Let z1- z2- z3 - C such that -z1- - -z2- - -z3- - 1- If z1 - z2 - z3 - 0 and z12 - z22 - z32 - 0- then -z1 - z2 - z3- is

(z_1 + z_2 + z_3)^2 = z_1^2 + z_2^2 + z_3^2 + 2 (z_1 z_2 + z_2 z_3 + z_3 z_1) (z_1 + z_2 + z_3)^2 = 2z_1z_2z_3 ( 1/z_3 + 1/z_1 + 1/z_2 ) (∵ ...

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Properties of Conjugate of a Complex Number

Let z1, z2,...

Question

Let $z_{1}, z_{2}, z_{3} \in$ C such that $| z_{1} | = | z_{2} | = | z_{3} | = 1$ . If $z_{1} + z_{2} + z_{3} \neq 0$ and $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 0$ , then $| z_{1} + z_{2} + z_{3} |$ is

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Z_{1} + Z_{2} + Z_{3} = 0

| Z_{1} | = | Z_{2} | = | Z_{3} | = 1

Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}

z_{1}, z_{2}

z_{3}

| z_{1} | = | z_{2} | = | z_{3} | = 1

\frac{z_{1}^{2}}{z_{2} z_{3}} + \frac{z_{2}^{2}}{z_{3} z_{1}} + \frac{z_{3}^{2}}{z_{1} z_{2}} + 1 = 0.

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

z_{1}, z_{2}, z_{3}

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

| z_{1} + z_{2} + z_{3} | = 21.

| z_{1} - z_{2} | = 2 \sqrt{3}, | z_{1} | = 3 \sqrt{3},

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

z_{1}, z_{2}, z_{3}

| z_{1} | = | z_{2} | = | z_{3} | = 1

\frac{z_{1}^{2}}{z_{2} z_{3}} + \frac{z_{2}^{2}}{z_{1} z_{3}} + \frac{z_{3}^{2}}{z_{1} z_{2}} = - 1.

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

z_{1}, z_{2}, z_{3}

z_{2} \neq z_{1}, a = | z_{1} |, b = | z_{2} | a n d c = | z_{3} |

∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣ = 0

(\frac{z_{3}}{z_{2}})

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