\[\boxed{\mathbf{1053.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[\mathbf{Дано:}\]
\[\overrightarrow{a} = 3\overrightarrow{p} - 2\overrightarrow{q};\]
\[\overrightarrow{b} = \overrightarrow{p} + 4\overrightarrow{q};\]
\[\overrightarrow{p}\bot\overrightarrow{q};\ \]
\[\left| \overrightarrow{p} \right| = \left| \overrightarrow{q} \right| = 1.\]
\[\mathbf{Найти:}\]
\[\overrightarrow{a} \bullet \overrightarrow{b} - ?\]
\[\mathbf{Решение.}\]
\[1)\ \overrightarrow{p}\bot\overrightarrow{q} \Longrightarrow \overrightarrow{p} \bullet \overrightarrow{q} = 0.\]
\[2)\ \overrightarrow{a} \bullet \overrightarrow{b} = \left( 3\overrightarrow{p} - 2\overrightarrow{q} \right)\left( \overrightarrow{p} + 4\overrightarrow{q} \right)\]
\[\overrightarrow{a} \bullet \overrightarrow{b} =\]
\[= 3\overrightarrow{p^{2}} + 12 \bullet \overrightarrow{p} \bullet \overrightarrow{q} - 2 \bullet \overrightarrow{p} \bullet \overrightarrow{q} - 8\overrightarrow{q^{2}}\]
\[\overrightarrow{a} \bullet \overrightarrow{b} = 3\overrightarrow{p^{2}} - 8\overrightarrow{q^{2}}.\]
\[3)\ \overrightarrow{a} \bullet \overrightarrow{b} = 3 - 8 = - 5.\]
\[\mathbf{Ответ:} - 5.\]
\[\boxed{\mathbf{1053.еуроки - ответы\ на\ пятёрку}}\]
\[\mathbf{Решение.}\]
\[1)\ O(0;0):\]
\[x^{2} + y^{2} = R^{2}.\ \]
\[2)\ B \in (O;R) \Longrightarrow OB = R.\]
\[3)\ OB =\]
\[= \sqrt{( - 1 - 0)^{2} + (3 - 0)^{2}} =\]
\[= \sqrt{1 + 9} = \sqrt{10}.\]
\[4)\ Уравнение\ окружности:\ \]
\[x^{2} + y^{2} = \left( \sqrt{10} \right)^{2}\]
\[x^{2} + y^{2} = 10.\]
\[Ответ:x^{2} + y^{2} = 10.\]