\(\begin{array}{l}A\left( {2; - 1;\,\,5} \right),\,\,\,B\left( {5; - 5;\,\,7} \right),\,\,C\left( {5;\,\,7;\,\,2} \right).\\ \Rightarrow \overrightarrow {AB} = \left( {3; - 4;\,\,2} \right).\end{array}\)
- a) Gọi \(D\left( {a;\,\,b;\,\,c} \right).\)
ABCD là hình bình hành \( \Leftrightarrow \overrightarrow {AB} = \overrightarrow {DC} \)
\(\begin{array}{l} \Leftrightarrow \left( {3; - 4;\,\,2} \right) = \left( {a - 5;\,\,b - 7;\,\,c - 2} \right)\\ \Leftrightarrow \left\{ \begin{array}{l}3 = a - 5\\ - 4 = b - 7\\2 = c - 2\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}a = 8\\b = 3\\c = 4\end{array} \right. \Rightarrow D\left( {8;\,\,3;\,\,4} \right).\end{array}\)
- b) Gọi \(M\left( {m;\,\,0;\,\,0} \right) \in Ox.\)
\( \Rightarrow \overrightarrow {AM} = \left( {m - 2;\,\,1;\,\,5} \right)\)
\(\begin{array}{l} \Rightarrow \left[ {\overrightarrow {AM} ,\,\,\overrightarrow {AB} } \right] = \left( {22;\,\,19 - 2m;\,\,5 - 4m} \right)\\ \Rightarrow {S_{ABM}} = \frac{1}{2}\left| {\left[ {\overrightarrow {AM} ,\,\,\overrightarrow {AB} } \right]} \right| = \frac{1}{2}\sqrt {{{22}^2} + {{\left( {19 - 2m} \right)}^2} + {{\left( {5 - 4m} \right)}^2}} \\ = \frac{1}{2}\sqrt {20{m^2} - 116m + 870} .\end{array}\)
Ta có: \({S_{ABM}}\,\,\,Min \Leftrightarrow S = 20{m^2} - 116m + 870\,\,\,Min\)
\( \Rightarrow S' = 40m - 116 \Rightarrow S' = 0 \Leftrightarrow m = \frac{{116}}{{40}} = \frac{{29}}{{10}}\)
\( \Rightarrow {S_{\min }} \Leftrightarrow m = \frac{{29}}{{10}} \Rightarrow M\left( {\frac{{29}}{{10}};\,\,0;\,\,0} \right).\)
