Geometry of a subset of $\mathbb{R}^3$ Let $(x_1,x_2,x_3)\in\mathbb{R}^3$.  How can I describe the geometry of vectors of the form
$$
  \left( \frac{x_1}{\sqrt{x_2^2+x_3^2}}, \frac{x_2}{\sqrt{x_2^2+x_3^2}}, \frac{x_3}{\sqrt{x_2^2+x_3^2}} \right) \, ?
$$
Thank you all!

To elaborate on user1551's answer a bit, notice that this locus you describe is a fibration of unit circles in the $yz$-plane over the $x$-axis. More precisely, you are interseted in the image of the map $$(x_1,x_2,x_3)\mapsto \left(\frac{x_1}{\sqrt{x_2^2+x_3^2}},\frac{x_2}{\sqrt{x_2^2+x_3^2}},\frac{x_3}{\sqrt{x_2^2+x_3^2}}\right),$$
which we will denote by $X$. Now consider the map $f:X\to \mathbb{R}$ given by $(x,y,z)\mapsto x$. Clearly the map $f$ is surjective and the fiber over a point $r\in \mathbb{R}$ (meaning $f^{-1}(r)$) is a circle in the $yz$-plane. 
