\[\boxed{\mathbf{1068.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[\mathbf{Дано:}\]
\[\left| \overrightarrow{a} \right| = 2;\]
\[\left| \overrightarrow{b} \right| = 5;\]
\[\widehat{\overrightarrow{a}\overrightarrow{b}} = 120{^\circ};\]
\[\overrightarrow{p} = x\overrightarrow{a} + 17\overrightarrow{b};\]
\[\overrightarrow{q} = 3\overrightarrow{a} - \overrightarrow{b};\]
\[\overrightarrow{p}\bot\overrightarrow{q}.\]
\[\mathbf{Найти:}\]
\[x - ?\]
\[\mathbf{Решение.}\]
\[1)\ \overrightarrow{p} \bullet \overrightarrow{q} =\]
\[= \left( x\overrightarrow{a} + 17\overrightarrow{b} \right)\left( 3\overrightarrow{a} - \overrightarrow{b} \right) =\]
\[= 3x{\overrightarrow{a}}^{2} - x\overrightarrow{a}\overrightarrow{b} + 51\overrightarrow{a}\overrightarrow{b} - 17{\overrightarrow{b}}^{2} =\]
\[= 12x + 5x - 225 - 425 =\]
\[= 17x - 680.\]
\[2)\ \overrightarrow{p}\bot\overrightarrow{q} \Longrightarrow \overrightarrow{p} \bullet \overrightarrow{q} = 0:\]
\[17x - 680 = 0\]
\[x = \frac{680}{17} = 40.\]
\[Ответ:\ x = 40.\]