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МатематикаРешебник по геометрии 11 класс. Атанасян ФГОС 624
624
\[\boxed{\mathbf{624.}ОК\ ГДЗ\ –\ домашка\ на\ 5}\]
\[Дано:\ \ \]
\[Решение.\]
\[\textbf{а)}\ \overrightarrow{\text{ON}} = \overrightarrow{\text{OM}} + \overrightarrow{\text{MN}}:\]
\[\overrightarrow{\text{MN}} = \overrightarrow{\text{ON}} - \overrightarrow{\text{OM}}.\]
\[\overrightarrow{\text{OP}} = \overrightarrow{\text{ON}} + \overrightarrow{\text{NP}} = \overrightarrow{\text{ON}} + 2\overrightarrow{\text{MN}};\]
\[\overrightarrow{\text{OP}} = \overrightarrow{\text{ON}} + 2\left( \overrightarrow{\text{ON}} - \overrightarrow{\text{OM}} \right) =\]
\[= 3\overrightarrow{\text{ON}} - 2\overrightarrow{\text{OM}}.\]
\[\textbf{б)}\ \overrightarrow{\text{MP}} = - \frac{1}{2}\overrightarrow{\text{PN}} = \frac{1}{2}\overrightarrow{\text{NP}}:\]
\[PM = MN;\ \ \overrightarrow{\text{PM}} = - \overrightarrow{\text{NM}}.\]
\[Получаем:\]
\[2\overrightarrow{\text{OM}} = \overrightarrow{\text{OP}} + \overrightarrow{\text{PM}} + \overrightarrow{\text{ON}} + \overrightarrow{\text{NM}}\]
\[\overrightarrow{\text{OP}} = 2\overrightarrow{\text{OM}} - \overrightarrow{\text{ON}}.\]
\[\textbf{в)}\ \overrightarrow{\text{ON}} = \overrightarrow{\text{OM}} + \overrightarrow{\text{MN}};\ \]
\[\ \overrightarrow{\text{MP}} = k \bullet \overrightarrow{\text{MN}}:\ \]
\[\overrightarrow{\text{OP}} = \overrightarrow{\text{ON}} + \overrightarrow{\text{NP}} =\]
\[= \overrightarrow{\text{OM}} + \overrightarrow{\text{MN}} + \overrightarrow{\text{NP}} =\]
\[= \overrightarrow{\text{OM}} + \overrightarrow{\text{MP}} = \overrightarrow{\text{OM}} + k \bullet \overrightarrow{\text{MN}}.\]
\[\overrightarrow{\text{MN}} = \overrightarrow{\text{MO}} + \overrightarrow{\text{ON}}:\]
\[\overrightarrow{\text{OP}} = \overrightarrow{\text{OM}} + k\overrightarrow{\text{MO}} + k\overrightarrow{\text{ON}} =\]
\[= k\overrightarrow{\text{ON}} + \overrightarrow{\text{OM}} - k\overrightarrow{\text{OM}} =\]
\[= k\overrightarrow{\text{ON}} + (1 - k)\overrightarrow{\text{OM}}.\]
\[Ответ:\ \ а)\ 3\overrightarrow{\text{ON}} - 2\overrightarrow{\text{OM}};\ \ \]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ б)\ 2\overrightarrow{\text{OM}} - \overrightarrow{\text{ON}};\ \ \]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ в)\ k\overrightarrow{\text{ON}} + (1 - k)\overrightarrow{\text{OM}}.\]
