$\begin{array}{l}
\overrightarrow {AM} = \overrightarrow {AB} + \overrightarrow {BM} = \overrightarrow {AB} + \frac{{BM}}{{BC}}.\overrightarrow {BC} = \overrightarrow {AB} + \frac{{BM}}{{BC}}\left( {\overrightarrow {AC} - \overrightarrow {AB} } \right)\\
= \overrightarrow {AB} - \frac{{BM}}{{BC}}\overrightarrow {AB} + \frac{{BM}}{{BC}}\overrightarrow {AC} = \frac{{BC - BM}}{{BC}}\overrightarrow {AB} + \frac{{BM}}{{BC}}\overrightarrow {AC} = \frac{{CM}}{{BC}}.\overrightarrow {AB} + \frac{{BM}}{{BC}}\overrightarrow {AC}
\end{array}$
