• Das Vektorprodukt
  • MNWeG
  • 04.02.2022
  • Mathematik
  • Vektoren
  • 12
  • Arbeitsblatt
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1
Berechne mithilfe des Vektorproduktes einen Vektor, der senkrecht auf den beiden Vektoren steht.

a) a=( -1-32)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ \text{-}1\\\text{-}3\\2\end{array} \right), b=( 0-12)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 0\\\text{-}1\\2\end{array} \right)

b) a=( 431)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ 4\\3\\1\end{array} \right), b=( 2-53)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 2\\\text{-}5\\3\end{array} \right)
c) a=( -531)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ \text{-}5\\3\\1\end{array} \right), b=( 002)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 0\\0\\2\end{array} \right)

d) a=( 100)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ 1\\0\\0\end{array} \right), b=( 010)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 0\\1\\0\end{array} \right)
a) a×b=(-421)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} \times \overrightarrow{b} = \left( \begin{array}{r} \text{-}4\\2\\1\end{array} \right)

b) a×b=(14-10-26)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} \times \overrightarrow{b} =\left( \begin{array}{r} 14\\\text{-}10\\\text{-}26\end{array} \right)

c) a×b=(6100)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} \times \overrightarrow{b} =\left( \begin{array}{r} 6\\10\\0\end{array} \right)

d) a×b=(001)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} \times \overrightarrow{b} =\left( \begin{array}{r} 0\\0\\1\end{array} \right)
2
Emilia sind beim Berechnen eines Vektorprodukts zwei Fehler unterlaufen. Prüfe ihre Rechnung und korrigiere die Fehler.

( 140)×( -1-32)=( 420(-3)0(-1)121(-3)4(-1))=( 8+302-34)=( 11-2-7)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \left( \begin{array}{r} \ 1\\4\\0\end{array} \right) \times \left( \begin{array}{r} \ \text{-}1\\\text{-}3\\2\end{array} \right)=\left( \begin{array}{c} \ 4 \cdot2-0\cdot(\text{-}3)\\0 \cdot (\text{-}1)-1\cdot2\\1\cdot(\text{-}3)-4\cdot(\text{-}1)\end{array} \right)=\left( \begin{array}{r} \ 8+3\\0-2\\\text{-}3-4\end{array} \right)=\left( \begin{array}{r} \ 11\\\text {-}2\\\text{-}7\end{array} \right)
( 140)×( -1-32)=( 420(-3)0(-1)121(-3)4(-1))=( 8+002-3+4)=( 8-21)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \left( \begin{array}{r} \ 1\\4\\0\end{array} \right) \times \left( \begin{array}{r} \ \text{-}1\\\text{-}3\\2\end{array} \right)=\left( \begin{array}{c} \ 4 \cdot2-0\cdot(\text{-}3)\\0 \cdot (\text{-}1)-1\cdot2\\1\cdot(\text{-}3)-4\cdot(\text{-}1)\end{array} \right)=\left( \begin{array}{r} \ 8+0\\0-2\\\text{-}3+4\end{array} \right)=\left( \begin{array}{r} \ 8\\\text {-}2\\1\end{array} \right)
3
Kreuze an, ob das Ergebnis der Berechnung eine Zahl, ein Vektor oder nicht definiert ist.

Zahl

Vektor

nicht definiert

ab\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} \cdot\overrightarrow{b}

x

a×b\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} \times \overrightarrow{b}

x

(ab)×c\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} (\overrightarrow{a} \cdot\overrightarrow{b})\times\overrightarrow{c}

x

a(b×c)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} \cdot(\overrightarrow{b}\times\overrightarrow{c})

x

a+(b×c)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} + (\overrightarrow{b}\times\overrightarrow{c})

x

a(b+c)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} \cdot(\overrightarrow{b} + \overrightarrow{c})

x

(ab)+c\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} (\overrightarrow{a} \cdot\overrightarrow{b})+\overrightarrow{c}

x

(ab)×(cd)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} (\overrightarrow{a} \cdot\overrightarrow{b})\times(\overrightarrow{c} \cdot\overrightarrow{d})

x

a(b×c)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} |\overrightarrow{a}| \cdot(\overrightarrow{b}\times\overrightarrow{c})

x

(ab)+c\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} (\overrightarrow{a} \cdot\overrightarrow{b})+|\overrightarrow{c}|

x