Determining Field Lines of a Vector Field in R^3

[Kork]

Homework Statement

Determine the field lines of the vector field f: R^3 --> R^3 defined by

f(x,y,z) = (x/(1+z²) , y/(x²+y²) , 0)

Homework Equations

The Attempt at a Solution

I know that my vector field is:

F(x,y,z) = (x/((1+z^2)*(x^2+y^2))*i + (y/((1+z^2)*(x^2+y^2))*j + 0k

I choose the point (1,1,1) and if I put that into F(1,1,1) I get

(1/8)i + (1/8)j + 0*k --> <1/8, 1/8>

I need a 3D graph right?

Well, first of all, I don't know how to plot (1/8,1/8) into a 3D graph. Second of all, what happens to the z-coordinate?
Third: I want more than one vector obviously, but when I try with (1,1,-1) I get ∞, which I cannot use to make a vector. Which coordinates can I use?

 

[MrCalc]

i would like to know this as well