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Câu hỏi: Mặt phẳng $\left( P \right)$ đi qua 2 điểm $A\left( 2;-1;4 \right),B\left( 3;2;1 \right)$ và vuông góc với $\left( \alpha \right):2x-y+3z-5=0$ là
A. $6x+9y+z+1=0.$
B. $6x+9y-7z+7=0.$
C. $6x+9y+7z+7=0.$
D. $6x-9y-7z+7=0.$
A. $6x+9y+z+1=0.$
B. $6x+9y-7z+7=0.$
C. $6x+9y+7z+7=0.$
D. $6x-9y-7z+7=0.$
$A\left( 2;-1;4 \right),B\left( 3;2;1 \right)\Rightarrow \overrightarrow{AB}=\left( 1;3;-3 \right)$
$\left( \alpha \right):2x-y+3z-5=0\Rightarrow \overrightarrow{{{n}_{\left( \alpha \right)}}}=\left( 2;-1;3 \right)$
$\Rightarrow \overrightarrow{{{n}_{\left( P \right)}}}=\left[ \overrightarrow{AB},\overrightarrow{{{n}_{\left( \alpha \right)}}} \right]=\left( 6;-9;-7 \right)\Rightarrow \left( P \right):6\left( x-2 \right)-9\left( y+1 \right)-7\left( z-4 \right)=0$
$\Leftrightarrow 6x-9y-7z+7=0.$
$\left( \alpha \right):2x-y+3z-5=0\Rightarrow \overrightarrow{{{n}_{\left( \alpha \right)}}}=\left( 2;-1;3 \right)$
$\Rightarrow \overrightarrow{{{n}_{\left( P \right)}}}=\left[ \overrightarrow{AB},\overrightarrow{{{n}_{\left( \alpha \right)}}} \right]=\left( 6;-9;-7 \right)\Rightarrow \left( P \right):6\left( x-2 \right)-9\left( y+1 \right)-7\left( z-4 \right)=0$
$\Leftrightarrow 6x-9y-7z+7=0.$
Đáp án D.