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Question Number 130818 by Study last updated on 29/Jan/21

if ∣z_1 ∣=∣z_2 ∣=∣z_3 ∣=1    and   ∣(1/z_1 )∣+∣(1/z_2 )∣+∣(1/z_3 )∣=1   find ∣z_1 +z_2 +z_3 ∣=?     z_1 ,z_2 ,z_3 ∈complex number

$${if}\:\mid{z}_{\mathrm{1}} \mid=\mid{z}_{\mathrm{2}} \mid=\mid{z}_{\mathrm{3}} \mid=\mathrm{1}\:\: \\ $$$${and}\:\:\:\mid\frac{\mathrm{1}}{{z}_{\mathrm{1}} }\mid+\mid\frac{\mathrm{1}}{{z}_{\mathrm{2}} }\mid+\mid\frac{\mathrm{1}}{{z}_{\mathrm{3}} }\mid=\mathrm{1} \\ $$$$\:{find}\:\mid{z}_{\mathrm{1}} +{z}_{\mathrm{2}} +{z}_{\mathrm{3}} \mid=?\:\:\: \\ $$$${z}_{\mathrm{1}} ,{z}_{\mathrm{2}} ,{z}_{\mathrm{3}} \in{complex}\:{number} \\ $$

Answered by TheSupreme last updated on 29/Jan/21

z_i =ρ_i e^(iθ_i )   ρ_1 =ρ_2 =ρ_3 =1  (1/ρ_1 )+(1/ρ_2 )+(1/ρ_3 )=1   3=1  impossible  we can evaluate  (1/z_1 )+(1/z_2 )+(1/z_3 )=1  e^(−iθ_1 ) +e^(−iθ_2 ) +e^(−iθ_3 ) =1  then  e^(iθ_1 ) +e^(iθ_2 ) +e^(iθ_3 ) = (e^(iθ_1 ) +e^(iθ_2 ) +e^(iθ_3 ) )^∗ =1^∗ =1

$${z}_{{i}} =\rho_{{i}} {e}^{{i}\theta_{{i}} } \\ $$$$\rho_{\mathrm{1}} =\rho_{\mathrm{2}} =\rho_{\mathrm{3}} =\mathrm{1} \\ $$$$\frac{\mathrm{1}}{\rho_{\mathrm{1}} }+\frac{\mathrm{1}}{\rho_{\mathrm{2}} }+\frac{\mathrm{1}}{\rho_{\mathrm{3}} }=\mathrm{1}\: \\ $$$$\mathrm{3}=\mathrm{1} \\ $$$${impossible} \\ $$$${we}\:{can}\:{evaluate} \\ $$$$\frac{\mathrm{1}}{{z}_{\mathrm{1}} }+\frac{\mathrm{1}}{{z}_{\mathrm{2}} }+\frac{\mathrm{1}}{{z}_{\mathrm{3}} }=\mathrm{1} \\ $$$${e}^{−{i}\theta_{\mathrm{1}} } +{e}^{−{i}\theta_{\mathrm{2}} } +{e}^{−{i}\theta_{\mathrm{3}} } =\mathrm{1} \\ $$$${then} \\ $$$${e}^{{i}\theta_{\mathrm{1}} } +{e}^{{i}\theta_{\mathrm{2}} } +{e}^{{i}\theta_{\mathrm{3}} } =\:\left({e}^{{i}\theta_{\mathrm{1}} } +{e}^{{i}\theta_{\mathrm{2}} } +{e}^{{i}\theta_{\mathrm{3}} } \right)^{\ast} =\mathrm{1}^{\ast} =\mathrm{1} \\ $$$$ \\ $$

Commented by Study last updated on 30/Jan/21

thanks sir

$${thanks}\:{sir} \\ $$

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