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In space \(Oxyz\)for two points \(M\left( {1;\; – 1; – \;1} \right)\) and \(N\left( {5;\;5;\;1} \right)\). Straight line \(MN\) whose equation is:
A. \(\left\{ \begin{array}{l}x = 5 + 2t\\y = 5 + 3t\\z = – 1 + t\end{array} \right.\)
B. \(\left\{ \begin{array}{l}x = 5 + t\\y = 5 + 2t\\z = 1 + 3t\end{array} \right.\)
C. \(\left\{ \begin{array}{l}x = 1 + 2t\\y = – 1 + 3t\\z = – 1 + t\end{array} \right.\)
D. \(\left\{ \begin{array}{l}x = 1 + 2t\\y = – 1 + t\\z = – 1 + 3t\end{array} \right.\)
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The answer:
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We have \(\overrightarrow {MN} = \left( {4;\;6;\;2} \right) = 2\left( {2;3;1} \right)\).
Straight line \(MN\) via \(M\left( {1;\; – 1; – \;1} \right)\) receive \(\overrightarrow {MN} = \left( {2;3;1} \right)\) make a direction vector with equation
\(\left\{ \begin{array}{l}x = 1 + 2t\\y = – 1 + 3t\\z = – 1 + t\end{array} \right.\).
