Magnetic field due to wire along $x$- axis at $A$

${\overrightarrow{\text{B}}}_1=\frac{{\mu }_0}{4\pi }\frac{i}{\text{R}}\left(-\hat{\text{k}}\right)\text{,}$

Magnetic field due to loop in $yz$- plane at $A$

${\overrightarrow{\text{B}}}_2=\frac{{\mu }_0}{4\pi }\frac{2\pi i}{\text{R}}\left(\hat{\text{i}}\right)$

Magnetic field due to wire along $z$-axis at $A$

${\overrightarrow{\text{B}}}_3=\frac{{\mu }_0}{4\pi }\frac{i}{\text{R}}\left(-\hat{\text{i}}\right)$

$\overrightarrow{\text{B}}={\overrightarrow{\text{B}}}_1{+\overrightarrow{\text{B}}}_2+{\overrightarrow{\text{B}}}_3$

    $=\frac{{\mu }_0}{4\pi }\frac{i}{\text{R}}\left(2\pi -1\right)\hat{\text{i}}+\frac{{\mu }_0}{4\pi }\frac{i}{\text{R}}\left(\hat{\text{k}}\right)$

$$\begin{gathered} \text{B}=\frac{{\mu }_0}{4\pi }\frac{i}{\text{R}}\sqrt{{\left(2\pi -1\right)}^2+1} \\ =\frac{{\mu }_0}{4\pi }\frac{i}{\text{R}}\sqrt{2\left(2{\pi }^2-2\pi +1\right)} \end{gathered}$$