$\begin{array}{l}
\overrightarrow {MG} = \overrightarrow {MB} + \overrightarrow {BG} = \dfrac{1}{2}\overrightarrow {CB} + \overrightarrow {BO} + \overrightarrow {OG} \\
= \dfrac{1}{2}\left( { - \overrightarrow {AD} } \right) + \dfrac{1}{2}\overrightarrow {BD} + \dfrac{1}{3}\overrightarrow {OD} \\
= - \dfrac{1}{2}\overrightarrow {AD} + \dfrac{1}{2}\overrightarrow {BD} + \dfrac{1}{3}.\dfrac{1}{2}\overrightarrow {BD} \\
= - \dfrac{1}{2}\overrightarrow {AD} + \dfrac{2}{3}\overrightarrow {BD} \\
= - \dfrac{1}{2}\overrightarrow {AD} + \dfrac{2}{3}\left( {\overrightarrow {BA} + \overrightarrow {BC} } \right)\\
= - \dfrac{1}{2}\overrightarrow {AD} - \dfrac{2}{3}\overrightarrow {AB} + \dfrac{2}{3}\overrightarrow {BC} \\
= - \dfrac{1}{2}\overrightarrow {AD} - \dfrac{2}{3}\overrightarrow {AB} + \dfrac{2}{3}\overrightarrow {AD} \\
= \dfrac{1}{6}\overrightarrow {AD} - \dfrac{2}{3}\overrightarrow {AB}
\end{array}$
