Let this be the question:
Suppose $\vec{v}$ and $\vec{w}$ are two vectors parallel to the plane $$x + 2y + 3z = 7.$$ Suppose furthermore that $\vec{v}$ is perpendicular to $\vec{w}$, $$‖v‖= 3, \ ‖w‖= 4.$$
How would you go about answering this question? I always reach a dead end when I try to solve $\vec{v}$ and $\vec{w}$ that satisfy the given information, I just don't know how to go about it. I tried to draw the plane and two parallel vectors, then I know the length of the cross product would be $12$. Then what do I do now? How could you solve this to find vectors $\vec{v}$ and $\vec{w}$?
You have: $\vec{u}\times \vec{v}= n(1,2,3)\implies |\vec{u}\times\vec{v}|=|n|\sqrt{1^2+2^2+3^2}=|n|\sqrt{14}=12\implies n = \pm\dfrac{12}{\sqrt{14}}\implies \vec{u}\times\vec{v}=\pm\dfrac{12}{\sqrt{14}}\left(1,2,3\right)$.