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.

  • $\begingroup$ Hint: From $f(x,y,z)= \left | \vec{v} \right |* \left | sin(\alpha ) \right |$, since $\vec{v}$ is constant, what can you say about what set of $(x,y,z)$ give some particular value? For example, try starting with $\alpha = \frac{\pi}{2}$. $\endgroup$ – rogerl Nov 4 '17 at 16:05
  • $\begingroup$ I already tryed to attack the question in this way and it go me nowhere. But, maybe i dont understand what this hint says. $\endgroup$ – danielt17 Nov 4 '17 at 16:06
  • $\begingroup$ If two vectors $\vec{x}_1$ and $\vec{x}_2$ make the same angle with $\vec{v}$, what can you say about $f(\vec{x}_1)$ and $f(\vec{x}_2)$? $\endgroup$ – rogerl Nov 4 '17 at 19:48
  • $\begingroup$ It means they have the same unit vector due I have no idea still on what should I do. could you please write a full answer when you have time $\endgroup$ – danielt17 Nov 5 '17 at 8:30
  • $\begingroup$ I won't write a full answer, because this seems to be a homework problem. But with $f(x,y,z) = |\vec{v}|\cdot|\sin\alpha|$, since $\vec{v}$ is a given constant, if $\vec{x_1}$ and $\vec{x_2}$ make the same angle $\alpha$ with $\vec{v}$, then $f(\vec{x_1}) = |\vec{v}|\cdot|\sin\alpha| = f(\vec{x_2})$. $\endgroup$ – rogerl Nov 5 '17 at 14:58

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