\[\boxed{\mathbf{800.}\mathbf{ОК\ ГДЗ - домашка\ на}\ 5}\]
\[\mathbf{Дано:}\]
\[\textbf{а)}\ \overrightarrow{m} \nearrow \nearrow \overrightarrow{n};\ \]
\[\textbf{б)}\ \overrightarrow{m} \nearrow \nearrow \overrightarrow{n}\ и\ \ \left| \overrightarrow{m} \right| \geq \overrightarrow{n}.\]
\[\mathbf{Доказать:}\]
\[\textbf{а)}\ \left| \overrightarrow{m} + \overrightarrow{n} \right| = \left| \overrightarrow{m} \right| + \left| \overrightarrow{n} \right|;\]
\[\textbf{б)}\ \left| \overrightarrow{m} + \overrightarrow{n} \right| = \left| \overrightarrow{m} \right| - \left| \overrightarrow{n} \right|.\]
\[\mathbf{Доказательство.}\]
\[\textbf{а)}\ \left| \overrightarrow{\text{AB}} \right| + \left| \overrightarrow{\text{BC}} \right| = \left| \overrightarrow{\text{AC}} \right|:\]
\[\ \left| \overrightarrow{m} + \overrightarrow{n} \right| = \left| \overrightarrow{m} \right| + \left| \overrightarrow{n} \right|.\]
\[\textbf{б)}\ AB = AC + BC:\]
\[\overrightarrow{\text{AC}} = \left| \overrightarrow{m} \right| + \left| \overrightarrow{n} \right|\ \]
\[\left| \overrightarrow{\text{AC}} \right| = \left| \overrightarrow{\text{AB}} \right| - \left| \overrightarrow{\text{BC}} \right|.\]
\[Следовательно:\]
\[\left| \overrightarrow{m} \right| + \left| \overrightarrow{n} \right| = \left| \overrightarrow{m} \right| - \left| \overrightarrow{n} \right|\]