Chọn phương án A.

\(\begin{aligned}
\left[\vec{a},\vec{b}\right]&=\left(\left|\begin{matrix}
2 & 1\\ m-2 & 2
\end{matrix}\right|;\left|\begin{matrix}
1 & m+1\\ 2 & 0
\end{matrix}\right|;\left|\begin{matrix}
m+1 & 2\\ 0 & m-2
\end{matrix}\right|\right)\\
&=\left(m-4;2m+1;-m^2-m+2\right).
\end{aligned}\)

Để \(\vec{a},\,\vec{b},\,\vec{c}\) đồng phẳng thì $$\begin{aligned}
&\left[\vec{a},\vec{b}\right]\cdot\vec{c}=0\\
\Leftrightarrow&(m-4)\cdot0+(2m+1)(m-2)+\left(-m^2-m+2\right)\cdot2=0\\
\Leftrightarrow&-5m+2=0\\
\Leftrightarrow&m=\dfrac{2}{5}.
\end{aligned}$$