Why are the vectors being subtracted in this equation?

[electron2]

the shown solution in the link is so different
http://i30.tinypic.com/35mfo94.gif

what those vectors represent
where are they
why they subtract them
i totally don't understand thing
did in a simpler way and got
a different result
where is the mistake

$$B=\frac{\mu _0 I}{4\pi}\int\frac{ds \hat{r}}{r^2}=\frac{\mu _0 I}{4\pi}\int_{0}^{\frac{2\pi}{3}}\frac{d\theta }{{R_1}^2}$$

 

[tiny-tim]

Hi electron2!

It looks as if you're subtracting them because the first line is for W₃, and the second line is for W₄, so when you add them, they're in opposite directions, so one comes out negative.