Решебник по геометрии 10 класс Атанасян ФГОС 757

\[\boxed{\mathbf{757.}ОК\ ГДЗ\ –\ домашка\ на\ 5}\]

\[Дано:\]

\[OA;OB;OC;OM - лучи;\]

\[\angle AOB = \angle BOC = \angle COA = 90{^\circ};\]

\[\angle AOM = \alpha_{1};\]

\[\angle BOM = \alpha_{2};\ \]

\[\angle COM = \alpha_{3}.\]

\[Доказать:\]

\[\text{co}s^{2}\alpha_{1} + cos^{2}\alpha_{2} + cos^{2}a_{3} = 1.\]

\[Доказательство.\]

\[\left| \overrightarrow{a} \right| = \sqrt{x^{2} + y^{2} + z^{2}};\ \ x;y;z -\]

\[координаты\ вектора\ \overrightarrow{a}.\]

\[Отсюда:\]

\[\left| \overrightarrow{a} \right|^{2} = x^{2} + y^{2} + z^{2} =\]

\[= \left| \overrightarrow{a} \right|^{2}\left( \text{co}s^{2}\alpha_{1} + cos^{2}\alpha_{2} + cos^{2}a_{3} \right)\]

\[\text{co}s^{2}\alpha_{1} + cos^{2}\alpha_{2} + cos^{2}a_{3} = 1.\]

\[Что\ и\ требовалось\ доказать.\]