\(\begin{array}{l} Goi\,\,\,M,\,\,N\,\,\,la\,\,\,trung\,\,diem\,\,cua\,\,\,BC,\,\,CA.\\ a)\,\,\,\overrightarrow {AH} = \overrightarrow {AG} + \overrightarrow {GH} \\ = \frac{2}{3}\overrightarrow {AM} + \overrightarrow {BG} \\ = \frac{2}{3}.\frac{1}{2}\left( {\overrightarrow {AB} + \overrightarrow {AC} } \right) + \frac{2}{3}\overrightarrow {BN} \\ = \frac{1}{3}\overrightarrow {AB} + \frac{1}{3}\overrightarrow {AC} + \frac{2}{3}.\frac{1}{3}\left( {\overrightarrow {BA} + \overrightarrow {BC} } \right)\\ = \frac{1}{3}\overrightarrow {AB} + \frac{1}{3}\overrightarrow {AC} + \frac{1}{3}\overrightarrow {BA} + \frac{1}{3}\overrightarrow {BA} + \frac{1}{3}\overrightarrow {AC} \\ = - \frac{1}{3}\overrightarrow {AB} + \frac{2}{3}\overrightarrow {AC} .\\ Tuong\,\,\,tu\,\,\,voi\,\,\,\overrightarrow {CH} .\\ b)\,\,\,\overrightarrow {MH} = \overrightarrow {MA} + \overrightarrow {AH} \\ = - \overrightarrow {AM} - \frac{1}{3}\overrightarrow {AB} + \frac{2}{3}\overrightarrow {AC} \\ = - \frac{1}{2}\left( {\overrightarrow {AB} + \overrightarrow {AC} } \right) - \frac{1}{3}\overrightarrow {AB} + \frac{2}{3}\overrightarrow {AC} \\ = - \frac{5}{6}\overrightarrow {AB} + \frac{1}{6}\overrightarrow {AC} . \end{array}\)
