Parametric equations of a straight line in the space
Consider the points $A = (2, 1,-2)$ and $B = (1,-2, 3)$, and find the parametric equations of the straight line that goes through $A$ anb $B$.
We will start computing a director vector of the straight line: $$\overrightarrow{AB}=B-A=(1,-2,3)-(2,1,-2)=(-1,-3,5)$$
Therefore, with the director vector and point $A$, we obtain the parametric equations: $$\left\{\begin{array}{l} x=2-k \\ y=1-3k \\ z=-2+5k \end{array}\right.$$
$\left\{\begin{array}{l} x=2-k \\ y=1-3k \\ z=-2+5k \end{array}\right.$