\[\boxed{\mathbf{786}\mathbf{.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[Рисунок\ по\ условию\mathbf{\ задачи:}\]
\[\mathbf{Дано:}\]
\[\mathrm{\Delta}\text{ABC};\]
\[AA_{1};BB_{1};CC_{1} - медианы.\]
\[\overrightarrow{a} = \overrightarrow{\text{AC}};\ \ \]
\[\overrightarrow{b} = \overrightarrow{\text{AB}}.\]
\[Выразить:\]
\[\overrightarrow{AA_{1}};\ \ \overrightarrow{BB_{1}\ };\ \overrightarrow{CC_{1}}\mathbf{.}\]
\[\mathbf{Решение.}\]
\[1)\ \overrightarrow{BB_{1}} = \overrightarrow{\text{BA}} + \ \overrightarrow{AB_{1}} =\]
\[= - \overrightarrow{\text{AB}} + \frac{1}{2}\overrightarrow{\text{AC}} = - \overrightarrow{b} + \frac{1}{2}\overrightarrow{a}.\]
\[2)\ \overrightarrow{CC_{1}} = \overrightarrow{\text{CB}} + \overrightarrow{BC_{1}} =\]
\[= \overrightarrow{\text{CA}} + \overrightarrow{AC_{1}} = - \overrightarrow{\text{AC}} + \frac{1}{2}\overrightarrow{\text{AB}} =\]
\[= \overrightarrow{a} + \frac{1}{2}\overrightarrow{b}.\]
\[3)\ \overrightarrow{\text{BC}} = \overrightarrow{\text{BA}} + \overrightarrow{\text{AC}} =\]
\[= - \overrightarrow{\text{AB}} + \overrightarrow{\text{AC}} = - \overrightarrow{b} + \overrightarrow{a}.\]
\[4)\ \overrightarrow{AA_{1}} = \overrightarrow{\text{AB}} + \overrightarrow{BA_{1}} =\]
\[= \overrightarrow{\text{AB}} + \frac{1}{2}\overrightarrow{\text{BC}} = \overrightarrow{b} + \frac{1}{2} \bullet \left( \overrightarrow{a} - \overrightarrow{b} \right) =\]
\[= \overrightarrow{b} + \frac{1}{2}\overrightarrow{a} - \frac{1}{2}\overrightarrow{b} =\]
\[= \frac{1}{2}\overrightarrow{b} + \frac{1}{2}\overrightarrow{a}.\]