Don't Guess

Let $\text{arg z}$ be the principal branch of argument, i.e, if $a = 2 + 2\sqrt{3}i = 4\cdot e^{\frac{i\pi}{3}} = 4\cdot e^{\frac{i\pi }{3} + 2\pi i} = ...$ then $\text{arg a} = \frac{\pi}{3}$ .

Then $A$ is the semi-line in the first quadrant in the Argand plane (cartesian plane) forming $60 º$ with real axis, i.e, $A = \{r\cdot e^{\frac{i\pi }{3}}; r \in \mathbb{R}, r >0\}$ and $A \cap B = \phi$ (the empty set) due to the sume of two vectors(apply the law of parallelogram), $z \in \mathbb{C}$ and $-2 - 2\sqrt{3}i$ will have an principal (main) branch of argument being $\frac{2\pi}{3}$ ,i.e, being $B = \{z = 2 + 2\sqrt{3}i + r\cdot e^{\frac{2i\pi }{3}}; r \in \mathbb{R}, r >0\}$ the semiline parallel to the semiline starting at the origin and forming $120 °$ with the real axis without cutting $A$