If z1,z2,......,zn are complex numbers such that |z1|=|z2=.....=|zn|=1, then |z1+z2+...+zn| is equal to

Question

If z1,z2,......,zn are complex numbers such that |z1|=|z2=.....=|zn|=1, then |z1+z2+...+zn| is equal to (A)  |z1z2z3.....zn|  (B)  |z1|+|z2|+....+|zn|  (C)  | (1/z1)+(1/z2)+....+(1/zn) |  (D)  n

Tardigrade Admin · Accepted Answer

We have, |z1|=|z2|=....=|zn|=1 ⇒ z1 overlinez1=z2 overlinez2=.....zn overlinezn=1 ⇒ overlinez1=(1/z1), overlinez2=(1/z2),.... overlinezn=(1/zn) Now, |z1+z2+....+zn| =| overlinez1+z2+....+zn| =| overlinez1+ overlinez2+....+ overlinezn|=| (1/z1)+(1/z2)+....+(1/zn) |

Q. If $z_{1}, z_{2}, ......, z_{n}$ are complex numbers such that $∣ z_{1} ∣ = ∣ z_{2} = ..... = ∣ z_{n} ∣ = 1,$ then $∣ z_{1} + z_{2} + ... + z_{n} ∣$ is equal to

$∣ z_{1} z_{2} z_{3} ..... z_{n} ∣$

$∣ z_{1} ∣ + ∣ z_{2} ∣ + .... + ∣ z_{n} ∣$

$∣ ∣ \frac{1}{z _{1}} + \frac{1}{z _{2}} + .... + \frac{1}{z _{n}} ∣ ∣$

$n$

$n$

Q. If z1​,z2​,......,zn​ are complex numbers such that ∣z1​∣=∣z2​=.....=∣zn​∣=1, then ∣z1​+z2​+...+zn​∣ is equal to

∣z1​z2​z3​.....zn​∣

∣z1​∣+∣z2​∣+....+∣zn​∣

∣∣​z1​1​+z2​1​+....+zn​1​∣∣​

n

n​

Q. If $z_{1}, z_{2}, ......, z_{n}$ are complex numbers such that $∣ z_{1} ∣ = ∣ z_{2} = ..... = ∣ z_{n} ∣ = 1,$ then $∣ z_{1} + z_{2} + ... + z_{n} ∣$ is equal to

$∣ z_{1} z_{2} z_{3} ..... z_{n} ∣$

$∣ z_{1} ∣ + ∣ z_{2} ∣ + .... + ∣ z_{n} ∣$

$∣ ∣ \frac{1}{z _{1}} + \frac{1}{z _{2}} + .... + \frac{1}{z _{n}} ∣ ∣$

$n$

$n$