 Solution

Place the wire on the $y$ axis, its endpoints at $-a$ and $b$, and the observation point at $(-d,0)$. With current along $+y$, $d{\mathbf{\ell }}^{\prime }\times \mathbf{r}$ points out of the page. Its magnitude givesIn the question's angle convention, $\cos \beta =b/\sqrt{d^2+b^2}$ and $\cos \alpha =-a/\sqrt{d^2+a^2}$. ThereforeThe minus sign in $\cos \alpha$ matters because $\alpha$ is the exterior obtuse angle.