${\displaystyle {\mathfrak {x}}\,=\,x_{1}{\mathfrak {a_{1}}}+x_{2}{\mathfrak {a_{2}}}+x_{3}{\mathfrak {a_{3}}}\;=\,{\mathfrak {a_{1}}}x_{1}+{\mathfrak {a_{2}}}x_{2}+{\mathfrak {a_{3}}}x_{3};}$

${\displaystyle {\underline {\mathfrak {a}}}=({\mathfrak {a}}_{1}\,{\mathfrak {a}}_{2}\,{\mathfrak {a}}_{3})^{\textsf {T}};}$

${\displaystyle {\underline {\mathfrak {b}}}=({\mathfrak {b}}_{1}\,{\mathfrak {b}}_{2}\,{\mathfrak {b}}_{3})^{\textsf {T}};}$

${\displaystyle {\underline {G}}_{\,{\underline {\mathfrak {a}}}}\,=\,{\underline {\mathfrak {a}}}\cdot {\underline {\mathfrak {a}}}^{\textsf {T}}=\,(g_{i,j})_{\,{\underline {\mathfrak {a}}}};}$

${\displaystyle {\underline {G}}_{\,{{\underline {\mathfrak {a}}},{\underline {\mathfrak {b}}}}}\,=\,{\underline {\mathfrak {a}}}\cdot {\underline {\mathfrak {b}}}^{\textsf {T}}=\,(g_{i,j})_{\,{{\underline {\mathfrak {a}}},{\underline {\mathfrak {b}}}}}\;}$

${\displaystyle {\text{ mit }}g_{i,j}\,=\,{\mathfrak {a}}_{i}\cdot {\mathfrak {b}}_{j};}$

${\displaystyle {\underline {x}}_{\,{\underline {\mathfrak {a}}}}\,=\,{(x_{1}\,x_{2}\,x_{3})_{\underline {\mathfrak {a}}}^{\textsf {T}}};}$

${\displaystyle {\underline {x}}_{\,{\underline {\mathfrak {a}}}}^{\textsf {T}}\,=\,{(x_{1}\,x_{2}\,x_{3})_{\underline {\mathfrak {a}}}};}$

${\displaystyle a\,{\stackrel {!}{=}}b}$

${\displaystyle (u\,v)'\,=\,{\stackrel {\downarrow }{u}}\,v\,+\,u\,{\stackrel {\downarrow }{v}}}$

${\displaystyle {\underline {\partial }}\,=\,(\partial _{1}\,\partial _{2}\,\partial _{3})^{\mathrm {T} }{\text{ mit }}\partial _{i}\,=\,{\frac {\partial }{\partial r_{i}}}}$

Reste ${\displaystyle \,\,\,{\mathsf {oder}}\,\,\,{\underline {\mathfrak {n}}}={\underline {N}}\,\,{\underline {\mathfrak {e}}}\,\,\,{\mathsf {bzw.}}\,\,\,{\underline {\mathfrak {n}}}^{\textsf {T}}={\underline {\mathfrak {e}}}^{\textsf {T}}\,{\underline {N}}^{\textsf {T}}}$ ........................ (5) (5a) (5b)

${\displaystyle {\begin{vmatrix}{\mathfrak {e}}_{1}&{\mathfrak {e}}_{2}&{\mathfrak {e}}_{3}\end{vmatrix}}{\begin{vmatrix}n_{11}&n_{12}&n_{13}\\n_{21}&n_{22}&n_{23}\\n_{31}&n_{32}&n_{33}\end{vmatrix}}}$ Resteende

Wenn wir die reziproken direkt aus den vorgegeben Vektoren ermitteln

${\displaystyle {\begin{aligned}{\underline {\mathfrak {n}}}^{*}&={\begin{pmatrix}{\mathfrak {n}}_{1}^{*}\\{\mathfrak {n}}_{2}^{*}\\{\mathfrak {n}}_{3}^{*}\end{pmatrix}}={\frac {1}{V_{\underline {\mathfrak {n}}}}}{\begin{pmatrix}{\mathfrak {n}}_{2}\times {\mathfrak {n}}_{3}\\{\mathfrak {n}}_{3}\times {\mathfrak {n}}_{1}\\{\mathfrak {n}}_{1}\times {\mathfrak {n}}_{2}\end{pmatrix}}={\frac {1}{V_{\underline {\mathfrak {n}}}}}{\begin{pmatrix}{\begin{vmatrix}{\mathfrak {e}}_{1}&{\mathfrak {e}}_{2}&{\mathfrak {e}}_{3}\\n_{21}&n_{22}&n_{23}\\n_{31}&n_{32}&n_{33}\end{vmatrix}}\\\,\\{\begin{vmatrix}n_{11}&n_{12}&n_{13}\\{\mathfrak {e}}_{1}&{\mathfrak {e}}_{2}&{\mathfrak {e}}_{3}\\n_{31}&n_{32}&n_{33}\end{vmatrix}}\\\,\\{\begin{vmatrix}n_{11}&n_{12}&n_{13}\\n_{21}&n_{22}&n_{23}\\{\mathfrak {e}}_{1}&{\mathfrak {e}}_{2}&{\mathfrak {e}}_{3}\end{vmatrix}}\end{pmatrix}}={\frac {1}{V_{\underline {\mathfrak {n}}}}}{\begin{pmatrix}N_{11}{\mathfrak {e}}_{1}+N_{12}{\mathfrak {e}}_{2}+N_{13}{\mathfrak {e}}_{3}\\N_{21}{\mathfrak {e}}_{1}+N_{22}{\mathfrak {e}}_{2}+N_{23}{\mathfrak {e}}_{3}\\N_{31}{\mathfrak {e}}_{1}+N_{32}{\mathfrak {e}}_{2}+N_{33}{\mathfrak {e}}_{3}\end{pmatrix}}\\&={\frac {1}{V_{\underline {\mathfrak {n}}}}}{\begin{pmatrix}N_{11}&N_{12}&N_{13}\\N_{21}&N_{22}&N_{23}\\N_{31}&N_{32}&N_{33}\end{pmatrix}}{\begin{pmatrix}{\mathfrak {e}}_{1}\\{\mathfrak {e}}_{2}\\{\mathfrak {e}}_{3}\end{pmatrix}}={\frac {1}{V_{\underline {\mathfrak {n}}}}}\,\,((N_{ij}))\,\,{\underline {\mathfrak {e}}}\end{aligned}}}$

${\displaystyle {\begin{vmatrix}n_{11}&n_{12}&n_{13}\\n_{21}&n_{22}&n_{23}\\n_{31}&n_{32}&n_{33}\end{vmatrix}}}$

${\displaystyle \mathbb {R} \;\mathbb {R} ^{3}\;\mathbb {N} \;\mathbb {N} }$ ${\displaystyle {\mathfrak {V}}_{\mathrm {O} }^{3}}$

Voraussetzungen

Vorbereitungen

Zahlenkörper

Geometrische Räume (Punkträume)

Vektorräume

Permutationen

Deltaistik / Epsilontik

Lineare Gleichungssysteme (bis zum "Hauptfall")

Determinanten

Lineare Substitutionen

Matrizen

146.52.35.152 19:13, 28. Mai 2016 (CEST)

${\displaystyle {\mathfrak {x}}\,=\,x_{1}{\mathfrak {a_{1}}}+x_{2}{\mathfrak {a_{2}}}+x_{3}{\mathfrak {a_{3}}}\;=\,{\mathfrak {a_{1}}}x_{1}+{\mathfrak {a_{2}}}x_{2}+{\mathfrak {a_{3}}}x_{3};}$

${\displaystyle {\underline {\mathfrak {a}}}=({\mathfrak {a}}_{1}\,{\mathfrak {a}}_{2}\,{\mathfrak {a}}_{3})^{\textsf {T}};}$

${\displaystyle {\underline {\mathfrak {b}}}=({\mathfrak {b}}_{1}\,{\mathfrak {b}}_{2}\,{\mathfrak {b}}_{3})^{\textsf {T}};}$

${\displaystyle {\underline {G}}_{\,{\underline {\mathfrak {a}}}}\,=\,{\underline {\mathfrak {a}}}\cdot {\underline {\mathfrak {a}}}^{\textsf {T}}=\,(g_{i,j})_{\,{\underline {\mathfrak {a}}}};}$

${\displaystyle {\underline {G}}_{\,{{\underline {\mathfrak {a}}},{\underline {\mathfrak {b}}}}}\,=\,{\underline {\mathfrak {a}}}\cdot {\underline {\mathfrak {b}}}^{\textsf {T}}=\,(g_{i,j})_{\,{{\underline {\mathfrak {a}}},{\underline {\mathfrak {b}}}}}\;}$

${\displaystyle {\text{ mit }}g_{i,j}\,=\,{\mathfrak {a}}_{i}\cdot {\mathfrak {b}}_{j};}$

${\displaystyle {\underline {x}}_{\,{\underline {\mathfrak {a}}}}\,=\,{(x_{1}\,x_{2}\,x_{3})_{\underline {\mathfrak {a}}}^{\textsf {T}}};}$

${\displaystyle {\underline {x}}_{\,{\underline {\mathfrak {a}}}}^{\textsf {T}}\,=\,{(x_{1}\,x_{2}\,x_{3})_{\underline {\mathfrak {a}}}};}$

stackrel-Verwendung

${\displaystyle a\,{\stackrel {!}{=}}b}$

${\displaystyle (u\,v)'\,=\,{\stackrel {\downarrow }{u}}\,v\,+\,u\,{\stackrel {\downarrow }{v}}}$

partial-Verwendung

${\displaystyle {\underline {\partial }}\,=\,(\partial _{1}\,\partial _{2}\,\partial _{3})^{\mathrm {T} }{\text{ mit }}\partial _{i}\,=\,{\frac {\partial }{\partial r_{i}}}}$

${\displaystyle }$${\displaystyle }$${\displaystyle }$

${\displaystyle {\mathfrak {a}}_{\mathfrak {b}}}$

${\displaystyle {\mathfrak {b}}_{\mathfrak {a}}}$

${\displaystyle {\mathfrak {a}}}$

${\displaystyle {\mathfrak {b}}}$

${\displaystyle \cos({\mathfrak {a}},{\mathfrak {b}})}$

${\displaystyle \mathbf {P} }$

\frac{\mathfrak b\,\mathfrak b}

--Staryvlk (Diskussion) 22:20, 24. Mär. 2015 (CET)

Aus Artikel "Einheitsvektor"

Die drei kanonischen Einheitsvektoren des dreidimensionalen Vektorraums ${\displaystyle \mathbb {R} ^{3}}$ werden in den angewandten Naturwissenschaften manchmal mit ${\displaystyle \mathbf {i} ,\,\mathbf {j} ,\,\mathbf {k} }$ bezeichnet:

${\displaystyle ==ende\mathbf {i} ={\vec {e}}_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad \mathbf {j} ={\vec {e}}_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad \mathbf {k} ={\vec {e}}_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}$

==sichtbare leerzeichen

␣␣␣␣

${\displaystyle abcd\mathrm {\textvisiblespace} efgh}$

${\displaystyle {\underline {a}}_{i\mathrm {\textvisiblespace} };{\underline {a}}_{\mathrm {\textvisiblespace} j};}$ ${\displaystyle {\underline {a}}_{i^{\mathrm {\textvisiblespace} }};{\underline {a}}_{^{\mathrm {\textvisiblespace} }j};}$

${\displaystyle {\underline {a}}_{i\diamond };{\underline {a}}_{\diamond j};}$

${\displaystyle {\underline {a}}_{i\Diamond };{\underline {a}}_{\Diamond j};}$ ${\displaystyle {\underline {a}}_{i\amalg };{\underline {a}}_{\amalg j};}$

${\displaystyle {\underline {a}}_{i\Box };{\underline {a}}_{\Box j};}$

${\displaystyle {\underline {a}}_{i\centerdot };{\underline {a}}_{\centerdot j};}$

${\displaystyle {\underline {a}}_{i\cup };{\underline {a}}_{\cup j};}$ ${\displaystyle {\underline {a}}_{i\bullet };{\underline {a}}_{\bullet j};}$

${\displaystyle {\underline {A}}^{\intercal }\,=}$ ==ende