If z1 and z2 are two complex numbers such that |z1+z2|=|z1|+|z2|, then arg (z1 ω)-(z2. i), where

step 1 In this question it is given that modulus of z1 plus z2 is equals to modulus of z1 plus modulus

If z1 and z2 are two complex numbers such that...

If $z_{1}$ and $z_{2}$ are two complex numbers such that $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$, then arg $\left(z_{1} \omega\right)-\arg \left(z_{2}\right.$ i), where $\omega$ is the complex cube root of unity, is (a) 0 (b) $\frac{\pi}{2}$ (c) $\frac{\pi}{3}$ (d) $\frac{\pi}{6}$

If $z_{1}$ and $z_{2}$ are two complex numbers such that $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$, then arg $\left(z_{1} \omega\right)-\arg \left(z_{2}\right.$ i), where $\omega$ is the complex cube root of unity, is
(a) 0
(b) $\frac{\pi}{2}$
(c) $\frac{\pi}{3}$
(d) $\frac{\pi}{6}$

Key Concepts

Properties of Arguments in Multiplication

The argument of the product of two complex numbers is equal to the sum of their arguments (up to an additive multiple of 2?). This additive property is fundamental in problems involving the determination of phase shifts and angle differences, especially when rotations and multiplications are involved.

Rotation in the Complex Plane via Multiplication

Multiplying a complex number by another complex number on the unit circle results in a rotation by the angle of the multiplier. In particular, when a complex number is multiplied by a cube root of unity, its argument is increased by the angle associated with that cube root, reflecting a rotational transformation.

Cube Roots of Unity

Cube roots of unity are the solutions to the equation ?^3 = 1. Apart from the trivial solution 1, the non-real cube roots of unity correspond to rotations by specific fixed angles (typically ±120° or ±2?/3 radians). Their multiplicative properties are used to introduce rotations in the complex plane.

Polar Representation of Complex Numbers

Complex numbers are often expressed in polar form as z = r·e^(i?), where r is the magnitude (or modulus) and ? is the argument (or angle). This form is especially useful for multiplication, division, and analyzing rotations in the complex plane.

Triangle Inequality and Its Equality Condition

The triangle inequality states that for any complex numbers z1 and z2, |z1 + z2| ? |z1| + |z2|. The equality holds if and only if z1 and z2 have the same argument, meaning they lie on the same ray from the origin. This condition is key in determining directional alignment in problems involving sums of complex numbers.

Transcript

Please give Ace some feedback