Cho 3 vectơ \(\overrightarrow{u}=\left( 2;-1;1 \right);\overrightarrow{v}=\left( m;3;-1 \right);\overrightarrow{w}=\left( 1;2;1 \right).\) Tìm m để 3 vectơ đồng phẳng.

* Hướng dẫn giải

Ta có:

\(\left[ \overrightarrow{u},\overrightarrow{v} \right]=\left( -2;m+2;m+6 \right)\\\Rightarrow \left[ \overrightarrow{u},\overrightarrow{v} \right].\overrightarrow{w}=-2+2m+4+m+6=3m+8\)

Ba vectơ \(\overrightarrow{u};\,\,\overrightarrow{v};\,\,\overrightarrow{w}\) đồng phẳng \(\Leftrightarrow 3m+8=0\Leftrightarrow m=-\frac{8}{3}.\)