If $z_{1}, z_{2}, z_{3}$ are three complex numbers, then
$| z_{2} + z_{3} |^{2} + | z_{3} + z_{1} |^{2} + | z_{1} + z_{2} |^{2}$ equal

2 | z_{1} + z_{2} + z_{3} |^{2}

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| z_{1} + z_{2} + z_{3} |^{2}

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2 | z_{1} + z_{2} + z_{3} |^{2} - | z_{1} |^{2} - | z_{2} |^{2} - | z_{3} |^{2}

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None of these

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Solution

The correct option is D None of these
${| z_{2} + z_{3} |}^{2} + {| z_{3} + z_{1} |}^{2} + {| z_{1} + z_{2} |}^{2} = {| z_{2} |}^{2} + {| z_{3} |}^{2} + 2 | z_{2} | | z_{3} | R e + {| z_{3} |}^{2} + {| z_{1} |}^{2} + 2 R e (| z_{1} | | z_{3} |) + {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 R e (| z_{1} | | z_{2} |)$
$= 2 ({| z_{1} |}^{2} + {| z_{2} |}^{2} + {| z_{3} |}^{2} + R e (z_{2} z_{3}) + R e (z_{1} z_{3}) + R e (z_{1} z_{2}))$
$∴$ Which is not equal to given options.
Hence, the answer is none of these.

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If z1,z2,z3 are three complex numbers, then|z2+z3|2+|z3+z1|2+|z1+z2|2 equal

The correct option is D None of these|z2+z3|2+|z3+z1|2+|z1+z2|2=|z2|2+|z3|2+2|z2||z3|Re+|z3|2+|z1|2+2Re(|z1||z3|)+|z1|2+|z2|2+2Re(|z1||z2|)=2(|z1|2+|z2|2+|z3|2+Re(z2z3)+Re(z1z3)+Re(z1z2))∴ Which is not equal to given options.Hence, the answer is none of these.

If $z_{1}, z_{2}, z_{3}$ are three complex numbers, then
$| z_{2} + z_{3} |^{2} + | z_{3} + z_{1} |^{2} + | z_{1} + z_{2} |^{2}$ equal