If the complex numbers z_1, z_2, z_3 represen | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the complex numbers $z_1, z_2, z_3$ denote the vertices $A, B, C$ of an equilateral triangle $ABC$.Since $|z_1| = |z_2| = |z_3|$,the origin $O

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) Let the complex numbers $z_1, z_2, z_3$ denote the vertices $A, B, C$ of an equilateral triangle $ABC$.Since $|z_1| = |z_2| = |z_3|$,the origin $O$ is equidistant from all vertices $A, B, C$.This implies that the origin $O$ is the circumcenter of the equilateral triangle $ABC$.For an equilateral triangle,the circumcenter,centroid,and orthocenter coincide.If $G$ is the centroid,then $G = \frac{z_1 + z_2 + z_3}{3}$.Since the centroid $G$ coincides with the circumcenter $O$ (which is the origin $0$),we have $\frac{z_1 + z_2 + z_3}{3} = 0$.Therefore,$z_1 + z_2 + z_3 = 0$.

If the complex numbers $z_1, z_2, z_3$ represent the vertices of an equilateral triangle such that $|z_1| = |z_2| = |z_3|$,then $z_1 + z_2 + z_3 = $

Similar Questions

If $|z+i|-|z-1|=|z|-2=0$ for a complex number $z$, then $z=$

If $z=x+iy$ represents a point $P$ in the Argand plane, then the area of the region represented by the inequality $2 < |z-(1+i)| < 3$ is (in $\pi$)

If the points $P_1$ and $P_2$ represent two complex numbers $z_1$ and $z_2$ respectively,then the point $P_3$ represents the number

Let $S = \{z = x + iy : |z - 1 + i| \geq |z|, |z| < 2, |z + i| = |z - 1|\}$. Then the set of all values of $x$,for which $w = 2x + iy \in S$ for some $y \in \mathbb{R}$,is:

The reflection of the complex number $(3 + 2i)$ in the straight line $z = -i \bar{z}$ is-

Vedclass Test Series

Exam Paper Generator

Online Exam Module