cant uderstand certain parts this another problem..

here is the problem and solution:
https://i30.tinypic.com/2w2mnap.jpg

why they divide into two cases?

then they find a magnetic field

but we dont have any electricity flowing??

the i recognized the formula for a magnetic field from a wire
but when we find a the formula from the biotsavart law
the oly direction is dlX[latex]\hat{r}[/latex]

there is no [latex]\vec{\theta}[/latex] there
because thus in Cartesian coordinates the direction always changes

how they get this theta direction thing??