Algebra> If the complex number z1,z2 ,z3 represent...

If the complex number z1,z2 ,z3 represent the vertices of an equilateral triangle such that |z1|=|z2|=|z3|, then find z1+z2+z3

Harshit , 9 Years ago

Grade 11

2 Answers

mycroft holmes

Here circumcentre is the origin O. Let the vertices be A,B,C. We have angles AOB = BOC = COA = 120^o and OA = OB = OC since its given that |z₁| = |z₂| = |z₃|

So we have $z_2 = z_1 \omega, z_3 = z_1 \omega^2$ where $\omega = e^{\frac{2\pi i}{3} }$ is the primitive cube root of infinity.

So $z_1+z_2+z_3 = z_1(1+\omega + \omega^2 ) = 0$

Last Activity: 9 Years ago

mycroft holmes

Shorter way to go about it is: Invoke the result that if the circumcenter is at the origin then the complex number corresponding to the orthocenter is z₁+z₂+z₃.

Since the triangle is equilateral, orthocenter and circumcenter coincide. So we have z₁+z₂+z₃ = 0

Last Activity: 9 Years ago