Check whether the lines given by (x-1)/2=(y-2)/3=(z-3)/4 and (x-4)/5=(y-1)/2=z are parallel or not. If parallel, find the distance between them,…

DRs of first line: $\displaystyle \langle2,3,4\rangle$

Any Point on first line: $\displaystyle (2\lambda+1, 3\lambda+2, 4\lambda+3)$;

For the lines to intersect, we must have some $\displaystyle \lambda$ and $\displaystyle \mu$ for which these coordinates coincide, i.e. $\displaystyle 2\lambda+1=5\mu+4$, $\displaystyle 3\lambda+2=2\mu+1$, $\displaystyle 4\lambda+3=\mu$, for some $\displaystyle \lambda$ and $\displaystyle \mu$

Solving the first two coordinate equations: $\displaystyle \lambda=-1$, $\displaystyle \mu=-1$; this satisfies the third equation too.

The lines intersect at $\displaystyle (-1,-1,-1)$