If z1- z2- z3 are complex numbers such that z 1- z 2- z 3-1 z 1-1 z 2-1 z 3-1- then find the value of z 1- z 2- z 3-

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Byju's Answer

Standard VIII

Physics

Static Friction

If z1, z2, z3...

Question

If z₁, z₂, z₃ are complex numbers such that $|z_{1}| = |z_{2}| = |z_{3}| = |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1$ , then find the value of $|z_{1} + z_{2} + z_{3}|$ .

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Solution

$|z_{1} + z_{2} + z_{3}| = |\frac{z_{1} \bar{z_{1}}}{\bar{z_{1}}} + \frac{z_{2} \bar{z_{2}}}{\bar{z_{2}}} + \frac{z_{3} \bar{z_{3}}}{\bar{z_{3}}}| = |\frac{{|z_{1}|}^{2}}{\bar{z_{1}}} + \frac{{|z_{2}|}^{2}}{\bar{z_{2}}} + \frac{{|z_{3}|}^{2}}{\bar{z_{3}}}| = |\frac{1}{\bar{z_{1}}} + \frac{1}{\bar{z_{2}}} + \frac{1}{\bar{z_{3}}}| [∵ |z_{1}| = |z_{2}| = |z_{3}| = 1] = \bar{|\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}|} = 1 [∵ |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1]$

Thus, the value of $|z_{1} + z_{2} + z_{3}|$ is 1.

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If z1, z2, z3 are complex numbers such that z1=z2=z3=1z1+1z2+1z3=1, then find the value of z1+z2+z3.

z1+z2+z3=z1z1¯z1¯+z2z2¯z2¯+z3z3¯z3¯ =z12z1¯+z22z2¯+z32z3¯ =1z1¯+1z2¯+1z3¯ [∵ z1=z2=z3=1] =1z1+1z2+1z3 ¯ =1 ∵ 1z1+1z2+1z3 =1 Thus, the value of z1+z2+z3 is 1.

If z₁, z₂, z₃ are complex numbers such that $|z_{1}| = |z_{2}| = |z_{3}| = |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1$ , then find the value of $|z_{1} + z_{2} + z_{3}|$ .