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МатематикаРешебник по геометрии 10 класс Атанасян ФГОС 633
633
\[\boxed{\mathbf{633.}ОК\ ГДЗ\ –\ домашка\ на\ 5}\]
\[Дано:\ \ \]
\[ABCD - тетраэдр;\ \ \]
\[BM = MC.\]
\[Разложить:\ \ \]
\[по\ векторам\ \overrightarrow{\text{AB}} = \overrightarrow{b},\ \overrightarrow{\text{AD}} = \overrightarrow{d}\ и\ \]
\[\overrightarrow{\text{AC}} = \overrightarrow{c}\ \]
\[векторы\ \overrightarrow{\text{BC}},\ \overrightarrow{\text{CD}},\overrightarrow{\text{DB}}\ и\ \overrightarrow{\text{DM}}.\]
\[Решение.\]
\[1)\ \overrightarrow{\text{BC}} = \overrightarrow{\text{BA}} + \overrightarrow{\text{AC}} = \overrightarrow{\text{AC}} - \overrightarrow{\text{AB}} =\]
\[= \overrightarrow{c} - \overrightarrow{b}.\]
\[2)\ \overrightarrow{\text{CD}} = \overrightarrow{\text{CA}} + \overrightarrow{\text{AD}} = \overrightarrow{\text{AD}} - \overrightarrow{\text{AC}} =\]
\[= \overrightarrow{d} - \overrightarrow{c}.\]
\[3)\ \overrightarrow{\text{DB}} = \overrightarrow{\text{DA}} + \overrightarrow{\text{AB}} = \overrightarrow{\text{AB}} - \overrightarrow{\text{AD}} =\]
\[= \overrightarrow{b} - \overrightarrow{d}.\]
\[4)\ \ \overrightarrow{\text{CM}} = - \overrightarrow{\text{BM}}:\]
\[\overrightarrow{\text{DM}} = \frac{1}{2}\left( \overrightarrow{\text{DB}} + \overrightarrow{\text{DC}} \right).\]
\[\overrightarrow{\text{DB}} = \overrightarrow{b} - \overrightarrow{d}\text{\ \ }и\ \ \overrightarrow{\text{DC}} = - \overrightarrow{\text{CD}} =\]
\[= \overrightarrow{c} - \overrightarrow{d}:\]
\[\overrightarrow{\text{DM}} = \frac{1}{2}\left( \overrightarrow{b} - \overrightarrow{d} + \overrightarrow{c} - \overrightarrow{d} \right) =\]
\[= \frac{1}{2}\left( \overrightarrow{b} + \overrightarrow{c} - 2\overrightarrow{d} \right) = \frac{1}{2}\overrightarrow{b} + \frac{1}{2}\overrightarrow{c} - \overrightarrow{d}.\]
\[Ответ:\ \ \overrightarrow{\text{BC}} = \overrightarrow{c} - \overrightarrow{b};\ \ \overrightarrow{\text{CD}} = \overrightarrow{d} - \overrightarrow{c};\ \ \]
\[\ \overrightarrow{\text{DB}} = \overrightarrow{b} - \overrightarrow{d};\ \ \overrightarrow{\text{DM}} =\]
\[= \frac{1}{2}\overrightarrow{b} + \frac{1}{2}\overrightarrow{c} - \overrightarrow{d}.\]
