Решебник по геометрии 11 класс. Атанасян ФГОС 581

\[\boxed{\mathbf{581.}ОК\ ГДЗ\ –\ домашка\ на\ 5}\]

\[Дано:\]

\[\text{ABC}A_{1}B_{1}C_{1} - куб;\]

\[O = AC_{1} \cap BD_{1}.\]

\[Найти:\]

\[число\ k.\]

\[Решение.\]

\[\textbf{а)}\ \overrightarrow{\text{AB}} = k \bullet \overrightarrow{\text{CD}}\text{.\ \ }\]

\[\overrightarrow{\text{AB}} = - \overrightarrow{\text{CD}}\ (противоположные\ \]

\[ребра\ куба\ равны\ и\ \]

\[параллельны:\]

\[\overrightarrow{\text{AB}} \uparrow \downarrow \overrightarrow{\text{CD}};\]

\[- \overrightarrow{\text{CD}} = k \bullet \overrightarrow{\text{CD}}\ \]

\[k = - 1.\]

\[\textbf{б)}\ \overrightarrow{AC_{1}} = k \bullet \overrightarrow{\text{AO}}.\]

\[\overrightarrow{AC_{1}} = \overrightarrow{\text{AO}} + \overrightarrow{OC_{1}} = 2\overrightarrow{\text{AO}};\]

\[k = 2.\]

\[\textbf{в)}\ \overrightarrow{OB_{1}} = k \bullet \overrightarrow{B_{1}D}.\]

\[OB_{1} = OD:\ \]

\[\overrightarrow{B_{1}D} = \overrightarrow{B_{1}O} + \overrightarrow{\text{OD}} = 2\overrightarrow{B_{1}O}\]

\[\overrightarrow{OB_{1}} = - \overrightarrow{B_{1}O}\]

\[\overrightarrow{OB_{1}} = k \bullet ( - 2\overrightarrow{OB_{1}})\]

\[k = - \overrightarrow{\frac{1}{2}}.\]

\[Ответ:\ \ а) - 1;\ \ б)\ 2;\ \ в) - \frac{1}{2}.\]