Vector equation of a straight line in the space
Consider the points $A = (2, 1,-2)$ and $B = (1,-2, 3)$, and find the vector equation of the straight line that goes through $A$ anb $B$.
We will start computing a director vector: $$\overrightarrow{AB}=B-A=(1,-2,3)-(2,1,-2)=(-1,-3,5)$$
Therefore the vector equation is: $$(x,y,z)=A+k\cdot\overrightarrow{AB}=(2,1,-2)+k\cdot(-1,-3,5)$$
$$(x,y,z)=A+k\cdot\overrightarrow{AB}=(2,1,-2)+k\cdot(-1,-3,5)$$