If z_1 , z_2 and z_3, z_4 are 2 pairs of comp | vedclass

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Question

Consider $\arg \left(\frac{z_{1}}{z_{4}}\right)+$ $\arg \left(\frac{z_{2}}{z_{3}}\right)$

$=\arg \left(z_{1}\right)-\arg \left(z_{4}\right)$ $+\arg

Vedclass · Accepted Answer

$0$

Vedclass · Answer

Consider $\arg \left(\frac{z_{1}}{z_{4}}ight)+$ $\arg \left(\frac{z_{2}}{z_{3}}ight)$

$=\arg \left(z_{1}ight)-\arg \left(z_{4}ight)$ $+\arg \left(z_{2}ight)-\arg \left(z_{3}ight)$

$=\left(\arg \left(z_{1}ight)+\arg \left(z_{2}ight)ight)$ $-\left(\arg \left(z_{3}ight)+\arg \left(z_{4}ight)ight)$

given $\left( {{z_2} = {{\bar z}_1} and {z_4} = {{\bar z}_3}} ight)$ 

$ = \left( {\arg \left( {{z_1}} ight) + \arg \left( {{{\bar z}_1}} ight)} ight) - $ $\left( {\arg \left( {{z_3}} ight) + \arg \left( {{{\bar z}_3}} ight)} ight)$

$\left\{ \begin{gathered}
  \operatorname{also} (\arg \left( {{{\bar z}_1}} ight) =  - \arg \left( {{z_1}} ight) \hfill \
  \arg \left( {{{\bar z}_3}} ight) =  - \arg \left( {{z_3}} ight) \hfill \ 
\end{gathered}  ight\}$

$=\left(\arg \left(z_{1}ight)-\arg \left(z_{1}ight)ight)-\left(\arg \left(z_{3}ight)-\arg \left(z_{3}ight)ight)$

$=0-0=0$

If $z_1 , z_2$ and $z_3, z_4$ are $2$ pairs of complex conjugate numbers, then $\arg \left( {\frac{{{z_1}}}{{{z_4}}}} \right) + \arg \left( {\frac{{{z_2}}}{{{z_3}}}} \right)$ equals

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