• Pythagoras und Skalarprodukt
  • anonym
  • 19.05.2019
  • Mathematik
  • 11
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  • Zueinander orthogonale Vektoren

    γ=\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \gamma =
    ab\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \vec{a}\perp \vec{b}

    Es gilt:

    Es gilt:

    a2+b2=\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} a^{2}+b^{2} =
    a2+b2=ab2\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \mid\vec{a}\mid ^{2}+\mid\vec{b}\mid ^{2}=\mid\vec{a}-\vec{b}\mid ^{2}

    Daher gilt auch:

    Daher soll gelten:

    a=\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \mid\vec{a}\mid =
    =>a2=\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} => \mid\vec{a}\mid ^{2} =
    =>b2=\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} => \mid\vec{b}\mid ^{2} =
    =>ab2=(a1b1)2+\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} => \mid\vec{a}-\vec{b}\mid ^{2} = \left ( {a_{1}}-{b_{1}} \right )^{2}+
    =>ab2=a122a1b1+b12+\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} => \mid\vec{a}-\vec{b}\mid ^{2} = {a_{1}}^{2}-2{a_{1}}{b_{1}}+{b_{1}}^{2} +

    Letztendlich gilt:

    ab2=a12+a22+b12+b222(a1b1+a2b2)\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \mid\vec{a}-\vec{b}\mid ^{2} = {a_{1}}^{2}+{a_{2}}^{2}+{b_{1}}^{2}+{b_{2}}^{2} -2({a_{1}}{b_{1}}+{a_{2}}{b_{2}})

    Ergänze (siehe oben):

    a2+b2=\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \mid\vec{a}\mid ^{2} +\mid\vec{b}\mid ^{2}=
    1
    Wann ist die Gleichung


    erfüllt?
    a2+b2=ab2\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \mid\vec{a}\mid ^{2}+\mid\vec{b}\mid ^{2}=\mid\vec{a}-\vec{b}\mid ^{2}
    Skalarprodukt

    Die Vektoren                      sind genau dann                                  zueinander, wenn gilt:

                                                                       

                                               wird als Skalarprodukt der Vektoren                    bezeichnet.

    a  und  b\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \vec{a} \; und \; \vec{b}
    ab=a1b1+a2b2=\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \vec{a}\cdot\vec{b} = {a_{1}}{b_{1}}+{a_{2}}{b_{2}} =
    ab=a1b1+a2b2\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \vec{a}\cdot\vec{b} = {a_{1}}{b_{1}}+{a_{2}}{b_{2}}
    a  und  b\gdef\cloze#1{\colorbox{dedede}{\color{transparent}{\large{$\displaystyle #1$}}}} \vec{a} \;und \;\vec{b}