Đáp án:

Gọi M là trung điểm BC

$\overrightarrow{AG}$ = $\frac{2}{3}$ $\overrightarrow{AM}$ = $\frac{1}{3}$ ($\overrightarrow{AB}$ + $\overrightarrow{AC}$ ) 2$\overrightarrow{IB}$ = 3$\overrightarrow{IB}$ <=> 2( $\overrightarrow{AC}$ - $\overrightarrow{AI}$) = -3 ( $\overrightarrow{AB}$ - $\overrightarrow{AI}$ ) <=> $\overrightarrow{AI}$ = $\frac{3}{5}$ $\overrightarrow{AB}$ + $\frac{2}{5}$ $\overrightarrow{AC}$

Tương tự <=> $\overrightarrow{AJ}$ = $\frac{5}{3}$ $\overrightarrow{AB}$ - $\frac{2}{3}$ $\overrightarrow{AC}$

Ta có hệ: $\begin{cases} \frac{3}{5} \overrightarrow{AB} + \frac{2}{5} \overrightarrow{AC} = \overrightarrow{AI}\\\frac{3}{5}\overrightarrow{AB} - \frac{2}{5} \overrightarrow{AC} = \overrightarrow{AI}\\ \end{cases}$ <=> $\begin{cases} \overrightarrow{AB} = \frac{5}{8}\overrightarrow{AI} + \frac{3}{5}\overrightarrow{AJ}\\\overrightarrow{AC} = \frac{25}{16}\overrightarrow{AI} - \frac{9}{16}\overrightarrow{AJ}\\ \end{cases}$ => $\overrightarrow{AG}$ = $\frac{1}{3}$ $\left[\begin{matrix} \frac{5}{8}\overrightarrow{AI} + \frac{3}{8}\overrightarrow{AJ}\\ +\frac{25}{16}\overrightarrow{AI} - \frac{9}{16}\overrightarrow{AJ}\end{matrix}\right.$ ]

= $\frac{35}{48}$ $\overrightarrow{AI}$ - $\frac{1}{16}$ $\overrightarrow{AJ}$