Решебник по геометрии 10 класс Атанасян ФГОС 685

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\[Пусть\ \ \ \overrightarrow{a}\left\{ x_{1};y_{1};z_{1} \right\};\ \ \]

\[\overrightarrow{b}\left\{ x_{2};y_{2};z_{2} \right\}.\]

\[Тогда:\]

\[\overrightarrow{a} \cdot \overrightarrow{b} = x_{1}x_{2} + y_{1}y_{2} + z_{1}z_{2}.\]

\[\overrightarrow{a}\left\{ 1; - 1;2 \right\};\ \ \overrightarrow{b}\left\{ - 1;1;1 \right\};\ \ \]

\[\overrightarrow{c}\left\{ 5;6;2 \right\}.\]

\[\overrightarrow{a} \cdot \overrightarrow{c} = 1 \cdot 5 + ( - 1) \cdot 6 + 2 \cdot 2 =\]

\[= 5 - 6 + 4 = 3.\]

\[\overrightarrow{a} \cdot \overrightarrow{b} = - 1 - 1 + 2 = 0.\]

\[\overrightarrow{b} \cdot \overrightarrow{c} = - 5 + 6 + 2 = 3.\]

\[\overrightarrow{a} \cdot \overrightarrow{a} = 1 + 1 + 4 = 6.\]

\[\sqrt{\overrightarrow{b} \cdot \overrightarrow{b}} = \sqrt{1 + 1 + 1} = \sqrt{3}.\]