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Version 24.1 von Daniel Stocker am 2024/02/06 11:55
Zeige letzte Bearbeiter
1 {{formula}}\overrightarrow{AB} = \overrightarrow{OB}-\overrightarrow{OA} = \left(\begin{array}{c} 0-2 \\ 9-(-1) \\ -3-4\end{array}\right) = \left(\begin{array}{c} -2 \\ 10 \\ -7\end{array}\right){{/formula}}
2
3 {{formula}}\overrightarrow{AC} = \overrightarrow{OC}-\overrightarrow{OA} = \left(\begin{array}{c} -2-2 \\ 5-(-1) \\ 1-4\end{array}\right) = \left(\begin{array}{c} -4 \\ 6 \\ -3\end{array}\right)
4 {{/formula}}
5
6 {{formula}} cos(\alpha) = \frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{\mid\overrightarrow{AB} \mid \cdot \mid \overrightarrow{AC} \mid} = \frac{(-2)\cdot (-4)+10\cdot 6 + (-7) \cdot (-3) }{\sqrt{(-2)^2+10^2+(-7)^2}\cdot \sqrt{(-4)^2+6^2+(-3)^2}} = \frac{89}{\sqrt{153} \cdot \sqrt{61}} = \frac{89}{\sqrt{9333}} {{/formula}}
7 {{formula}} \Rightarrow \alpha \approx 22,89^{\circ}{{/formula}}
8
9 Fläche = {{formula}} \frac{1}{2} \cdot \mid \overrightarrow{AB} \mid \cdot \mid \overrightarrow{AC} \mid \cdot sin(\alpha) = \frac{1}{2} \cdot \sqrt{153} \cdot \sqrt{61}\cdot sin(22,89^{\circ}) \approx 18,79 \text{ FE}{{/formula}}