Let $0\neq \vec{v}\in \mathbb{R}$ a constant vector. For every $(x,y,z)\neq 0$ let $f(x,y,z)=\frac{\left |(x,y,z)\times \vec{v} \right |}{\left |(x,y,z) \right |}$. Find all of the level sets of $f$. Are there between all of the level sets a plane and a line which make a $90^{\circ}$ angle?
I have no idea how I should solve this question, i tryed setting $\vec{v}=(a,b,c)$ and opening the cross multiplaction but nothing seems to work. I also tryed setting it like that $f(x,y,z)= \left | \vec{v} \right |* \left | sin(\alpha ) \right |$ but this gets me nowhere because i cant find a connection with alpha.
Anybody has any hints on how to attack this question because i currently have no idea.