Two lines $r_1$ and $r_2$ are defined per the following systems of equations $$r_1:\quad x+2y=3,y+2z=3$$ $$r_2: \quad x+y+z=6,x-2z=5$$
The lines are defined as sets of points, which satisfy both of their equations. I want to know the distance between these lines. This distance is equal to the distance between two points on the lines, if the direction vector between these points is perpendicular to both lines. Finding two general points on these lines is easy. To do this, however, I need the direction vectors for both lines. How can I obtain these vectors?
Well, first of all bring the lines into the standard vector form as ${\vec r}={\vec a}+{\lambda}{\vec b}$. Then you may cross the two $\vec b$ to obtain the direction vector of the required line.