Đáp án:
$\begin{array}{l}
\overrightarrow {AG} \\
= \overrightarrow {AI} + \overrightarrow {IG} \\
= \frac{1}{2}\left( {\overrightarrow {AB} + \overrightarrow {AC} } \right) + \frac{2}{3}.\overrightarrow {IM} \\
= \frac{1}{2}\overrightarrow a + \frac{1}{2}.\left( {\overrightarrow {AB} + \overrightarrow {AD} } \right) + \frac{2}{3}.\frac{1}{2}.\overrightarrow {BD} \\
= \frac{1}{2}\overrightarrow a + \frac{1}{2}.\overrightarrow a + \frac{1}{2}.\overrightarrow b + \frac{1}{3}.\left( {\overrightarrow {BA} + \overrightarrow {BC} } \right)\\
= \overrightarrow a + \frac{1}{2}\overrightarrow b - \frac{1}{2}\overrightarrow a + \frac{1}{3}\overrightarrow b \\
= \frac{1}{2}\overrightarrow a + \frac{5}{6}\overrightarrow b
\end{array}$
