5\\3{,}5\end{pmatrix}}\cdot \left({\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-4{,}8\\-3{,}5\\2{,}5\end{pmatrix}}\right)\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-4{,}8\\-3{,}5\\2{,}5\end{pmatrix}}\right|}}={\frac {\left|{\begin{pmatrix}1{,}5\\3{,}5\\3{,}5\end{pmatrix}}\cdot {\begin{pmatrix}11{,}25\\11{,}25\\37{,}35\end{pmatrix}}\right|}{\left|{\begin{pmatrix}11{,}25\\11{,}25\\37{,}35\end{pmatrix}}\right|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>25</mn> </mtd> </mtr> <mtr> <mtd> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>25</mn> </mtd> </mtr> <mtr> <mtd> <mn>37</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>35</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>25</mn> </mtd> </mtr> <mtr> <mtd> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>25</mn> </mtd> </mtr> <mtr> <mtd> <mn>37</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>35</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(g_{1},g_{2})={\frac {\left|({\vec {p_{3}}}-{\vec {p_{1}}})\cdot (({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}}))\right|}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}})\right|}}={\frac {\left|{\begin{pmatrix}1{,}5\\3{,}5\\3{,}5\end{pmatrix}}\cdot \left({\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-4{,}8\\-3{,}5\\2{,}5\end{pmatrix}}\right)\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-4{,}8\\-3{,}5\\2{,}5\end{pmatrix}}\right|}}={\frac {\left|{\begin{pmatrix}1{,}5\\3{,}5\\3{,}5\end{pmatrix}}\cdot {\begin{pmatrix}11{,}25\\11{,}25\\37{,}35\end{pmatrix}}\right|}{\left|{\begin{pmatrix}11{,}25\\11{,}25\\37{,}35\end{pmatrix}}\right|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ae699a36a6e2ae243edf0721cdf1cddcee9a588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:113.448ex; height:19.509ex;" alt="{\displaystyle d(g_{1},g_{2})={\frac {\left|({\vec {p_{3}}}-{\vec {p_{1}}})\cdot (({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}}))\right|}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{4}}}-{\vec {p_{3}}})\right|}}={\frac {\left|{\begin{pmatrix}1{,}5\\3{,}5\\3{,}5\end{pmatrix}}\cdot \left({\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-4{,}8\\-3{,}5\\2{,}5\end{pmatrix}}\right)\right|}{\left|{\begin{pmatrix}-4{,}5\\4{,}5\\0\end{pmatrix}}\times {\begin{pmatrix}-4{,}8\\-3{,}5\\2{,}5\end{pmatrix}}\right|}}={\frac {\left|{\begin{pmatrix}1{,}5\\3{,}5\\3{,}5\end{pmatrix}}\cdot {\begin{pmatrix}11{,}25\\11{,}25\\37{,}35\end{pmatrix}}\right|}{\left|{\begin{pmatrix}11{,}25\\11{,}25\\37{,}35\end{pmatrix}}\right|}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {\left|186{,}975\right|}{\left|{\sqrt {11{,}25^{2}+11{,}25^{2}+37{,}35^{2}}}\right|}}=4{,}605\ldots \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>|</mo> <mn>186,975</mn> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <msup> <mn>25</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>11</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <msup> <mn>25</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>37</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <msup> <mn>35</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4,605</mn> <mo>…</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {\left|186{,}975\right|}{\left|{\sqrt {11{,}25^{2}+11{,}25^{2}+37{,}35^{2}}}\right|}}=4{,}605\ldots \;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dbacf4ddefdd7069a77e296b283e0126ab9e7a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:43.791ex; height:9.009ex;" alt="{\displaystyle ={\frac {\left|186{,}975\right|}{\left|{\sqrt {11{,}25^{2}+11{,}25^{2}+37{,}35^{2}}}\right|}}=4{,}605\ldots \;}">[LE].</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Abstand_zwischen_Punkt_und_Ebene">Abstand zwischen Punkt und Ebene</h4>[<a href="/w/index.php?title=Abstand&veaction=edit&section=7" title="Abschnitt bearbeiten: Abstand zwischen Punkt und Ebene" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Abstand zwischen Punkt und Ebene">Quelltext bearbeiten</a>]</div> Der Abstand zwischen dem <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2728b2a274122fbaf50539fa2dd9c885afca413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}"> und der <a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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}"> mit der <a href="/wiki/Koordinatenform#Koordinatenform_einer_Ebenengleichung" title="Koordinatenform">Koordinatenform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+by+cz-f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mi>z</mi> <mo>−</mo> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+by+cz-f=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77041352e46ea5634ffdb3cf27a8414abf106d2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.868ex; height:2.509ex;" alt="{\displaystyle ax+by+cz-f=0}"><a href="#cite_note-Konstante_f-6">[A 1]</a> beträgt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\;(1)\;\;d(P_{0},E)={\frac {|ax_{0}+by_{0}+cz_{0}-f|}{\sqrt {a^{2}+b^{2}+c^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>c</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\;(1)\;\;d(P_{0},E)={\frac {|ax_{0}+by_{0}+cz_{0}-f|}{\sqrt {a^{2}+b^{2}+c^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8749d36e6c590a3df2bb12bfd29e7ba96cf6c851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.909ex; height:7.009ex;" alt="{\displaystyle \;\;(1)\;\;d(P_{0},E)={\frac {|ax_{0}+by_{0}+cz_{0}-f|}{\sqrt {a^{2}+b^{2}+c^{2}}}}}"><a href="#cite_note-Konstante_f-6">[A 1]</a></dd></dl> Für die einzusetzenden Werte gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;{\begin{aligned}&\left(2\right)a=y_{1}z_{2}-y_{2}z_{1}+y_{2}z_{3}-y_{3}z_{2}+y_{3}z_{1}-y_{1}z_{3}\\&\left(3\right)b=z_{1}x_{2}-z_{2}x_{1}+z_{2}x_{3}-z_{3}x_{2}+z_{3}x_{1}-z_{1}x_{3}\\&\left(4\right)c=x_{1}y_{2}-x_{2}y_{1}+x_{2}y_{3}-x_{3}y_{2}+x_{3}y_{1}-x_{1}y_{3}\\&\left(5\right)f=x_{1}y_{2}z_{3}-x_{1}y_{3}z_{2}+x_{2}y_{3}z_{1}-x_{2}y_{1}z_{3}+x_{3}y_{1}z_{2}-x_{3}y_{2}z_{1}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>a</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mi>b</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mi>c</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mi>f</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;{\begin{aligned}&\left(2\right)a=y_{1}z_{2}-y_{2}z_{1}+y_{2}z_{3}-y_{3}z_{2}+y_{3}z_{1}-y_{1}z_{3}\\&\left(3\right)b=z_{1}x_{2}-z_{2}x_{1}+z_{2}x_{3}-z_{3}x_{2}+z_{3}x_{1}-z_{1}x_{3}\\&\left(4\right)c=x_{1}y_{2}-x_{2}y_{1}+x_{2}y_{3}-x_{3}y_{2}+x_{3}y_{1}-x_{1}y_{3}\\&\left(5\right)f=x_{1}y_{2}z_{3}-x_{1}y_{3}z_{2}+x_{2}y_{3}z_{1}-x_{2}y_{1}z_{3}+x_{3}y_{1}z_{2}-x_{3}y_{2}z_{1}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d52c907c49083f16cd3fa671b5ee8ab7919f801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:63.611ex; height:12.509ex;" alt="{\displaystyle \;{\begin{aligned}&\left(2\right)a=y_{1}z_{2}-y_{2}z_{1}+y_{2}z_{3}-y_{3}z_{2}+y_{3}z_{1}-y_{1}z_{3}\\&\left(3\right)b=z_{1}x_{2}-z_{2}x_{1}+z_{2}x_{3}-z_{3}x_{2}+z_{3}x_{1}-z_{1}x_{3}\\&\left(4\right)c=x_{1}y_{2}-x_{2}y_{1}+x_{2}y_{3}-x_{3}y_{2}+x_{3}y_{1}-x_{1}y_{3}\\&\left(5\right)f=x_{1}y_{2}z_{3}-x_{1}y_{3}z_{2}+x_{2}y_{3}z_{1}-x_{2}y_{1}z_{3}+x_{3}y_{1}z_{2}-x_{3}y_{2}z_{1}\\\end{aligned}}}"></dd></dl> Wenn drei Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a846e6741a44ffc62819d8842ef8677e888bf0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{1}=(x_{1},y_{1},z_{1})}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec3b67f42fe95e4cf78de50e3a82fdcd66aca95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{2}=(x_{2},y_{2},z_{2})}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}=(x_{3},y_{3},z_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}=(x_{3},y_{3},z_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2842a5d94e8010623e5112ee2548de52a996af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.235ex; height:2.843ex;" alt="{\displaystyle P_{3}=(x_{3},y_{3},z_{3})}"> gegeben sind, die eine Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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}"> bestimmen (siehe <a href="/wiki/Dreipunkteform" title="Dreipunkteform">Dreipunkteform</a>) dann lässt sich der Abstand mithilfe der Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}},\;{\vec {p_{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}},\;{\vec {p_{3}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98cc3d2ccaac7ad20e49cd85adb44fd93bdd7f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.039ex; width:10.369ex; height:3.343ex;" alt="{\displaystyle {\vec {p_{1}}},\;{\vec {p_{2}}},\;{\vec {p_{3}}}}"> mit folgender <a href="/wiki/Formel" title="Formel">Formel</a> berechnen: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\;(6)\;\;d(P_{0},E)=\left|{\frac {({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{3}}}-{\vec {p_{1}}})}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{3}}}-{\vec {p_{1}}})\right|}}\cdot ({\vec {p_{0}}}-{\vec {p_{1}}})\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\;(6)\;\;d(P_{0},E)=\left|{\frac {({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{3}}}-{\vec {p_{1}}})}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{3}}}-{\vec {p_{1}}})\right|}}\cdot ({\vec {p_{0}}}-{\vec {p_{1}}})\right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93b00847914137883acaab924c4131392091872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:52.867ex; height:9.509ex;" alt="{\displaystyle \;\;(6)\;\;d(P_{0},E)=\left|{\frac {({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{3}}}-{\vec {p_{1}}})}{\left|({\vec {p_{2}}}-{\vec {p_{1}}})\times ({\vec {p_{3}}}-{\vec {p_{1}}})\right|}}\cdot ({\vec {p_{0}}}-{\vec {p_{1}}})\right|}"><a href="#cite_note-7">[6]</a><a href="#cite_note-8">[A 2]</a></dd></dl> Dabei steht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"> für das <a href="/wiki/Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukt</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" 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 }"> für das <a href="/wiki/Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\quad \right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mspace width="1em" /> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\quad \right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f0a27d435f017bc00e28b99de480f494948dc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.616ex; height:2.843ex;" alt="{\displaystyle \left|\quad \right|}"> für den <a href="/wiki/Vektor#Länge/Betrag_eines_Vektors" title="Vektor">Betrag des Vektors</a>. Beispiel <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:01_Abstand_Raum_Punkt-Ebene.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/01_Abstand_Raum_Punkt-Ebene.png/330px-01_Abstand_Raum_Punkt-Ebene.png" decoding="async" width="330" height="268" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/01_Abstand_Raum_Punkt-Ebene.png/495px-01_Abstand_Raum_Punkt-Ebene.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/01_Abstand_Raum_Punkt-Ebene.png/660px-01_Abstand_Raum_Punkt-Ebene.png 2x" data-file-width="870" data-file-height="707" /></a><figcaption>Beispiel: Konstruktion des Abstandes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32338c4e5e0ea50fe57124c394d1f962d623b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.58ex; height:2.843ex;" alt="{\displaystyle d(P,E)}"> zwischen dem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" 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}"> und der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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}"> im Raum.</figcaption></figure> Konstruktion des Abstandes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,E)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,E)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32338c4e5e0ea50fe57124c394d1f962d623b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.58ex; height:2.843ex;" alt="{\displaystyle d(P,E)}"><a href="#cite_note-9">[7]</a> Gegeben seien die Koordinaten der drei Punkte der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377c615c7b86c01c5a1fcfab1f5b3791a7a16f1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.421ex; height:2.176ex;" alt="{\displaystyle E\;}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left(1\left|0\right|0\right),\;B=\left(2\left|1\right|1\right),\;C=\left(3\left|0\right|2\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mrow> <mo>|</mo> <mn>0</mn> <mo>|</mo> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>B</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <mo>|</mo> <mn>1</mn> <mo>|</mo> </mrow> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mi>C</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <mo>|</mo> <mn>0</mn> <mo>|</mo> </mrow> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left(1\left|0\right|0\right),\;B=\left(2\left|1\right|1\right),\;C=\left(3\left|0\right|2\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c78cb81474453b1d6b9df386e77d4a0da441cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.795ex; height:2.843ex;" alt="{\displaystyle A=\left(1\left|0\right|0\right),\;B=\left(2\left|1\right|1\right),\;C=\left(3\left|0\right|2\right)}"> sowie des außerhalb liegenden Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\left(4\left|5\right|-3\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mrow> <mo>|</mo> <mn>5</mn> <mo>|</mo> </mrow> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\left(4\left|5\right|-3\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce7ffa5c8e9168a8982c88060d8c3363f193958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.696ex; height:2.843ex;" alt="{\displaystyle P=\left(4\left|5\right|-3\right).}"> Nach dem Eintragen der Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,\;B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,\;B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed23246542b8cd3fec05ea404cd81034fcbe78a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.186ex; height:2.509ex;" alt="{\displaystyle A,\;B}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> sowie des außerhalb liegenden Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd35af9d5901e795c83d9f519ac73264e74fa595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.392ex; height:2.509ex;" alt="{\displaystyle P,}"> kann die Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E:2x-2z-2=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>:</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E:2x-2z-2=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e58c72e22c3cc30b5f515bc6a7fbb28b1b23d492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.56ex; height:2.343ex;" alt="{\displaystyle E:2x-2z-2=0}"> generiert werden. Anschließend fällt man das Lot vom Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"> des <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinatenursprungs</a> auf die Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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}"> mit dem Fußpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f173f63969b15d29f4efe8403a0a7032ec8f8186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.571ex; height:2.176ex;" alt="{\displaystyle D.}"> Durch die Punkte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> verläuft auch der, aus der Parameterdarstellung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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}"> ermittelbare, <a href="/wiki/Normalenvektor" title="Normalenvektor">Normalenvektor</a> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}=\left(2\left|0\right|-2\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <mo>|</mo> <mn>0</mn> <mo>|</mo> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {n}}=\left(2\left|0\right|-2\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de4cbc86a228d21ba9f9dbec1eb3c8f63e9c4bcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.345ex; height:2.843ex;" alt="{\displaystyle {\vec {n}}=\left(2\left|0\right|-2\right).}"> Abschließend liefert die Parallele zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {OD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>O</mi> <mi>D</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {OD}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e3f1afa9739fa3023770a4b85936cf654ecced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.813ex; height:3.009ex;" alt="{\displaystyle {\overline {OD}}}"> ab dem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" 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}"> bis zur Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" 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}"> den Abstand: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,E)=3\cdot {\sqrt {2}}\;=4{,}2426\ldots \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thickmathspace" /> <mo>=</mo> <mn>4,242</mn> <mn>6</mn> <mo>…</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,E)=3\cdot {\sqrt {2}}\;=4{,}2426\ldots \;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64c046f61a530001555fd04b8bd1ec762727df0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.577ex; height:3.176ex;" alt="{\displaystyle d(P,E)=3\cdot {\sqrt {2}}\;=4{,}2426\ldots \;}">[LE]. Nachrechnung Ermittlung der einzusetzenden Werte für Formel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25115739469707c4758b189fe310a750092a80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (1)}"> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;{\begin{aligned}&\left(2\right)a=0\cdot 1-1\cdot 0+1\cdot 2-0\cdot 1+0\cdot 0-0\cdot 2=2\\&\left(3\right)b=0\cdot 2-1\cdot 1+1\cdot 3-2\cdot 2+2\cdot 1-0\cdot 3=0\\&\left(4\right)c=1\cdot 1-2\cdot 0+2\cdot 0-3\cdot 1+3\cdot 0-1\cdot 0=-2\\&\left(5\right)f=1\cdot 1\cdot 2-1\cdot 0\cdot 1+2\cdot 0\cdot 0-2\cdot 0\cdot 2+3\cdot 0\cdot 1-3\cdot 1\cdot 0=2\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>−</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <mn>0</mn> <mo>−</mo> <mn>0</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mo>⋅</mo> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mo>−</mo> <mn>0</mn> <mo>⋅</mo> <mn>3</mn> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mn>0</mn> <mo>−</mo> <mn>3</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mn>0</mn> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mn>0</mn> <mo>⋅</mo> <mn>0</mn> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mn>0</mn> <mo>⋅</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo>−</mo> <mn>3</mn> <mo>⋅</mo> <mn>1</mn> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;{\begin{aligned}&\left(2\right)a=0\cdot 1-1\cdot 0+1\cdot 2-0\cdot 1+0\cdot 0-0\cdot 2=2\\&\left(3\right)b=0\cdot 2-1\cdot 1+1\cdot 3-2\cdot 2+2\cdot 1-0\cdot 3=0\\&\left(4\right)c=1\cdot 1-2\cdot 0+2\cdot 0-3\cdot 1+3\cdot 0-1\cdot 0=-2\\&\left(5\right)f=1\cdot 1\cdot 2-1\cdot 0\cdot 1+2\cdot 0\cdot 0-2\cdot 0\cdot 2+3\cdot 0\cdot 1-3\cdot 1\cdot 0=2\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61da39389ba6793b27d77f0342e61f14650a65e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:68.669ex; height:12.509ex;" alt="{\displaystyle \;{\begin{aligned}&\left(2\right)a=0\cdot 1-1\cdot 0+1\cdot 2-0\cdot 1+0\cdot 0-0\cdot 2=2\\&\left(3\right)b=0\cdot 2-1\cdot 1+1\cdot 3-2\cdot 2+2\cdot 1-0\cdot 3=0\\&\left(4\right)c=1\cdot 1-2\cdot 0+2\cdot 0-3\cdot 1+3\cdot 0-1\cdot 0=-2\\&\left(5\right)f=1\cdot 1\cdot 2-1\cdot 0\cdot 1+2\cdot 0\cdot 0-2\cdot 0\cdot 2+3\cdot 0\cdot 1-3\cdot 1\cdot 0=2\\\end{aligned}}}"></dd></dl> Diese Werte eingesetzt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25115739469707c4758b189fe310a750092a80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (1)}"> ergeben schließlich <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\;(1)\;\;d(P_{0},E)={\frac {|2\cdot 4+0\cdot 5+(-2)\cdot (-3)-2|}{\sqrt {2^{2}+0^{2}+(-2)^{2}}}}=3\cdot {\sqrt {2}}\;=4{,}2426\ldots \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <mn>5</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msqrt> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mspace width="thickmathspace" /> <mo>=</mo> <mn>4,242</mn> <mn>6</mn> <mo>…</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\;(1)\;\;d(P_{0},E)={\frac {|2\cdot 4+0\cdot 5+(-2)\cdot (-3)-2|}{\sqrt {2^{2}+0^{2}+(-2)^{2}}}}=3\cdot {\sqrt {2}}\;=4{,}2426\ldots \;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28364da8efce313beb7413febb5d4017ed6a75ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:71.089ex; height:8.509ex;" alt="{\displaystyle \;\;(1)\;\;d(P_{0},E)={\frac {|2\cdot 4+0\cdot 5+(-2)\cdot (-3)-2|}{\sqrt {2^{2}+0^{2}+(-2)^{2}}}}=3\cdot {\sqrt {2}}\;=4{,}2426\ldots \;}">[LE].</dd></dl> Das Ergebnis gleicht dem des Beispiels. <div class="mw-heading mw-heading2"><h2 id="Andere_Definitionen">Andere Definitionen</h2>[<a href="/w/index.php?title=Abstand&veaction=edit&section=8" title="Abschnitt bearbeiten: Andere Definitionen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Andere Definitionen">Quelltext bearbeiten</a>]</div> Die Definition des <a href="/wiki/Euklidischer_Abstand" title="Euklidischer Abstand">euklidischen Abstands</a> kann mithilfe von <a href="/wiki/Metrik_(Mathematik)" class="mw-redirect" title="Metrik (Mathematik)">Metriken</a> verallgemeinert werden. Der euklidische Abstand ist der <a href="/wiki/Euklidische_Norm" title="Euklidische Norm">euklidischen Norm</a> (2-Norm) eines <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraums</a>, z. B. des <a href="/wiki/Dreidimensional" class="mw-redirect" title="Dreidimensional">dreidimensionalen</a> <a href="/wiki/Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raums</a>, zugeordnet, siehe <a href="/wiki/Metrischer_Raum#Beispiele" title="Metrischer Raum">Metrischer Raum - Beispiele</a>. <div class="mw-heading mw-heading3"><h3 id="Manhattan-Metrik">Manhattan-Metrik</h3>[<a href="/w/index.php?title=Abstand&veaction=edit&section=9" title="Abschnitt bearbeiten: Manhattan-Metrik" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Manhattan-Metrik">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Manhattan-Metrik" title="Manhattan-Metrik">Manhattan-Metrik</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Manhattan_distance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/220px-Manhattan_distance.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/330px-Manhattan_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/440px-Manhattan_distance.svg.png 2x" data-file-width="283" data-file-height="283" /></a><figcaption>Die Linien in rot, blau und gelb sind drei Beispiele für die Manhattan-Metrik zwischen den zwei schwarzen Punkten (je 12 Einheiten lang). Die grüne Linie stellt zum Vergleich den euklidischen Abstand dar, der eine Länge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6{\sqrt {2}}\approx 8{,}5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6{\sqrt {2}}\approx 8{,}5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49f16e2d67f5b84450a9d8e97f9239a32b3499b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.331ex; height:3.009ex;" alt="{\displaystyle 6{\sqrt {2}}\approx 8{,}5}"> Einheiten hat.</figcaption></figure> Die sogenannte Manhattan-Metrik ist eine <a href="/wiki/Metrischer_Raum" title="Metrischer Raum">Metrik</a>, in der der Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> zwischen zwei <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> als die <a href="/wiki/Summe" title="Summe">Summe</a> der <a href="/wiki/Absolutbetrag" class="mw-redirect" title="Absolutbetrag">absoluten</a> <a href="/wiki/Subtraktion" title="Subtraktion">Differenzen</a> ihrer Einzel<a href="/wiki/Koordinaten" class="mw-redirect" title="Koordinaten">koordinaten</a> definiert wird:<a href="#cite_note-10">[8]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)=\sum _{i}\left|A_{i}-B_{i}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)=\sum _{i}\left|A_{i}-B_{i}\right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76c674c9acab4e21a6df7c755679bc6f3601b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.647ex; height:5.509ex;" alt="{\displaystyle d(A,B)=\sum _{i}\left|A_{i}-B_{i}\right|}"></dd></dl> Die Manhattan-Metrik ist die von der <a href="/wiki/Summennorm" title="Summennorm">Summennorm</a> (1-Norm) eines <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraums</a> erzeugte Metrik. Weil die Wege zwischen zwei Punkten immer <a href="/wiki/Rechtwinklig" class="mw-redirect" title="Rechtwinklig">rechtwinklig</a> entlang den horizontalen und vertikalen Linien (Straßen) verlaufen, aber nicht durch die quadratischen „Gebäudeblöcke“, ist der Abstand zwischen zwei Punkten nicht kleiner und im Allgemeinen größer als der <a href="/wiki/Euklidischer_Abstand" title="Euklidischer Abstand">euklidischen Abstand</a>. Der Abstand zwischen zwei Punkten mit ganzzahligen <a href="/wiki/Koordinatensystem" title="Koordinatensystem">Koordinaten</a> (Kreuzungen) ist immer eine <a href="/wiki/Ganze_Zahl" title="Ganze Zahl">ganze Zahl</a>. So ist beispielsweise in der nebenstehenden Grafik die Manhattan-Metrik in einem <a href="/wiki/Zweidimensional" class="mw-redirect" title="Zweidimensional">zweidimensionalen</a> <a href="/wiki/Raum_(Mathematik)" title="Raum (Mathematik)">Raum</a>, sodass sich <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)=\left|A_{1}-B_{1}\right|+\left|A_{2}-B_{2}\right|=\left|0-6\right|+\left|0-6\right|=\left|-6\right|+\left|-6\right|=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mn>0</mn> <mo>−</mo> <mn>6</mn> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mn>0</mn> <mo>−</mo> <mn>6</mn> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <mn>6</mn> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <mn>6</mn> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)=\left|A_{1}-B_{1}\right|+\left|A_{2}-B_{2}\right|=\left|0-6\right|+\left|0-6\right|=\left|-6\right|+\left|-6\right|=12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858affe1cea51e7e343fae4b9083831778f3fd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.752ex; height:2.843ex;" alt="{\displaystyle d(A,B)=\left|A_{1}-B_{1}\right|+\left|A_{2}-B_{2}\right|=\left|0-6\right|+\left|0-6\right|=\left|-6\right|+\left|-6\right|=12}"></dd></dl> ergibt, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(A_{1},A_{2})=(0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(A_{1},A_{2})=(0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e30b8cbe97fafe68e90aecbecc9b6c88add26041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.546ex; height:2.843ex;" alt="{\displaystyle A=(A_{1},A_{2})=(0,0)}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(B_{1},B_{2})=(6,6)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(B_{1},B_{2})=(6,6)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69972774678d2ec55b44fa0bb6e21b45f3fc3b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.609ex; height:2.843ex;" alt="{\displaystyle B=(B_{1},B_{2})=(6,6)}"> die schwarz markierten Punkte sind. <div class="mw-heading mw-heading2"><h2 id="Abstandsmessung_auf_gekrümmten_Flächen">Abstandsmessung auf gekrümmten Flächen</h2>[<a href="/w/index.php?title=Abstand&veaction=edit&section=10" title="Abschnitt bearbeiten: Abstandsmessung auf gekrümmten Flächen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Abstandsmessung auf gekrümmten Flächen">Quelltext bearbeiten</a>]</div> Auf der <a href="/wiki/Kugeloberfl%C3%A4che" class="mw-redirect" title="Kugeloberfläche">Kugeloberfläche</a> wird der Abstand entlang von <a href="/wiki/Gro%C3%9Fkreis" title="Großkreis">Großkreisen</a> bestimmt und im <a href="/wiki/Gradma%C3%9F" class="mw-redirect" title="Gradmaß">Gradmaß</a> oder <a href="/wiki/Bogenma%C3%9F" class="mw-redirect" title="Bogenmaß">Bogenmaß</a> angegeben. Zur Berechnung des Abstandes siehe <a href="/wiki/Orthodrome" title="Orthodrome">Orthodrome</a>. Auf dem <a href="/wiki/Erdellipsoid" class="mw-redirect" title="Erdellipsoid">Erdellipsoid</a> oder anderen <a href="/wiki/Konvexe_Menge" title="Konvexe Menge">konvexen</a> Flächen benutzt man die <a href="/wiki/Geod%C3%A4tische_Linie" class="mw-redirect" title="Geodätische Linie">geodätische Linie</a> oder den <a href="/w/index.php?title=Normalschnitt&action=edit&redlink=1" class="new" title="Normalschnitt (Seite nicht vorhanden)">Normalschnitt</a>. In der <a href="/wiki/Geod%C3%A4sie" title="Geodäsie">Geodäsie</a> und den <a href="/wiki/Geowissenschaften" title="Geowissenschaften">Geowissenschaften</a> spricht man eher von Distanz oder Entfernung, die metrisch angegeben wird. <div class="mw-heading mw-heading2"><h2 id="Dichtestes_Punktpaar">Dichtestes Punktpaar</h2>[<a href="/w/index.php?title=Abstand&veaction=edit&section=11" title="Abschnitt bearbeiten: Dichtestes Punktpaar" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Dichtestes Punktpaar">Quelltext bearbeiten</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Closest_pair_of_points.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Closest_pair_of_points.svg/220px-Closest_pair_of_points.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Closest_pair_of_points.svg/330px-Closest_pair_of_points.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Closest_pair_of_points.svg/440px-Closest_pair_of_points.svg.png 2x" data-file-width="256" data-file-height="256" /></a><figcaption>Die zwei Punkte mit dem kleinsten Abstand sind rot markiert.</figcaption></figure> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Dichtestes_Punktpaar" title="Dichtestes Punktpaar">Dichtestes Punktpaar</a></div> Das Problem des dichtesten Punktpaares (englisch closest pair of points problem) ist die Suche nach den zwei am dichtesten beieinander liegenden <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkten</a> in einer <a href="/wiki/Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a>. Gegeben ist eine beliebige <a href="/wiki/Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> von Punkten in der Ebene und gesucht sind zwei dieser Punkte, sodass der <a href="/wiki/Euklidischer_Abstand" title="Euklidischer Abstand">euklidische Abstand</a> minimal ist. Ein ähnliches Problem ist die Suche nach den zwei am weitesten voneinander entfernten Punkten in der Ebene, also den zwei Punkten mit dem maximalen <a href="/wiki/Euklidischer_Abstand" title="Euklidischer Abstand">euklidischen Abstand</a>. Der <a href="/wiki/Brute_Force" class="mw-redirect" title="Brute Force">Brute-force</a>-Algorithmus berechnet die Abstände zwischen allen möglichen Punktpaaren und wählt das Punktpaar mit dem kleinsten Abstand aus. Die <a href="/wiki/Laufzeit_(Informatik)" title="Laufzeit (Informatik)">Laufzeit</a> des <a href="/wiki/Algorithmus" title="Algorithmus">Algorithmus</a> ist quadratisch und liegt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd9594a16cb898b8f2a2dff9227a385ec183392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{2})}">. Ein <a href="/wiki/Teile_und_herrsche_(Informatik)" class="mw-redirect" title="Teile und herrsche (Informatik)">Divide-and-conquer</a>-Algorithmus hat eine <a href="/wiki/Laufzeit_(Informatik)" title="Laufzeit (Informatik)">Laufzeit</a>, die in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n\cdot \log n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>⋅</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n\cdot \log n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837218b6d28ce003c0f81f7af156da3ede782fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.41ex; height:2.843ex;" alt="{\displaystyle O(n\cdot \log n)}"> liegt. <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2>[<a href="/w/index.php?title=Abstand&veaction=edit&section=12" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Siehe auch">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Entfernungsma%C3%9F" class="mw-redirect" title="Entfernungsmaß">Entfernungsmaß</a></li> <li><a href="/wiki/Entfernungsmessung" title="Entfernungsmessung">Entfernungsmessung</a></li> <li><a href="/wiki/Geod%C3%A4tische_Distanz" title="Geodätische Distanz">Geodätische Distanz</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2>[<a href="/w/index.php?title=Abstand&veaction=edit&section=13" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Weblinks">Quelltext bearbeiten</a>]</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"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></div><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Distance_between_two_points?uselang=de">Commons: Abstand</a> – Sammlung von Bildern</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/16px-Wiktfavicon_en.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/24px-Wiktfavicon_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/32px-Wiktfavicon_en.svg.png 2x" data-file-width="16" data-file-height="16" /><a href="https://de.wiktionary.org/wiki/Abstand" class="extiw" title="wikt:Abstand">Wiktionary: Abstand</a> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/13px-Wikiquote-logo.svg.png" decoding="async" width="13" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/20px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/27px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></div><a href="https://de.wikiquote.org/wiki/Abstand" class="extiw" title="q:Abstand">Wikiquote: Abstand</a> – Zitate </div> <div class="mw-heading mw-heading2"><h2 id="Anmerkungen">Anmerkungen</h2>[<a href="/w/index.php?title=Abstand&veaction=edit&section=14" title="Abschnitt bearbeiten: Anmerkungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Anmerkungen">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-Konstante_f-6">↑ <a href="#cite_ref-Konstante_f_6-0">a</a> <a href="#cite_ref-Konstante_f_6-1">b</a> Um eine Doppelbezeichnung der Konstante <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> zu vermeiden wurde mit passendem Vorzeichen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0edfedee3fca0a26dd6f515e7ed9517a4e2cd04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.087ex; height:2.509ex;" alt="{\displaystyle -f}"> gewählt. </li> <li id="cite_note-8"><a href="#cite_ref-8">↑</a> Im Gegensatz zur Formel aus dem englischen Sprachraum wurde für den Abstand die Bezeichnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> anstatt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> gewählt. </li> </ol> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Abstand&veaction=edit&section=15" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Abstand&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="/wiki/Petra_Stein" title="Petra Stein">Petra Stein</a>, Sven Vollnhals: <a rel="nofollow" class="external text" href="https://www.uni-due.de/imperia/md/content/soziologie/stein/skript_clusteranalyse_sose2011.pdf#page=18&zoom=80,-502,847">3.5.1 Spezialfälle der Minkowski-Metrik: Das euklidische Distanzmaß.</a> 3.5 Distanz- und Ähnlichkeitsmaße für metrische Variablen. In: Grundlagen clusteranalytischer Verfahren. Universität Duisburg-Essen, 1. April 2011, S. 15, abgerufen am 19. Oktober 2018.<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%3AAbstand&rft.title=3.5.1+Spezialf%C3%A4lle+der+Minkowski-Metrik%3A+Das+euklidische+Distanzma%C3%9F&rft.description=3.5.1+Spezialf%C3%A4lle+der+Minkowski-Metrik%3A+Das+euklidische+Distanzma%C3%9F&rft.identifier=https%3A%2F%2Fwww.uni-due.de%2Fimperia%2Fmd%2Fcontent%2Fsoziologie%2Fstein%2Fskript_clusteranalyse_sose2011.pdf%23page%3D18%26zoom%3D80%2C-502%2C847&rft.creator=%5B%5BPetra+Stein%5D%5D%2C+Sven+Vollnhals&rft.publisher=Universit%C3%A4t+Duisburg-Essen">  </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> Klaus Hefft: <a rel="nofollow" class="external text" href="https://www.thphys.uni-heidelberg.de/~hefft/vk1/#913d">9.1.3 Euklidischer Raum.</a> 9.1 Dreidimensionaler euklidischer Raum. In: MATHEMATISCHER VORKURS zum Studium der Physik. Universität Heidelberg, 8. Juli 2018, abgerufen am 19. Oktober 2018.<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%3AAbstand&rft.title=9.1.3+Euklidischer+Raum&rft.description=9.1.3+Euklidischer+Raum&rft.identifier=https%3A%2F%2Fwww.thphys.uni-heidelberg.de%2F%7Ehefft%2Fvk1%2F%23913d&rft.creator=Klaus+Hefft&rft.publisher=Universit%C3%A4t+Heidelberg">  </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> Wolfram MathWorld: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html">Point-Line Distance--2-Dimensional</a> </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> Wolfram MathWorld: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html">Point-Line Distance--3-Dimensional</a> </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> Wolfram MathWorld: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Line-LineDistance.html">Line-Line Distance</a> </li> <li id="cite_note-7"><a href="#cite_ref-7">↑</a> Wolfram MathWorld: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Point-PlaneDistance.html">Point-Plane Distance</a> </li> <li id="cite_note-9"><a href="#cite_ref-9">↑</a> R. Verfürth: <a rel="nofollow" class="external text" href="https://www.ruhr-uni-bochum.de/num1/files/lectures/MBBI1.pdf#page=37&zoom=auto,-13,632">I.5.7. Parameterfreie Darstellungen einer Ebene.; Beispiel I.5.6.</a> Mathematik für Maschinenbauer, Bauingenieure und Umwelttechniker I. Ruhr-Universität Bochum, Dezember 2006, S. 37―39, abgerufen am 22. Mai 2021.<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%3AAbstand&rft.title=I.5.7.+Parameterfreie+Darstellungen+einer+Ebene.%3B+Beispiel+I.5.6.&rft.description=I.5.7.+Parameterfreie+Darstellungen+einer+Ebene.%3B+Beispiel+I.5.6.&rft.identifier=https%3A%2F%2Fwww.ruhr-uni-bochum.de%2Fnum1%2Ffiles%2Flectures%2FMBBI1.pdf%23page%3D37%26zoom%3Dauto%2C-13%2C632&rft.creator=R.+Verf%C3%BCrth&rft.publisher=Ruhr-Universit%C3%A4t+Bochum&rft.date=2006-12">  </li> <li id="cite_note-10"><a href="#cite_ref-10">↑</a> Wolfram MathWorld: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TaxicabMetric.html">Taxicab Metric</a> </li> </ol> <div class="hintergrundfarbe1 rahmenfarbe1 navigation-not-searchable normdaten-typ-s" style="border-style: solid; border-width: 1px; clear: left; margin-bottom:1em; margin-top:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="normdaten"> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div> Normdaten (Sachbegriff): <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4228463-6">4228463-6</a> (<a rel="nofollow" class="external text" href="https://lobid.org/gnd/4228463-6">lobid</a>, <a rel="nofollow" class="external text" href="https://swb.bsz-bw.de/DB=2.104/SET=1/TTL=1/CMD?retrace=0&trm_old=&ACT=SRCHA&IKT=2999&SRT=RLV&TRM=4228463-6">OGND</a>, <a rel="nofollow" class="external text" href="https://prometheus.lmu.de/gnd/4228463-6">AKS</a>) </div> </div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Abstand&oldid=243778450">https://de.wikipedia.org/w/index.php?title=Abstand&oldid=243778450</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Euklidische_Geometrie" title="Kategorie:Euklidische Geometrie">Euklidische Geometrie</a></li><li><a href="/wiki/Kategorie:Navigation" title="Kategorie:Navigation">Navigation</a></li><li><a href="/wiki/Kategorie:Dimensionale_Messtechnik" title="Kategorie:Dimensionale Messtechnik">Dimensionale Messtechnik</a></li><li><a href="/wiki/Kategorie:Messgr%C3%B6%C3%9Fe" title="Kategorie:Messgröße">Messgröße</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > Meine Werkzeuge </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item">Nicht angemeldet</li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n">Diskussionsseite</a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y">Beiträge</a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Abstand" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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id="p-coll-print_export-label" class="vector-menu-heading " > Drucken/exportieren </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Spezial:DownloadAsPdf&page=Abstand&action=show-download-screen">Als PDF herunterladen</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Abstand&printable=yes" title="Druckansicht dieser Seite [p]" accesskey="p">Druckversion</a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > In anderen Projekten </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Distance" hreflang="en">Commons</a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://de.wikiquote.org/wiki/Abstand" hreflang="de">Wikiquote</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q126017" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g">Wikidata-Datenobjekt</a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > In anderen Sprachen </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Afstand" title="Afstand – Afrikaans" lang="af" hreflang="af" data-title="Afstand" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%AD%E1%89%80%E1%89%B5" title="ርቀት – Amharisch" lang="am" hreflang="am" data-title="ርቀት" data-language-autonym="አማርኛ" data-language-local-name="Amharisch" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Distancia" title="Distancia – Aragonesisch" lang="an" hreflang="an" data-title="Distancia" data-language-autonym="Aragonés" data-language-local-name="Aragonesisch" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D9%81%D8%A9" title="مسافة – Arabisch" lang="ar" hreflang="ar" data-title="مسافة" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%A8%D8%B9%D9%88%D8%AF%D9%8A%D8%A9" title="بعودية – Marokkanisches Arabisch" lang="ary" hreflang="ary" data-title="بعودية" data-language-autonym="الدارجة" data-language-local-name="Marokkanisches Arabisch" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distancia" title="Distancia – Asturisch" lang="ast" hreflang="ast" data-title="Distancia" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C9%99saf%C9%99" title="Məsafə – Aserbaidschanisch" lang="az" hreflang="az" data-title="Məsafə" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D9%81%D8%AA" title="مسافت – Südaserbaidschanisch" lang="azb" hreflang="azb" data-title="مسافت" data-language-autonym="تۆرکجه" data-language-local-name="Südaserbaidschanisch" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BB%D1%8B%D2%AB%D0%BB%D1%8B%D2%A1" title="Алыҫлыҡ – Baschkirisch" lang="ba" hreflang="ba" data-title="Алыҫлыҡ" data-language-autonym="Башҡортса" data-language-local-name="Baschkirisch" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Distansya" title="Distansya – Zentralbikolano" lang="bcl" hreflang="bcl" data-title="Distansya" data-language-autonym="Bikol Central" data-language-local-name="Zentralbikolano" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B4%D0%BB%D0%B5%D0%B3%D0%BB%D0%B0%D1%81%D1%86%D1%8C" title="Адлегласць – Belarussisch" lang="be" hreflang="be" data-title="Адлегласць" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%B4%D0%BB%D0%B5%D0%B3%D0%BB%D0%B0%D1%81%D1%8C%D1%86%D1%8C" title="Адлегласьць – Weißrussisch (Taraschkewiza)" lang="be-tarask" hreflang="be-tarask" data-title="Адлегласьць" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Weißrussisch (Taraschkewiza)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5" title="Разстояние – Bulgarisch" lang="bg" hreflang="bg" data-title="Разстояние" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%A6%E0%A5%81%E0%A4%B0%E0%A4%A4%E0%A5%8D%E0%A4%B5" title="दुरत्व – Bhojpuri" lang="bh" hreflang="bh" data-title="दुरत्व" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A7%82%E0%A6%B0%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="দূরত্ব – Bengalisch" lang="bn" hreflang="bn" data-title="দূরত্ব" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%97%D0%B0%D0%B9" title="Зай – Russisches Burjatisch" lang="bxr" hreflang="bxr" data-title="Зай" data-language-autonym="Буряад" data-language-local-name="Russisches Burjatisch" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Dist%C3%A0ncia" title="Distància – Katalanisch" lang="ca" hreflang="ca" data-title="Distància" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AF%D9%88%D9%88%D8%B1%DB%8C" title="دووری – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="دووری" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vzd%C3%A1lenost" title="Vzdálenost – Tschechisch" lang="cs" hreflang="cs" data-title="Vzdálenost" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BD%C3%A7%C4%95%D1%88" title="Инçĕш – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Инçĕш" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Afstandsformlen" title="Afstandsformlen – Dänisch" lang="da" hreflang="da" data-title="Afstandsformlen" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%80%CF%8C%CF%83%CF%84%CE%B1%CF%83%CE%B7_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Απόσταση (γεωμετρία) – Griechisch" lang="el" hreflang="el" data-title="Απόσταση (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Distance" title="Distance – Englisch" lang="en" hreflang="en" data-title="Distance" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Distanco" title="Distanco – Esperanto" lang="eo" hreflang="eo" data-title="Distanco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distancia" title="Distancia – Spanisch" lang="es" hreflang="es" data-title="Distancia" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kaugus" title="Kaugus – Estnisch" lang="et" hreflang="et" data-title="Kaugus" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Distantzia" title="Distantzia – Baskisch" lang="eu" hreflang="eu" data-title="Distantzia" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87" title="فاصله – Persisch" lang="fa" hreflang="fa" data-title="فاصله" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/V%C3%A4limatka" title="Välimatka – Finnisch" lang="fi" hreflang="fi" data-title="Välimatka" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Distance_(math%C3%A9matiques)" title="Distance (mathématiques) – Französisch" lang="fr" hreflang="fr" data-title="Distance (mathématiques)" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distancia" title="Distancia – Galicisch" lang="gl" hreflang="gl" data-title="Distancia" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%A7" title="מרחק – Hebräisch" lang="he" hreflang="he" data-title="מרחק" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%82%E0%A4%B0%E0%A5%80" title="दूरी – Hindi" lang="hi" hreflang="hi" data-title="दूरी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Daljina" title="Daljina – Kroatisch" lang="hr" hreflang="hr" data-title="Daljina" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Distans" title="Distans – Haiti-Kreolisch" lang="ht" hreflang="ht" data-title="Distans" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haiti-Kreolisch" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%A1vols%C3%A1g" title="Távolság – Ungarisch" lang="hu" hreflang="hu" data-title="Távolság" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A5%D5%BC%D5%A1%D5%BE%D5%B8%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D5%A5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6)" title="Հեռավորություն (երկրաչափություն) – Armenisch" lang="hy" hreflang="hy" data-title="Հեռավորություն (երկրաչափություն)" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Distantia" title="Distantia – Interlingua" lang="ia" hreflang="ia" data-title="Distantia" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Penyauh" title="Penyauh – Iban" lang="iba" hreflang="iba" data-title="Penyauh" data-language-autonym="Jaku Iban" data-language-local-name="Iban" class="interlanguage-link-target">Jaku Iban</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Jarak" title="Jarak – Indonesisch" lang="id" hreflang="id" data-title="Jarak" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Disto" title="Disto – Ido" lang="io" hreflang="io" data-title="Disto" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fjarl%C3%A6g%C3%B0" title="Fjarlægð – Isländisch" lang="is" hreflang="is" data-title="Fjarlægð" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distanza_(matematica)" title="Distanza (matematica) – Italienisch" lang="it" hreflang="it" data-title="Distanza (matematica)" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%B7%9D%E9%9B%A2" title="距離 – Japanisch" lang="ja" hreflang="ja" data-title="距離" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%9C%E1%83%AB%E1%83%98%E1%83%9A%E1%83%98" title="მანძილი – Georgisch" lang="ka" hreflang="ka" data-title="მანძილი" data-language-autonym="ქართული" data-language-local-name="Georgisch" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Ameccaq" title="Ameccaq – Kabylisch" lang="kab" hreflang="kab" data-title="Ameccaq" data-language-autonym="Taqbaylit" data-language-local-name="Kabylisch" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%80%D0%B0%D2%9B%D0%B0%D1%88%D1%8B%D2%9B%D1%82%D1%8B%D2%9B" title="Арақашықтық – Kasachisch" lang="kk" hreflang="kk" data-title="Арақашықтық" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A6%E0%B3%82%E0%B2%B0" title="ದೂರ – Kannada" lang="kn" hreflang="kn" data-title="ದೂರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B1%B0%EB%A6%AC" title="거리 – Koreanisch" lang="ko" hreflang="ko" data-title="거리" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-krc mw-list-item"><a href="https://krc.wikipedia.org/wiki/%D0%A3%D0%B7%D0%B0%D0%BA%D1%8A%D0%BB%D1%8B%D0%BA%D1%8A" title="Узакълыкъ – Karatschaiisch-Balkarisch" lang="krc" hreflang="krc" data-title="Узакълыкъ" data-language-autonym="Къарачай-малкъар" data-language-local-name="Karatschaiisch-Balkarisch" class="interlanguage-link-target">Къарачай-малкъар</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/D%C3%BBrah%C3%AE" title="Dûrahî – Kurdisch" lang="ku" hreflang="ku" data-title="Dûrahî" data-language-autonym="Kurdî" data-language-local-name="Kurdisch" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%B0%D0%BB%D1%8B%D0%BA" title="Аралык – Kirgisisch" lang="ky" hreflang="ky" data-title="Аралык" data-language-autonym="Кыргызча" data-language-local-name="Kirgisisch" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Ofstand" title="Ofstand – Luxemburgisch" lang="lb" hreflang="lb" data-title="Ofstand" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxemburgisch" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Aafstandj" title="Aafstandj – Limburgisch" lang="li" hreflang="li" data-title="Aafstandj" data-language-autonym="Limburgs" data-language-local-name="Limburgisch" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atstumas" title="Atstumas – Litauisch" lang="lt" hreflang="lt" data-title="Atstumas" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Att%C4%81lums" title="Attālums – Lettisch" lang="lv" hreflang="lv" data-title="Attālums" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%82%D0%BE%D1%98%D0%B0%D0%BD%D0%B8%D0%B5" title="Растојание – Mazedonisch" lang="mk" hreflang="mk" data-title="Растојание" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%A4%E0%A4%B0" title="अंतर – Marathi" lang="mr" hreflang="mr" data-title="अंतर" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Jarak" title="Jarak – Malaiisch" lang="ms" hreflang="ms" data-title="Jarak" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%80%E1%80%BD%E1%80%AC%E1%80%A1%E1%80%9D%E1%80%B1%E1%80%B8" title="အကွာအဝေး – Birmanisch" lang="my" hreflang="my" data-title="အကွာအဝေး" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Birmanisch" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Afstand" title="Afstand – Niederländisch" lang="nl" hreflang="nl" data-title="Afstand" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Avstand" title="Avstand – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Avstand" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Avstand" title="Avstand – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Avstand" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%88%E0%A8%82%E0%A8%A1%E0%A8%BE" title="ਪੈਂਡਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਪੈਂਡਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Odleg%C5%82o%C5%9B%C4%87" title="Odległość – Polnisch" lang="pl" hreflang="pl" data-title="Odległość" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%88%D8%A7%D9%BC%D9%86" title="واټن – Paschtu" lang="ps" hreflang="ps" data-title="واټن" data-language-autonym="پښتو" data-language-local-name="Paschtu" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Dist%C3%A2ncia" title="Distância – Portugiesisch" lang="pt" hreflang="pt" data-title="Distância" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5" title="Расстояние – Russisch" lang="ru" hreflang="ru" data-title="Расстояние" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Distance" title="Distance – Schottisch" lang="sco" hreflang="sco" data-title="Distance" data-language-autonym="Scots" data-language-local-name="Schottisch" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rastojanje" title="Rastojanje – Serbokroatisch" lang="sh" hreflang="sh" data-title="Rastojanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Distance" title="Distance – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Distance" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vzdialenos%C5%A5" title="Vzdialenosť – Slowakisch" lang="sk" hreflang="sk" data-title="Vzdialenosť" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Razdalja" title="Razdalja – Slowenisch" lang="sl" hreflang="sl" data-title="Razdalja" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nhambwe" title="Nhambwe – Shona" lang="sn" hreflang="sn" data-title="Nhambwe" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Distanca" title="Distanca – Albanisch" lang="sq" hreflang="sq" data-title="Distanca" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%82%D0%BE%D1%98%D0%B0%D1%9A%D0%B5" title="Растојање – Serbisch" lang="sr" hreflang="sr" data-title="Растојање" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Avst%C3%A5nd" title="Avstånd – Schwedisch" lang="sv" hreflang="sv" data-title="Avstånd" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Umbali" title="Umbali – Suaheli" lang="sw" hreflang="sw" data-title="Umbali" data-language-autonym="Kiswahili" data-language-local-name="Suaheli" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%82%E0%AE%B0%E0%AE%AE%E0%AF%8D" title="தூரம் – Tamil" lang="ta" hreflang="ta" data-title="தூரம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A6%E0%B1%82%E0%B0%B0%E0%B0%82" title="దూరం – Telugu" lang="te" hreflang="te" data-title="దూరం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%81%D0%BE%D1%84%D0%B0" title="Масофа – Tadschikisch" lang="tg" hreflang="tg" data-title="Масофа" data-language-autonym="Тоҷикӣ" data-language-local-name="Tadschikisch" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%A2%E0%B8%B0%E0%B8%97%E0%B8%B2%E0%B8%87" title="ระยะทาง – Thailändisch" lang="th" hreflang="th" data-title="ระยะทาง" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distansiya" title="Distansiya – Tagalog" lang="tl" hreflang="tl" data-title="Distansiya" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mesafe" title="Mesafe – Türkisch" lang="tr" hreflang="tr" data-title="Mesafe" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Mpfhuka" title="Mpfhuka – Tsonga" lang="ts" hreflang="ts" data-title="Mpfhuka" data-language-autonym="Xitsonga" data-language-local-name="Tsonga" class="interlanguage-link-target">Xitsonga</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D1%96%D0%B4%D1%81%D1%82%D0%B0%D0%BD%D1%8C" title="Відстань – Ukrainisch" lang="uk" hreflang="uk" data-title="Відстань" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%81%D8%A7%D8%B5%D9%84%DB%81" title="فاصلہ – Urdu" lang="ur" hreflang="ur" data-title="فاصلہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Masofa" title="Masofa – Usbekisch" lang="uz" hreflang="uz" data-title="Masofa" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Distansa" title="Distansa – Venetisch" lang="vec" hreflang="vec" data-title="Distansa" data-language-autonym="Vèneto" data-language-local-name="Venetisch" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kho%E1%BA%A3ng_c%C3%A1ch" title="Khoảng cách – Vietnamesisch" lang="vi" hreflang="vi" data-title="Khoảng cách" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Distansiya" title="Distansiya – Waray" lang="war" hreflang="war" data-title="Distansiya" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%B7%9D%E7%A6%BB" title="距离 – Wu" lang="wuu" hreflang="wuu" data-title="距离" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Umgama" title="Umgama – Xhosa" lang="xh" hreflang="xh" data-title="Umgama" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%95%D7%95%D7%99%D7%99%D7%98%D7%A7%D7%99%D7%99%D7%98" title="ווייטקייט – Jiddisch" lang="yi" hreflang="yi" data-title="ווייטקייט" data-language-autonym="ייִדיש" data-language-local-name="Jiddisch" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B7%9D%E7%A6%BB" title="距离 – Chinesisch" lang="zh" hreflang="zh" data-title="距离" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%B7%9D" title="距 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="距" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/K%C5%AB-l%C4%AB" title="Kū-lī – Min Nan" lang="nan" hreflang="nan" data-title="Kū-lī" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Min Nan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%B7%9D%E9%9B%A2" title="距離 – Kantonesisch" lang="yue" hreflang="yue" data-title="距離" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target">粵語</a></li> </ul> <div class="after-portlet after-portlet-lang"><a 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Abstand – Wikipedia