\[\boxed{\mathbf{1052.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[\mathbf{Дано:}\]
\[\left| \overrightarrow{a} \right| = 5;\left| \overrightarrow{b} \right| = 2;\]
\[\left| \overrightarrow{c} \right| = 4;\overrightarrow{a}\bot\overrightarrow{b};\]
\[\overrightarrow{p} = \overrightarrow{a} - \overrightarrow{b} - \overrightarrow{c};\]
\[\overrightarrow{q} = \overrightarrow{a} - \overrightarrow{b} + \overrightarrow{c}.\]
\[\mathbf{Найти:}\]
\[\overrightarrow{p} \bullet \overrightarrow{q} - ?\]
\[\mathbf{Решение.}\]
\[1)\ \overrightarrow{a}\bot\overrightarrow{b} \Longrightarrow \overrightarrow{a} \bullet \overrightarrow{b} = 0.\]
\[2)\ \overrightarrow{p} \bullet \overrightarrow{q} =\]
\[= \left( \overrightarrow{a} - \overrightarrow{b} - \overrightarrow{c} \right)\left( \overrightarrow{a} - \overrightarrow{b} + \overrightarrow{c} \right)\]
\[\overrightarrow{p} \bullet \overrightarrow{q} = \overrightarrow{a^{2}} + \overrightarrow{b^{2}} - \overrightarrow{c^{2}}.\]
\[3)\ \overrightarrow{p} \bullet \overrightarrow{q} = 5^{2} + 2^{2} - 4^{2} =\]
\[= 25 + 4 - 16 = 13.\]
\[\mathbf{Ответ:}13.\]