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Question Number 147764 by puissant last updated on 23/Jul/21

Answered by Olaf_Thorendsen last updated on 23/Jul/21

$$\mathrm{Le}\mathrm{plan}\pi \mathrm{a}\mathrm{pour}\mathrm{equation}:$$$$x-y+z=2$$$$\mathrm{Le}\mathrm{vecteur}\overrightarrow{\mathrm{N}}\left(\begin{array}{c}+1\\ -1\\ +1\end{array}\right)\mathrm{est}\mathrm{un}\mathrm{vecteur}$$$$\mathrm{normal}\mathrm{a}\pi .$$$$\mathrm{La}\mathrm{droite}\mathrm{I}\mathrm{a}\mathrm{pour}\mathrm{equation}:$$$$\frac{x-2}{2}=\frac{y+1}{2}=\frac{z}{1}$$$$\mathrm{Le}\mathrm{vecteur}\overrightarrow{u}\left(\begin{array}{c}+2\\ +2\\ +1\end{array}\right)\mathrm{est}\mathrm{un}\mathrm{vecteur}$$$$\mathrm{directeur}\mathrm{de}\mathrm{I}.$$$$$$$$\overrightarrow{u}\bullet \overrightarrow{\mathrm{N}}=\mid \mid \overrightarrow{u}\mid \mid \times \mid \mid \overrightarrow{\mathrm{N}}\mid \mid \times \cos (\overrightarrow{u},\overrightarrow{\mathrm{N}})$$$$\overrightarrow{u}\bullet \overrightarrow{\mathrm{N}}=3\times \sqrt{3}\times \cos (\overrightarrow{u},\overrightarrow{\mathrm{N}})\left(1\right)$$$$$$$$\overrightarrow{u}\bullet \overrightarrow{\mathrm{N}}=x_{\overrightarrow{u}}{x}_{\overrightarrow{\mathrm{N}}}+y_{\overrightarrow{u}}{y}_{\overrightarrow{\mathrm{N}}}+z_{\overrightarrow{u}}{z}_{\overrightarrow{\mathrm{N}}}$$$$\overrightarrow{u}\bullet \overrightarrow{\mathrm{N}}=(+2)(+1)+(+2)(-1)+(+1)(+1)$$$$\overrightarrow{u}\bullet \overrightarrow{\mathrm{N}}=1\left(2\right)$$$$$$$$\left(1\right)\mathrm{et}\left(2\right):\cos (\overrightarrow{u},\overrightarrow{\mathrm{N}})=\frac{1}{3\sqrt{3}}$$$$$$$${\mathrm{L}}^{\prime }\mathrm{angle}\mathrm{entre}\mathrm{la}\mathrm{droite}\mathrm{I}\mathrm{est}\mathrm{le}\mathrm{plan}\pi \mathrm{est}$$$$\mathrm{le}\mathrm{complement}\mathrm{a}\frac{\pi }{2}\mathrm{de}{\mathrm{l}}^{\prime }\mathrm{angle}\mathrm{entre}$$$$\mathrm{la}\mathrm{droite}\mathrm{I}\mathrm{et}\mathrm{un}\mathrm{vecteur}\mathrm{normal}\mathrm{au}$$$$\mathrm{plan}\pi .\mathrm{Le}\mathrm{sinus}\mathrm{du}\mathrm{premier}\mathrm{est}\mathrm{donc}$$$$\mathrm{egal}\mathrm{au}\mathrm{cosinus}\mathrm{du}\mathrm{second}.$$$$$$$$\mathrm{Le}\mathrm{sinus}\mathrm{est}\mathrm{donc}\frac{1}{3\sqrt{3}}.$$

Commented by puissant last updated on 23/Jul/21

$$\mathrm{thanks}$$

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