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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Normalenform</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Normalenform</b><sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>, <b>Normalform</b><sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> oder <b>Normalengleichung</b><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> eine spezielle Form einer <a href="Geradengleichung" title="Geradengleichung">Geradengleichung</a> oder <a href="Ebenengleichung" title="Ebenengleichung">Ebenengleichung</a>. In der Normalenform wird eine <a href="Gerade" title="Gerade">Gerade</a> in der <a href="Euklidische_Ebene" class="mw-redirect" title="Euklidische Ebene">euklidischen Ebene</a> oder eine <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a> im <a href="Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a> durch einen <a href="St%C3%BCtzvektor" class="mw-redirect" title="Stützvektor">Stützvektor</a> und einen <a href="Normalenvektor" title="Normalenvektor">Normalenvektor</a> dargestellt. Eine Gerade oder Ebene besteht dann aus denjenigen Punkten in der Ebene oder im Raum, für die der Differenzvektor aus Ortsvektor und Stützvektor <a href="Orthogonalit%C3%A4t" title="Orthogonalität">senkrecht</a> zum Normalenvektor steht. Die Normalenform ist damit eine spezielle <a href="Implizite_Funktion" class="mw-redirect" title="Implizite Funktion">implizite Darstellung</a> der Gerade oder Ebene.
</p><p>Einen Spezialfall der Normalenform stellt die <a href="Hessesche_Normalform" title="Hessesche Normalform">hessesche Normalform</a> dar, bei der der Normalenvektor <a href="Einheitsvektor" title="Einheitsvektor">normiert</a> und <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">orientiert</a> ist und statt des Stützvektors der <a href="Abstand" title="Abstand">Abstand</a> vom <a href="Koordinatenursprung" class="mw-redirect" title="Koordinatenursprung">Koordinatenursprung</a> verwendet wird.
</p>
<div class="mw-heading mw-heading2"><h2 id="Normalenform_einer_Geradengleichung">Normalenform einer Geradengleichung</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Darstellung">Darstellung</h3></div>
<p>In der Normalenform wird eine Gerade in der Ebene durch einen Stützvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> und einen Normalenvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"></span> beschrieben (siehe Bild 1). Eine Gerade besteht dann aus denjenigen Punkten in der Ebene, deren Ortsvektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">
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</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {n}}=0\quad }">
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<annotation encoding="application/x-tex">{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {n}}=0\quad }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b0f899f64e213baaafa0641adca1d8975434b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.962ex; height:2.843ex;" alt="{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {n}}=0\quad }" loading="lazy"></span>bzw. äquivalent hierzu<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}">
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<annotation encoding="application/x-tex">{\displaystyle \quad {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e17eb19e1d84b88cb9dae51d6206dbb451a7fc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.224ex; height:2.676ex;" alt="{\displaystyle \quad {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}" loading="lazy"></span></dd></dl>
<p>erfüllen. Hierbei bezeichnet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"></span> das <a href="Skalarprodukt" title="Skalarprodukt">Skalarprodukt</a> zweier Vektoren, welches null ist, wenn die Vektoren <a href="Orthogonalit%C3%A4t" title="Orthogonalität">senkrecht</a> aufeinander stehen. Der Stützvektor ist der <a href="Ortsvektor" title="Ortsvektor">Ortsvektor</a> eines beliebigen Punkts auf der Gerade, der auch als <i>Stützpunkt</i> oder <i>Aufpunkt</i> bezeichnet wird. Der Normalenvektor ist ein Vektor, der mit der Gerade einen <a href="Rechter_Winkel" title="Rechter Winkel">rechten Winkel</a> bildet. In der Normalenform werden demnach die Punkte der Geraden implizit dadurch definiert, dass der Differenzvektor aus Ortsvektor und Stützvektor senkrecht zum Normalenvektor der Gerade steht.
</p><p>Legt man ein ebenes rechtwinkliges Koordinatensystem zugrunde und stellt die Vektoren mithilfe ihrer Komponenten dar, so liest sich die Geradengleichung als
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}-{\begin{pmatrix}p_{1}\\p_{2}\end{pmatrix}}\right)\cdot {\begin{pmatrix}n_{1}\\n_{2}\end{pmatrix}}=0}">
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<annotation encoding="application/x-tex">{\displaystyle \left({\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}-{\begin{pmatrix}p_{1}\\p_{2}\end{pmatrix}}\right)\cdot {\begin{pmatrix}n_{1}\\n_{2}\end{pmatrix}}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d2f878d18d02ea1cd6aa7bce73f730e9ff5c117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.776ex; height:6.176ex;" alt="{\displaystyle \left({\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}-{\begin{pmatrix}p_{1}\\p_{2}\end{pmatrix}}\right)\cdot {\begin{pmatrix}n_{1}\\n_{2}\end{pmatrix}}=0}" loading="lazy"></span>.</dd></dl>
<p>Schreibt man diese Gleichung aus, so erhält man die Normalenform als Koordinatengleichung:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}-p_{1})\cdot n_{1}+(x_{2}-p_{2})\cdot n_{2}=0\quad }">
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<annotation encoding="application/x-tex">{\displaystyle (x_{1}-p_{1})\cdot n_{1}+(x_{2}-p_{2})\cdot n_{2}=0\quad }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60ec2aa4717b2b563b6091135dd530ba46f93d51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.195ex; height:2.843ex;" alt="{\displaystyle (x_{1}-p_{1})\cdot n_{1}+(x_{2}-p_{2})\cdot n_{2}=0\quad }" loading="lazy"></span>bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad x_{1}n_{1}+x_{2}n_{2}=p_{1}n_{1}+p_{2}n_{2}}">
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<annotation encoding="application/x-tex">{\displaystyle \quad x_{1}n_{1}+x_{2}n_{2}=p_{1}n_{1}+p_{2}n_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb6d0afac99a52ff6a59df105c727f8ed40eb527.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.113ex; height:2.343ex;" alt="{\displaystyle \quad x_{1}n_{1}+x_{2}n_{2}=p_{1}n_{1}+p_{2}n_{2}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Beispiel">Beispiel</h3></div>
<p>Im Bild 2 ist der Ortsvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\begin{pmatrix}4\\1\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\begin{pmatrix}4\\1\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/680c0dce568dfb79ced98e5edbae5413dfd48b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.763ex; height:6.176ex;" alt="{\displaystyle {\vec {x}}={\begin{pmatrix}4\\1\end{pmatrix}}}" loading="lazy"></span>, der Stützvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}={\begin{pmatrix}3\\3\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}3\\3\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5294f4af7258cc6054d9dfa223ae408a73f3974.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:9.848ex; height:6.176ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}3\\3\end{pmatrix}}}" loading="lazy"></span> und der Normalenvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\begin{pmatrix}2\\1\end{pmatrix}}}">
<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 class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\begin{pmatrix}2\\1\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f5756d3933b11877b5c8bccec12b4d2caa02164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.828ex; height:6.176ex;" alt="{\displaystyle {\vec {n}}={\begin{pmatrix}2\\1\end{pmatrix}}}" loading="lazy"></span>. Durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4-3)\cdot 2+(1-3)\cdot 1=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mn>2</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (4-3)\cdot 2+(1-3)\cdot 1=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adb3880fb3211504eafe0e037fa56106b68588d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.734ex; height:2.843ex;" alt="{\displaystyle (4-3)\cdot 2+(1-3)\cdot 1=0}" loading="lazy"></span></dd></dl>
<p>wird bestätigt, dass der Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(4|1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X(4|1)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4969459f8d6d6cb993c52c5e294ce50ca6d147a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.761ex; height:2.843ex;" alt="{\displaystyle X(4|1)}" loading="lazy"></span> auf der Geraden liegt.
</p><p>Die Normalenform in Koordinatendarstellung ist
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;2x_{1}+x_{2}=9.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thickmathspace"></mspace>
<mn>2</mn>
<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>
<mo>=</mo>
<mn>9.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \;2x_{1}+x_{2}=9.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3dceff688043898bb5b3c9211c9cda5d0bcffaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.324ex; height:2.509ex;" alt="{\displaystyle \;2x_{1}+x_{2}=9.}" loading="lazy"></span></dd></dl>
<p>Jede Wahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}|x_{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>1</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x_{1}|x_{2})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3ec30e3a2fdfb183c0c94a675d8adca3f401c34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.224ex; height:2.843ex;" alt="{\displaystyle (x_{1}|x_{2})}" loading="lazy"></span>, die diese Gleichung erfüllt, beispielsweise <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2|5)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>5</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (2|5)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42382a2abae493c0d4f1580cf709216957a5458c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.781ex; height:2.843ex;" alt="{\displaystyle (2|5)}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3{,}5|2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>3</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>,</mo>
</mrow>
<mn>5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (3{,}5|2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e627ab226dc11ea00ca282f0341a8d91e546d05a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.59ex; height:2.843ex;" alt="{\displaystyle (3{,}5|2)}" loading="lazy"></span>, entspricht dann einem Geradenpunkt.
</p>
<div class="mw-heading mw-heading3"><h3 id="Eigenschaften">Eigenschaften</h3></div>
<p>Ein Punkt, dessen Ortsvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> die Normalengleichung nicht erfüllt, liegt für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}\cdot {\vec {n}}>{\vec {p}}\cdot {\vec {n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}\cdot {\vec {n}}>{\vec {p}}\cdot {\vec {n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9910bf6022be1c677503da374424b193cb829f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.901ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}\cdot {\vec {n}}>{\vec {p}}\cdot {\vec {n}}}" loading="lazy"></span> auf derjenigen Seite der Gerade, in die der Normalenvektor zeigt, und ansonsten auf der anderen Seite.
</p>
<div class="mw-heading mw-heading3"><h3 id="Berechnung">Berechnung</h3></div>
<div class="mw-heading mw-heading4"><h4 id="Aus_der_Parameterform">Aus der Parameterform</h4></div>
<p>Aus der <a href="Parameterform#Parameterform_einer_Geradengleichung" title="Parameterform">Parameterform einer Geradengleichung</a> lässt sich ein Normalenvektor der Geraden bestimmen, indem die beiden Komponenten des <a href="Richtungsvektor" class="mw-redirect" title="Richtungsvektor">Richtungsvektors</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>u</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c41e9cf70c5e5b56e2128a136985a75f90ba43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}" loading="lazy"></span> der Geraden vertauscht werden und bei einer der beiden Komponenten das Vorzeichen geändert wird, das heißt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\begin{pmatrix}-u_{2}\\u_{1}\end{pmatrix}}}">
<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 class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\begin{pmatrix}-u_{2}\\u_{1}\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c9ed53afad73ad84c28ce1e481dd6d2b4aa7352.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.858ex; height:6.176ex;" alt="{\displaystyle {\vec {n}}={\begin{pmatrix}-u_{2}\\u_{1}\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>
<p>Der Stützvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> kann aus der Parameterform übernommen werden.
</p>
<div class="mw-heading mw-heading4"><h4 id="Aus_der_Zweipunkteform">Aus der Zweipunkteform</h4></div>
<p>Aus der <a href="Zweipunkteform" title="Zweipunkteform">Zweipunkteform</a> einer Geradengleichung wird zunächst ein Richtungsvektor der Geraden als Differenzvektor zwischen den Ortsvektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>q</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {q}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a4e063f8ee7dae2488859c45a4e645db5148085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}" loading="lazy"></span> der beiden Punkte ermittelt und dann wie bei der Parameterform verfahren, also
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\begin{pmatrix}-(q_{2}-p_{2})\\q_{1}-p_{1}\end{pmatrix}}}">
<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 class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\begin{pmatrix}-(q_{2}-p_{2})\\q_{1}-p_{1}\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8fb66c34cc84bcf6df872242f8158920a22fa29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.439ex; height:6.176ex;" alt="{\displaystyle {\vec {n}}={\begin{pmatrix}-(q_{2}-p_{2})\\q_{1}-p_{1}\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>
<p>Als Stützvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> kann der Ortsvektor einer der Punkte verwendet werden.
</p>
<div class="mw-heading mw-heading4"><h4 id="Aus_der_allgemeinen_Koordinatenform">Aus der allgemeinen Koordinatenform</h4></div>
<p>Aus der <a href="Allgemeine_Koordinatenform#Allgemeine_Koordinatenform_einer_Geradengleichung" title="Allgemeine Koordinatenform">allgemeinen Koordinatenform einer Geradengleichung</a> mit den Parametern <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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>
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a,b}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/181523deba732fda302fd176275a0739121d3bc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> lässt sich ein Normalenvektor der Gerade direkt als
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\begin{pmatrix}a\\b\end{pmatrix}}}">
<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 class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\begin{pmatrix}a\\b\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94ea782a3e7207430a5e1680cc635dd8c5e2e045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.896ex; height:6.176ex;" alt="{\displaystyle {\vec {n}}={\begin{pmatrix}a\\b\end{pmatrix}}}" loading="lazy"></span></dd></dl>
<p>ablesen. Einen Stützvektor der Gerade erhält man, je nachdem ob <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> ungleich null ist, durch Wahl von
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}={\begin{pmatrix}c/a\\0\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}c/a\\0\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a7c5faf3483ff44333c59401ed3fa77cd59fb05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:12.085ex; height:6.176ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}c/a\\0\end{pmatrix}}}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}={\begin{pmatrix}0\\c/b\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>b</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}0\\c/b\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5449b95103fe83ca7668f76d043c94f97357dc5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:11.853ex; height:6.176ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}0\\c/b\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>
<p>Analog lässt sich auf diese Weise auch aus der <a href="Achsenabschnittsform" title="Achsenabschnittsform">Achsenabschnittsform</a> einer Geradengleichung ein Normalenvektor und ein Stützvektor ermitteln.
</p>
<div class="mw-heading mw-heading2"><h2 id="Normalenform_einer_Ebenengleichung">Normalenform einer Ebenengleichung</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Darstellung_2">Darstellung</h3></div>
<p>Analog wird eine Ebene im dreidimensionalen Raum in der Normalenform ebenfalls durch einen Stützvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> und einen Normalenvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">
<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>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"></span> beschrieben (siehe Bild 3). Eine Ebene besteht dann aus denjenigen Punkten im Raum, deren Ortsvektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> die Gleichung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {n}}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {n}}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd99b9ea5de9026093eab07f60bc1fa57728890d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.639ex; height:2.843ex;" alt="{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {n}}=0}" loading="lazy"></span></dd></dl>
<p>erfüllen. Der Stützvektor ist dabei wiederum der Ortsvektor eines beliebigen Punkts in der Ebene und der Normalenvektor ist ein Vektor, der senkrecht auf der Ebene steht. Das bedeutet, dass der Normalenvektor mit allen Geraden der Ebene, die durch den Stützpunkt verlaufen, einen rechten Winkel bildet. Eine äquivalente Darstellung der Normalenform ist wiederum
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7c9d9b60113925b0c7a0e192d8b518c1011b0f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.901ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}" loading="lazy"></span></dd></dl>
<p>und ein Punkt, dessen Ortsvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> die Normalengleichung erfüllt, liegt auf der Ebene. Gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}\cdot {\vec {n}}>{\vec {p}}\cdot {\vec {n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}\cdot {\vec {n}}>{\vec {p}}\cdot {\vec {n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9910bf6022be1c677503da374424b193cb829f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.901ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}\cdot {\vec {n}}>{\vec {p}}\cdot {\vec {n}}}" loading="lazy"></span>, dann liegt der Punkt auf derjenigen Seite der Ebene, in die der Normalenvektor zeigt, ansonsten auf der anderen Seite.
</p>
<div class="mw-heading mw-heading3"><h3 id="Beispiel_2">Beispiel</h3></div>
<p>Ausgedrückt in kartesischen Koordinaten lautet die Normalenform einer Ebenengleichung,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}-p_{1})\cdot n_{1}+(x_{2}-p_{2})\cdot n_{2}+(x_{3}-p_{3})\cdot n_{3}=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>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>n</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>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>n</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>3</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x_{1}-p_{1})\cdot n_{1}+(x_{2}-p_{2})\cdot n_{2}+(x_{3}-p_{3})\cdot n_{3}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46158206bee740446046caa6d1bcec1a427b5fc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.098ex; height:2.843ex;" alt="{\displaystyle (x_{1}-p_{1})\cdot n_{1}+(x_{2}-p_{2})\cdot n_{2}+(x_{3}-p_{3})\cdot n_{3}=0}" loading="lazy"></span>.</dd></dl>
<p>Im Bild 4 ist der Ortsvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\begin{pmatrix}4\\2\\2\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\begin{pmatrix}4\\2\\2\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a93ed5a5632036e89b1fa97249e440cde70cba31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:10.409ex; height:9.176ex;" alt="{\displaystyle {\vec {x}}={\begin{pmatrix}4\\2\\2\end{pmatrix}}}" loading="lazy"></span>, der Stützvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}={\begin{pmatrix}3\\3\\4\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}3\\3\\4\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4b20e360368536f5c3adc049fa809e2e0b1430d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:10.494ex; height:9.176ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}3\\3\\4\end{pmatrix}}}" loading="lazy"></span> und der Normalenvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\begin{pmatrix}1\\3\\-1\end{pmatrix}}}">
<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 class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\begin{pmatrix}1\\3\\-1\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ade778365dc141ff194c05a96a5b397c8f809d47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.282ex; height:9.176ex;" alt="{\displaystyle {\vec {n}}={\begin{pmatrix}1\\3\\-1\end{pmatrix}}}" loading="lazy"></span>. Durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4-3)\cdot 1+(2-3)\cdot 3+(2-4)\cdot (-1)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mn>1</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mn>3</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (4-3)\cdot 1+(2-3)\cdot 3+(2-4)\cdot (-1)=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d3120860851bf9ee9c7ec605312a95f40790ce8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.008ex; height:2.843ex;" alt="{\displaystyle (4-3)\cdot 1+(2-3)\cdot 3+(2-4)\cdot (-1)=0}" loading="lazy"></span></dd></dl>
<p>wird bestätigt, dass der Punkt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(4|2|2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>2</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X(4|2|2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3605057d7d563836ce1915a125f7f6f6bdc9461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.57ex; height:2.843ex;" alt="{\displaystyle X(4|2|2)}" loading="lazy"></span> auf der Ebene liegt.
</p><p>Die Normalenform (in <a href="Koordinatenform" class="mw-redirect" title="Koordinatenform">Koordinatendarstellung</a>) ist
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}+3x_{2}-x_{3}=8.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mn>3</mn>
<msub>
<mi>x</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>
<mo>=</mo>
<mn>8.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1}+3x_{2}-x_{3}=8.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c09125011de6b5b17c7d3a8cbf118ec406e448e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.903ex; height:2.509ex;" alt="{\displaystyle x_{1}+3x_{2}-x_{3}=8.}" loading="lazy"></span></dd></dl>
<p>Jede Wahl von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}|x_{2}|x_{3})}">
<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>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x_{1}|x_{2}|x_{3})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/396118d027bc31e4610510ab732b40114e490274.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.255ex; height:2.843ex;" alt="{\displaystyle (x_{1}|x_{2}|x_{3})}" loading="lazy"></span>, die diese Gleichung erfüllt, entspricht dann einem Ebenenpunkt. Beispielsweise folgt mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(1|3|2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>3</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B(1|3|2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c930fd93871635720b6e77dfbc24acca85385b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:2.843ex;" alt="{\displaystyle B(1|3|2)}" loading="lazy"></span> ebenfalls
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+3\cdot 3-2=8.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>+</mo>
<mn>3</mn>
<mo>⋅<!-- ⋅ --></mo>
<mn>3</mn>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>=</mo>
<mn>8.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1+3\cdot 3-2=8.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03407e5eea8ca8745351a01b6fcf741fbb6a4a0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.917ex; height:2.343ex;" alt="{\displaystyle 1+3\cdot 3-2=8.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Berechnung_2">Berechnung</h3></div>
<div class="mw-heading mw-heading4"><h4 id="Aus_der_Parameterform_2">Aus der Parameterform</h4></div>
<p>Aus der <a href="Parameterform#Parameterform_einer_Ebenengleichung" title="Parameterform">Parameterform einer Ebenengleichung</a> mit den beiden Richtungsvektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {u}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>u</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c41e9cf70c5e5b56e2128a136985a75f90ba43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>v</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85820588abd7333ef4d0c56539cb31c20e730753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" loading="lazy"></span> lässt sich ein Normalenvektor der Ebene durch Berechnung des <a href="Kreuzprodukt" title="Kreuzprodukt">Kreuzprodukts</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\vec {u}}\times {\vec {v}}}">
<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 class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>u</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>v</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\vec {u}}\times {\vec {v}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e36017bf7516276e888940eeace892b5dea66641.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.839ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}={\vec {u}}\times {\vec {v}}}" loading="lazy"></span></dd></dl>
<p>bestimmen. Der Stützvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> kann aus der Parameterform übernommen werden.
</p>
<div class="mw-heading mw-heading4"><h4 id="Aus_der_Dreipunkteform">Aus der Dreipunkteform</h4></div>
<p>Aus der <a href="Dreipunkteform" title="Dreipunkteform">Dreipunkteform</a> einer Ebenengleichung werden zunächst zwei Richtungsvektoren als Differenzvektoren zwischen den Ortsvektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>q</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {q}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a4e063f8ee7dae2488859c45a4e645db5148085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy"></span> jeweils zweier Punkte ermittelt und dann wie bei der Parameterform das Kreuzprodukt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}=({\vec {q}}-{\vec {p}})\times ({\vec {r}}-{\vec {p}})}">
<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>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>q</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<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>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}=({\vec {q}}-{\vec {p}})\times ({\vec {r}}-{\vec {p}})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55dd593208f70a4938077930ab38418bca2c9cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.815ex; height:2.843ex;" alt="{\displaystyle {\vec {n}}=({\vec {q}}-{\vec {p}})\times ({\vec {r}}-{\vec {p}})}" loading="lazy"></span></dd></dl>
<p>berechnet. Als Stützvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"></span> kann der Ortsvektor einer der Punkte verwendet werden.
</p>
<div class="mw-heading mw-heading4"><h4 id="Aus_der_allgemeinen_Koordinatenform_2">Aus der allgemeinen Koordinatenform</h4></div>
<p>Aus der <a href="Allgemeine_Koordinatenform#Allgemeine_Koordinatenform_einer_Ebenengleichung" title="Allgemeine Koordinatenform">allgemeinen Koordinatenform einer Ebenengleichung</a> mit den Parametern <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo>,</mo>
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" 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}" loading="lazy"></span> lässt sich ein Normalenvektor der Ebene als
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\begin{pmatrix}a\\b\\c\end{pmatrix}}}">
<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 class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\begin{pmatrix}a\\b\\c\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe790b3fc6e4596872a840950eb35bf494a49d55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:10.541ex; height:9.176ex;" alt="{\displaystyle {\vec {n}}={\begin{pmatrix}a\\b\\c\end{pmatrix}}}" loading="lazy"></span></dd></dl>
<p>ablesen. Einen Stützvektor erhält man, je nachdem welche der Zahlen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo>,</mo>
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13f068df656c1b1911ae9f81628c49a6181194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}" loading="lazy"></span> ungleich null ist, durch Wahl von
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}={\begin{pmatrix}d/a\\0\\0\end{pmatrix}},~{\vec {p}}={\begin{pmatrix}0\\d/b\\0\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>,</mo>
<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}d/a\\0\\0\end{pmatrix}},~{\vec {p}}={\begin{pmatrix}0\\d/b\\0\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86361c08d94274feb34732efc6c2bbfb03f276e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.089ex; width:27.172ex; height:9.509ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}d/a\\0\\0\end{pmatrix}},~{\vec {p}}={\begin{pmatrix}0\\d/b\\0\end{pmatrix}}}" loading="lazy"></span> oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}={\begin{pmatrix}0\\0\\d/c\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>p</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>c</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}={\begin{pmatrix}0\\0\\d/c\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/664c61b67cc226298033d75665fa7df6015cc01f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-left: -0.089ex; width:12.717ex; height:9.509ex;" alt="{\displaystyle {\vec {p}}={\begin{pmatrix}0\\0\\d/c\end{pmatrix}}}" loading="lazy"></span>.</dd></dl>
<p>Analog lässt sich auf diese Weise auch aus der <a href="Achsenabschnittsform" title="Achsenabschnittsform">Achsenabschnittsform</a> einer Ebenengleichung ein Normalenvektor und ein Stützvektor ermitteln.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verallgemeinerung">Verallgemeinerung</h2></div>
<p>Allgemein wird durch eine Normalengleichung eine <a href="Hyperebene" title="Hyperebene">Hyperebene</a> im <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensionalen euklidischen Raum beschrieben. Im <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensionalen euklidischen Raum besteht eine Hyperebene entsprechend aus denjenigen Punkten, deren Ortsvektoren <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> die Gleichung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {n}}=0\quad }">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b0f899f64e213baaafa0641adca1d8975434b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.962ex; height:2.843ex;" alt="{\displaystyle ({\vec {x}}-{\vec {p}})\cdot {\vec {n}}=0\quad }" loading="lazy"></span>bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}">
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<annotation encoding="application/x-tex">{\displaystyle \quad {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e17eb19e1d84b88cb9dae51d6206dbb451a7fc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.224ex; height:2.676ex;" alt="{\displaystyle \quad {\vec {x}}\cdot {\vec {n}}={\vec {p}}\cdot {\vec {n}}}" loading="lazy"></span></dd></dl>
<p>erfüllen. Es wird dabei lediglich mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-komponentigen statt mit zwei- oder dreikomponentigen Vektoren gerechnet. Eine solche Gleichung wird in der Literatur auch <i>Hessesche Normalform</i> genannt, selbst wenn der Normalenvektor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"></span> nicht normiert ist.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
</p><p>Eine Hyperebene teilt den <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensionalen Raum in zwei Teile, die <a href="Halbraum" title="Halbraum">Halbräume</a> genannt werden. Gilt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}\cdot {\vec {n}}>{\vec {p}}\cdot {\vec {n}}}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9910bf6022be1c677503da374424b193cb829f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.901ex; height:2.676ex;" alt="{\displaystyle {\vec {x}}\cdot {\vec {n}}>{\vec {p}}\cdot {\vec {n}}}" loading="lazy"></span>, dann liegt der Punkt in demjenigen Halbraum, in den der Normalenvektor zeigt, ansonsten in dem anderen. Ein Punkt, dessen Ortsvektor die Normalengleichung erfüllt, liegt genau auf der Hyperebene.
</p>
<div class="mw-heading mw-heading2"><h2 id="Lösbarkeit_von_linearen_Gleichungssystemen"><span id="L.C3.B6sbarkeit_von_linearen_Gleichungssystemen"></span>Lösbarkeit von linearen Gleichungssystemen</h2></div>
<p>Jede Gleichung eines <a href="Lineares_Gleichungssystem" title="Lineares Gleichungssystem">linearen Gleichungssystems</a> lässt sich als Normalenform einer Hyperebene in einem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-dimensionalen <a href="Vektorraum" title="Vektorraum">Vektorraum</a> deuten, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> die Anzahl der Variablen bzw. Unbekannten ist. Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy"></span> sind dies Geraden in der Ebene, für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c5a5a42ced00df920fad4ab2d4acdb960a4105b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}" loading="lazy"></span> Ebenen im Raum. Damit lässt sich die <a href="Lineares_Gleichungssystem#Lösbarkeit" title="Lineares Gleichungssystem">Lösbarkeit eines linearen Gleichungssystems</a> zurückführen auf ein Schnittproblem von Hyperebenen: Gesucht ist die Menge der gemeinsamen Punkte aller Hyperebenen. Aus der Lage der Normalenvektoren und damit der Hyperebenen zueinander kann auf die Lösbarkeit des linearen Gleichungssystems und auf die Anzahl der Lösungen geschlossen werden.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Lothar_Papula" title="Lothar Papula">Lothar Papula</a>: <cite style="font-style:italic">Mathematische Formelsammlung: Für Ingenieure und Naturwissenschaftler</cite>. Springer, 2009, ISBN 978-3-8348-9598-1.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Normalenform&rft.au=Lothar+Papula&rft.btitle=Mathematische+Formelsammlung%3A+F%C3%BCr+Ingenieure+und+Naturwissenschaftler&rft.date=2009&rft.genre=book&rft.isbn=9783834895981&rft.pub=Springer" style="display:none"> </span></li>
<li><a href="Harald_Scheid" title="Harald Scheid">Harald Scheid</a>, Wolfgang Schwarz: <cite style="font-style:italic">Elemente der linearen Algebra und der Analysis</cite>. Springer, 2009, ISBN 978-3-8274-2255-2.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Normalenform&rft.au=Harald+Scheid%2C+Wolfgang+Schwarz&rft.btitle=Elemente+der+linearen+Algebra+und+der+Analysis&rft.date=2009&rft.genre=book&rft.isbn=9783827422552&rft.pub=Springer" style="display:none"> </span></li>
<li><a href="Edmund_Weitz" title="Edmund Weitz">Edmund Weitz</a>: <cite style="font-style:italic">Konkrete Mathematik (nicht nur) für Informatiker</cite>. 2. Auflage. Springer Spektrum, 2021, ISBN 978-3-662-62617-7, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>379–383</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Normalenform&rft.au=Edmund+Weitz&rft.btitle=Konkrete+Mathematik+%28nicht+nur%29+f%C3%BCr+Informatiker&rft.date=2021&rft.edition=2.&rft.genre=book&rft.isbn=9783662626177&rft.pages=379-383&rft.pub=Springer+Spektrum" style="display:none"> </span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><span class="cite"><a rel="nofollow" class="external text" href="https://de.serlo.org/mathe/1893/ebene-von-parameterform-in-normalform-umwandeln"><i>Ebene von Parameterform in Normalform umwandeln.</i></a> In: <i>Serlo.</i><span class="Abrufdatum"> Abgerufen am 23. Februar 2014</span>.</span><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%3ANormalenform&rft.title=Ebene+von+Parameterform+in+Normalform+umwandeln&rft.description=Ebene+von+Parameterform+in+Normalform+umwandeln&rft.identifier=https%3A%2F%2Fde.serlo.org%2Fmathe%2F1893%2Febene-von-parameterform-in-normalform-umwandeln"> </span></li>
<li><span class="cite"><a rel="nofollow" class="external text" href="https://de.serlo.org/mathe/1905/ebene-von-koordinatenform-in-normalform-umwandeln"><i>Ebene von Koordinatenform in Normalform umwandeln.</i></a> In: <i>Serlo.</i><span class="Abrufdatum"> Abgerufen am 23. Februar 2014</span>.</span><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%3ANormalenform&rft.title=Ebene+von+Koordinatenform+in+Normalform+umwandeln&rft.description=Ebene+von+Koordinatenform+in+Normalform+umwandeln&rft.identifier=https%3A%2F%2Fde.serlo.org%2Fmathe%2F1905%2Febene-von-koordinatenform-in-normalform-umwandeln"> </span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-:0-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:0_1-0">a</a></sup> <sup><a href="#cite_ref-:0_1-1">b</a></sup></span> <span class="reference-text"><cite style="font-style:italic">Schülerduden Die Mathematik II</cite>. 3. Auflage. Dudenverlag, Mannheim / Wien / Zürich 1991, ISBN 3-411-04273-7, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>149</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Normalenform&rft.btitle=Sch%C3%BClerduden+Die+Mathematik+II&rft.date=1991&rft.edition=3.&rft.genre=book&rft.isbn=3411042737&rft.pages=149&rft.place=Mannheim+%2F+Wien+%2F+Z%C3%BCrich&rft.pub=Dudenverlag" style="display:none"> </span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Jens Kunath: <cite style="font-style:italic">Analytische Geometrie und Lineare Algebra zwischen Abitur und Studium I</cite>. 2. Auflage. Springer Spektrum, Berlin / Heidelberg 2023, ISBN 978-3-662-67811-4, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>170</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Normalenform&rft.au=Jens+Kunath&rft.btitle=Analytische+Geometrie+und+Lineare+Algebra+zwischen+Abitur+und+Studium+I&rft.date=2023&rft.edition=2.&rft.genre=book&rft.isbn=9783662678114&rft.pages=170&rft.place=Berlin+%2F+Heidelberg&rft.pub=Springer+Spektrum" style="display:none"> </span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Dörte Haftendorn, Dieter Riebesehl, Hubert Dammer: <cite style="font-style:italic">Höhere Mathematik sehen und verstehen</cite>. 2. Auflage. Springer, Berlin / Heidelberg 2024, ISBN 978-3-662-69291-2, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>155<span style="display:inline-block;width:.2em"> </span>f</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Normalenform&rft.au=D%C3%B6rte+Haftendorn%2C+Dieter+Riebesehl%2C+Hubert+Dammer&rft.btitle=H%C3%B6here+Mathematik+sehen+und+verstehen&rft.date=2024&rft.edition=2.&rft.genre=book&rft.isbn=9783662692912&rft.pages=155+f.&rft.place=Berlin+%2F+Heidelberg&rft.pub=Springer" style="display:none"> </span></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite style="font-style:italic">Hessesche Normalform</cite>. In: <cite style="font-style:italic">Lexikon der Mathematik: Band 2: Eig bis Inn</cite>. 2. Auflage. Springer Spektrum, Berlin / Heidelberg 2017, ISBN 978-3-662-53503-5, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>401</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Normalenform&rft.atitle=Hessesche+Normalform&rft.btitle=Lexikon+der+Mathematik%3A+Band+2%3A+Eig+bis+Inn&rft.date=2017&rft.edition=2.&rft.genre=book&rft.isbn=9783662535035&rft.pages=401&rft.place=Berlin+%2F+Heidelberg&rft.pub=Springer+Spektrum" style="display:none"> </span></span>
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