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Câu hỏi: Trong không gian với hệ tọa độ Oxyz, cho ba mặt phẳng $\left( \alpha \right):x+y+2z+1=0$ ; $\left( \beta \right):x+y-z+2=0$ ; $\left( \gamma \right):x-y+5=0$. Mệnh đề nào sau đây sai?
A. $\left( \alpha \right)\bot \left( \gamma \right)$.
B. $\left( \alpha \right)//\left( \gamma \right)$.
C. $\left( \gamma \right)//\left( \beta \right)$.
D. $\left( \alpha \right)\bot \left( \beta \right)$.
A. $\left( \alpha \right)\bot \left( \gamma \right)$.
B. $\left( \alpha \right)//\left( \gamma \right)$.
C. $\left( \gamma \right)//\left( \beta \right)$.
D. $\left( \alpha \right)\bot \left( \beta \right)$.
Ta có $\left\{ \begin{aligned}
& \overrightarrow{{{n}_{\alpha }}}=\left( 1;1;2 \right) \\
& \overrightarrow{{{n}_{\beta }}}=\left( 1;1;-1 \right) \\
& \overrightarrow{{{n}_{\gamma }}}=\left( 1;-1;0 \right) \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \overrightarrow{{{n}_{\alpha }}}.\overrightarrow{{{n}_{\gamma }}}=1-1+0=0\Rightarrow \left( \alpha \right)\bot \left( \gamma \right) \\
& \overrightarrow{{{n}_{\beta }}}.\overrightarrow{{{n}_{\gamma }}}=1-1+0=0\Rightarrow \left( \beta \right)\bot \left( \gamma \right) \\
& \overrightarrow{{{n}_{\alpha }}}.\overrightarrow{{{n}_{\beta }}}=1+1-2=0\Rightarrow \left( \alpha \right)\bot \left( \beta \right) \\
\end{aligned} \right.\Rightarrow $
& \overrightarrow{{{n}_{\alpha }}}=\left( 1;1;2 \right) \\
& \overrightarrow{{{n}_{\beta }}}=\left( 1;1;-1 \right) \\
& \overrightarrow{{{n}_{\gamma }}}=\left( 1;-1;0 \right) \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& \overrightarrow{{{n}_{\alpha }}}.\overrightarrow{{{n}_{\gamma }}}=1-1+0=0\Rightarrow \left( \alpha \right)\bot \left( \gamma \right) \\
& \overrightarrow{{{n}_{\beta }}}.\overrightarrow{{{n}_{\gamma }}}=1-1+0=0\Rightarrow \left( \beta \right)\bot \left( \gamma \right) \\
& \overrightarrow{{{n}_{\alpha }}}.\overrightarrow{{{n}_{\beta }}}=1+1-2=0\Rightarrow \left( \alpha \right)\bot \left( \beta \right) \\
\end{aligned} \right.\Rightarrow $
Đáp án B.