Ta có
\(d:\left\{ \begin{align}
& M\left( 2;\,1;\,0 \right)\in d \\
& \overrightarrow{{{u}_{d}}}=\left( 1;\,2;\,-1 \right) \\
\end{align} \right.\).
Do \(A\in Ox,\,B\in Oy\Rightarrow AB\subset \left( Oxy \right)\)\( \Rightarrow \overrightarrow{{{u}_{AB}}}\bot \overrightarrow{k}=\left( 0;\,0;\,1 \right)\).
Đường thẳng \(AB\bot d\)\( \Rightarrow \overrightarrow{{{u}_{AB}}}\bot \overrightarrow{{{u}_{d}}}\).
Suy ra \(\overrightarrow{{{u}_{AB}}}\)\( =\left[ \overrightarrow{k},\,\overrightarrow{{{u}_{d}}} \right]=\left( -2;\,1;\,0 \right)\).
Do \(\left\{ \begin{align}
& d\subset \left( P \right) \\
& AB\subset \left( P \right) \\
\end{align} \right.\)
\(\Rightarrow \overrightarrow{{{n}_{P}}}\)\( =\left[ \overrightarrow{{{u}_{AB}}},\,\overrightarrow{{{u}_{d}}} \right]=\left( -1;\,-2;\,-5 \right)\).
Phương trình mặt phẳng \(\left( P \right)\) qua \(M\left( 2;\,1;\,0 \right)\) và nhận véctơ \(\overrightarrow{{{n}_{P}}}=\left( -1;\,-2;\,-5 \right)\) làm một véctơ pháp tuyến là \(\left( P \right):-1\left( x-2 \right)-2\left( y-1 \right)-5\left( z-0 \right)=0\)\( \Leftrightarrow x+2y+5z-4=0\).