If $|z_{1}+z_{2}|=|z_{1}-z_{2}|$, then the different in the amplitudes of $z_{1}$ and $z_{2}$ is

Find the value of $\theta$ if $\frac{\left(3+2i\sin\theta\right)}{\left(1-2i\sin\theta\ \right)}$ Is purely imaginary.

The modulus of the complex number $z$ such that $\left| z + 3 - i\right | = 1$ and $\arg{z} = \pi$ is equal to

Argument of the complex number  $\left( \dfrac { - 1 - 3 i } { 2 + i } \right).$

$z_{1}=-1-i \sqrt{3}, z_{2}=\sqrt{3}-i$.