If $${{{z_1}}}$$ and $${{{z_2}}}$$ are two nonzero complex numbers such that $$\left| {{z_1}\, + {z_2}} \right| = \left| {{z_1}} \right|\, + \left| {{z_2}} \right|\,$$, then Arg $${z_1}$$ - Arg $${z_2}$$ is equal to

The value of $$\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})$$ is

Let $${z_1}$$ and $${z_2}$$ be complex numbers such that $${z_1}$$ $$ \ne $$ $${z_2}$$ and $$\left| {{z_1}} \right| =\,\left| {{z_2}} \right|$$. If $${z_1}$$ has positive real and $${z_2}$$ has negative imaginary part, then $${{{z_1}\, + \,{z_2}} \over {{z_1}\, - \,{z_2}}}$$ may be

If $${z_1}$$ = a + ib and $${z_2}$$ = c + id are complex numbers such that $$\left| {{z_1}} \right| = \left| {{z_2}} \right| = 1$$ and $${\mathop{\rm Re}\nolimits} ({z_1}\,{\overline z _2}) = 0$$, then the pair of complex numbers $${w_1}$$ = a + ic and $${w_2}$$ = b+ id satisfies -