\[\overrightarrow {AK\,} = \,\frac{1}{2}\left( {\overrightarrow {AM\,} + \overrightarrow {AN\,} } \right) = \,\frac{1}{2}\left( {\frac{1}{2}\overrightarrow {AB\,} + \frac{2}{3}\overrightarrow {AC\,} } \right) = \frac{1}{4}\overrightarrow {AB\,} + \frac{1}{3}\overrightarrow {AC\,} \].