{left| {{z_1} + {z_2}} right|^2} + {left| {{z | vedclass

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(B) We use the property of the modulus of complex numbers: $\left| z \right|^2 = z \bar{z}$.Expanding the terms:${\left| {{z_1} + {z_2}} \right|^2} =

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$2(\left| {{z_1}} \right|^2 + \left| {{z_2}} \right|^2)$

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(B) We use the property of the modulus of complex numbers: $\left| z \right|^2 = z \bar{z}$.Expanding the terms:${\left| {{z_1} + {z_2}} \right|^2} = (z_1 + z_2)(\bar{z_1} + \bar{z_2}) = z_1\bar{z_1} + z_1\bar{z_2} + z_2\bar{z_1} + z_2\bar{z_2} = \left| {{z_1}} \right|^2 + \left| {{z_2}} \right|^2 + z_1\bar{z_2} + z_2\bar{z_1}$.Similarly:${\left| {{z_1} - {z_2}} \right|^2} = (z_1 - z_2)(\bar{z_1} - \bar{z_2}) = z_1\bar{z_1} - z_1\bar{z_2} - z_2\bar{z_1} + z_2\bar{z_2} = \left| {{z_1}} \right|^2 + \left| {{z_2}} \right|^2 - z_1\bar{z_2} - z_2\bar{z_1}$.Adding these two expressions:${\left| {{z_1} + {z_2}} \right|^2} + {\left| {{z_1} - {z_2}} \right|^2} = (\left| {{z_1}} \right|^2 + \left| {{z_2}} \right|^2 + z_1\bar{z_2} + z_2\bar{z_1}) + (\left| {{z_1}} \right|^2 + \left| {{z_2}} \right|^2 - z_1\bar{z_2} - z_2\bar{z_1})$$= 2\left| {{z_1}} \right|^2 + 2\left| {{z_2}} \right|^2 = 2(\left| {{z_1}} \right|^2 + \left| {{z_2}} \right|^2)$.

${\left| {{z_1} + {z_2}} \right|^2} + {\left| {{z_1} - {z_2}} \right|^2}$ is equal to

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