Đáp án:
$\overrightarrow{AG}= \dfrac49\overrightarrow{AB}+\dfrac13\overrightarrow{AC} $
Giải thích các bước giải:
Gọi $E$ là trung điểm $BM$
$\to \begin{cases}AM = ME = EB =\dfrac13AB\\CG = \dfrac23CE\end{cases}$
$\to \begin{cases}\overrightarrow{AE}=\dfrac23\overrightarrow{AB}\\\overrightarrow{CG} = \dfrac23\overrightarrow{CE}\end{cases}$
Ta có:
$\quad \overrightarrow{AG}= \overrightarrow{AC} +\overrightarrow{CG}$
$\to \overrightarrow{AG}= \overrightarrow{AC} +\dfrac23\overrightarrow{CE}$
$\to \overrightarrow{AG}= \overrightarrow{AC} +\dfrac23(\overrightarrow{CA} + \overrightarrow{AE})$
$\to \overrightarrow{AG}= \overrightarrow{AC} -\dfrac23\overrightarrow{AC}+ \dfrac23\overrightarrow{AE}$
$\to \overrightarrow{AG}= \dfrac13\overrightarrow{AC} + \dfrac49\overrightarrow{AB}$
Vậy $\overrightarrow{AG}= \dfrac49\overrightarrow{AB}+\dfrac13\overrightarrow{AC} $
