\[\boxed{\mathbf{1056.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[Рисунок\ по\ условию\ задачи:\]
\[\mathbf{Дано:}\]
\[ABCD - ромб.\]
\[\mathbf{Доказать:}\]
\[AC\bot BD.\]
\[\mathbf{Доказательство.}\]
\[1)\ \overrightarrow{\text{BD}} = \overrightarrow{\text{BA}} + \overrightarrow{\text{BC}};\]
\[\overrightarrow{\text{AC}} = \overrightarrow{\text{BC}} - \overrightarrow{\text{BA}}.\]
\[2)\ \overrightarrow{\text{BD}} \bullet \overrightarrow{\text{AC}} =\]
\[= \left( \overrightarrow{\text{BA}} + \overrightarrow{\text{BC}} \right)\left( \overrightarrow{\text{BC}} - \overrightarrow{\text{BA}} \right) =\]
\[= \overrightarrow{BC^{2}} - \overrightarrow{BA^{2}}.\]
\[3)\ \left| \overrightarrow{\text{CB}} \right| = \left| \overrightarrow{\text{BA}} \right| = a:\]
\[\overrightarrow{\text{BD}} \bullet \overrightarrow{\text{AC}} = a^{2} - a^{2} = 0 \Longrightarrow\]
\[\Longrightarrow AC\bot BD.\]
\[Что\ и\ требовалось\ доказать.\]