\[\boxed{\mathbf{802.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[Рисунок\ по\ условию\mathbf{\ задачи:}\]
\[\mathbf{Дано:}\]
\[\mathrm{\Delta}ABC;\]
\[BN = 2NC;\]
\[\overrightarrow{a} = \overrightarrow{\text{BA}};\ \]
\[\overrightarrow{b} = \overrightarrow{\text{BC}}.\]
\[\mathbf{Выразить:}\]
\[\overrightarrow{\text{AN}}\ через\ \overrightarrow{a}\ и\ \overrightarrow{b}.\]
\[\mathbf{Решение.}\]
\[1)\ \overrightarrow{\text{BN}} = 2\overrightarrow{\text{NC}}:\]
\[\overrightarrow{\text{BN}} = \frac{2}{3}\overrightarrow{\text{BC}} = \frac{2}{3}\overrightarrow{b}.\]
\[2)\ По\ правилу\ треугольника:\ \ \]
\[\overrightarrow{\text{AN}} = \overrightarrow{\text{AB}} + \overrightarrow{\text{BN}} = - \overrightarrow{\text{BA}} + \overrightarrow{\text{BN}} =\]
\[= - \overrightarrow{a} + \frac{2}{3}\overrightarrow{b}.\]