Solution
= Solution
The common ratio in the two <plane> equations can be denoted by $u$. Their intersection then has $x=1+4u$, $y=2+3u$, and $z=-1+2u$. The direction vector $(4,3,2)$ has <Euclidean norm> $\sqrt{29}$, so the requested <unit vector> form is
$$
\boxed{\mathbf r=(1,2,-1)+\lambda\frac{(4,3,2)}{\sqrt{29}},\qquad\lambda\in\mathbb R}.
$$
Substituting this expression into both equations gives the same ratio $\lambda/\sqrt{29}$, verifying the entire line.