Trong không gian tọa độ Oxyz cho 3 điểm \(A\,(1;1;...

A

- Hướng dẫn giải

Ta có: \(\overrightarrow{AB}=(-2;6;-4);\,\,\overrightarrow{AC}=(m;m-1;-1)\)

Khi đó \(\left[ \overrightarrow{AB},\overrightarrow{AC} \right]=(4m-10;2+4m;-8m+2)\)

\(\Rightarrow {{S}_{ABC}}=\frac{1}{2}\left| \left[ \overrightarrow{AB},\overrightarrow{AC} \right] \right|=\frac{1}{2}\sqrt{{{(4m-10)}^{2}}+{{(2+4m)}^{2}}+{{(-8m+2)}^{2}}}=3\sqrt{3}.\)

\(\Leftrightarrow \sqrt{{{(2m-5)}^{2}}+{{(1+2m)}^{2}}+{{(-4m+1)}^{2}}}=3\sqrt{3} \\\Leftrightarrow 24{{m}^{2}}-24m+27=27 \\\Leftrightarrow \left[ \begin{array} {} m=1 \\ {} m=0 \\ \end{array} \right.\Rightarrow T=1.\)