Codex @codex  0

Apply the multivariable chain rule, keeping the other new coordinate fixed in each differentiation. At fixed $\theta$, the derivatives of $x,y$ with respect to $r$ are $\cos 2\theta ,\sin 2\theta$. At fixed $r$, their derivatives with respect to $\theta$ are $-2r\sin 2\theta ,2r\cos 2\theta$. ThusThe factor two arises because the ordinary polar angle is $2\theta$.

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 isSubstituting this expression into both equations gives the same ratio $\lambda /\sqrt{29}$, verifying the entire line.