• Das Skalarprodukt
  • MNWeG
  • 04.02.2022
  • Mathematik
  • Vektoren
  • 12
  • Arbeitsblatt
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1
Prüfe mithilfe des Skalarproduktes, ob die Vektoren senkrecht aufeinander stehen.

a) a=( -2-44)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ \text{-}2\\\text{-}4\\4\end{array} \right), b=( 8-22)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 8\\\text{-}2\\2\end{array} \right)

b) a=( 320)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ 3\\2\\0\end{array} \right), b=( 2-34)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 2\\\text{-}3\\4\end{array} \right)
c) a=( -431)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ \text{-}4\\3\\1\end{array} \right), b=( 3-16)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 3\\\text{-}1\\6\end{array} \right)

d) a=( 020)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ 0\\2\\0\end{array} \right), b=( -104)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ \text{-}1\\0\\4\end{array} \right)
a) ab=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} ·\overrightarrow{b}= 0
\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \rArr Die Vektoren stehen senkrecht aufeinander.

b) ab=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} ·\overrightarrow{b}= 0
\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \rArr Die Vektoren stehen senkrecht aufeinander.

c) ab=9\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} ·\overrightarrow{b}= 9
\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \rArr Die Vektoren stehen nicht senkrecht aufeinander.

d) ab=0\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} ·\overrightarrow{b}= 0
\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \rArr Die Vektoren stehen senkrecht aufeinander.
2
Bestimme die fehlende Koordinate, sodass a\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} senkrecht zu b\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} ist.
a) a=( 315)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ 3\\1\\5\end{array} \right), b=( b18-4)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ b_1\\8\\\text{-}4\end{array} \right)
b) a=( -5a21)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ \text{-}5\\a_2\\1\end{array} \right), b=( 106-4)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 10\\6\\\text{-}4\end{array} \right)
a) b1=4\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} b_1 = 4

b) a2=9\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} a_2=9
3
Bestimme einen Vektor, der zu a\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} und zub\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} senkrecht ist.
a=( 124)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{a} = \left( \begin{array}{r} \ 1\\2\\4\end{array} \right), b=( 33-2)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{b} = \left( \begin{array}{r} \ 3\\3\\\text{-}2\end{array} \right)
z. B. c=( -1614-3)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} \overrightarrow{c} = \left( \begin{array}{r} \ \text{-}16\\14\\\text{-}3\end{array} \right)

4
Gegeben ist die Ebene E ⁣:x=( 142)+r ( -304)+s ( -21-2)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} E\!: \overrightarrow{x} = \left( \begin{array}{r} \ 1\\4\\2\end{array} \right) + r\ · \left( \begin{array}{r} \ \text{-}3\\0\\4\end{array} \right)+s\ · \left( \begin{array}{r} \ \text{-}2\\1\\\text{-}2\end{array} \right).

Bestimme die Gleichung einer Geraden g\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} g, die die Ebene E\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} E im Punkt P(142)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} P (1|4|2) senkrecht schneidet.
z. B. g:x=( 142)+t( 4143)\gdef\cloze#1{{\raisebox{-.05em}{\colorbox{none}{\color{526060}{\large{$\displaystyle #1$}}}}}} g:\overrightarrow{x} = \left( \begin{array}{r} \ 1\\4\\2\end{array} \right)+t· \left( \begin{array}{r} \ 4\\14\\3\end{array} \right)