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 $$B$$.
See development and solution
Development:
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: $$$(x,y,z)=A+k\cdot\overrightarrow{AB}=(2,1,-2)+k\cdot(-1,-3,5)$$$
Solution:
$$(x,y,z)=A+k\cdot\overrightarrow{AB}=(2,1,-2)+k\cdot(-1,-3,5)$$