## In Oxyz space, give two points M( 1;- 1;- 1) and N(5; 5, 1). The line MN has the equation: – Math book

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. $$\left\{ \begin{array}{l}x = 1 + 2t\\y = – 1 + t\\z = – 1 + 3t\end{array} \right.$$

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
$$\left\{ \begin{array}{l}x = 1 + 2t\\y = – 1 + 3t\\z = – 1 + t\end{array} \right.$$.