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Question: If the complex numbers \( {z_1},{z_2},{z_3} \) represent the vertices of an equilateral triangle suc...

If the complex numbers z1,z2,z3{z_1},{z_2},{z_3} represent the vertices of an equilateral triangle such that ∣z1∣=∣z2∣=∣z3∣\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| then z1+z2+z3{z_1} + {z_2} + {z_3} equal to
A)0A)0
B)1B)1
C)−1C) - 1
D)D) None of these

Explanation

Solution

First, complex numbers are the real and imaginary combined numbers in the form of z=x+iyz = x + iy , where xx and yy are the real numbers and ii are the imaginary.
Imaginary ii can be also represented into the real values only if, i2=−1{i^2} = - 1
The equilateral triangle is a triangle with all the sides are exactly equal, and the vertices are represented in the question as z1,z2,z3{z_1},{z_2},{z_3}

Complete step by step answer:
Since from the given that z1,z2,z3{z_1},{z_2},{z_3} is represented as the vertices of an equilateral triangle.
So, there are 33 vertices points given and assume the point O as the origin point of the equilateral triangle.
Hence, if O is the origin point then we can say the vertices as OA=z1OA = {z_1} (the first vertex point in the equilateral triangle with point A)
Similarly, say the vertices as OB=z2OB = {z_2} (the second vertex point in the equilateral triangle with the point B) and say the vertices as OC=z3OC = {z_3} (the third vertex point in the equilateral triangle with the point C)
Also, from the given question we have ∣z1∣=∣z2∣=∣z3∣\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| vertices and apply the vertices according to the given points as OA=z1,OB=z2,OC=z3OA = {z_1},OB = {z_2},OC = {z_3}
Thus, we get, ∣z1∣=∣z2∣=∣z3∣⇒∣OA∣=∣OB∣=∣OC∣\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| \Rightarrow \left| {OA} \right| = \left| {OB} \right| = \left| {OC} \right|
Taking out the modulus value we get, ∣z1∣=∣z2∣=∣z3∣⇒OA=OB=OC\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| \Rightarrow OA = OB = OC
Since O is the origin point of the equilateral triangle., and which is also the circumcenter of the given triangle ABC.
Thus, the only possibility of adding the vertices is z1+z2+z3=0{z_1} + {z_2} + {z_3} = 0 (as O is the circumcenters of △ABC\vartriangle ABC )
Which is the centroid of the given circumcenter with the origin point O.
Thus, we get z1+z2+z33=0⇒z1+z2+z3=0\dfrac{{{z_1} + {z_2} + {z_3}}}{3} = 0 \Rightarrow {z_1} + {z_2} + {z_3} = 0

So, the correct answer is “Option A”.

Note: Since the centroid of the triangle is the point of intersection of the three vertices and which can be represented as [(x1+x2+x33),(y1+y2+y33)][(\dfrac{{{x_1} + {x_2} + {x_3}}}{3}),(\dfrac{{{y_1} + {y_2} + {y_3}}}{3})]for real line, where x1,x2,x3{x_1},{x_2},{x_3}are the coordinate points in the x-axis andy1,y2,y3{y_1},{y_2},{y_3}are the coordinate points in the y-axis.
In complex numbers, the centroid of the triangle can be expressed as (z1+z2+z33)(\dfrac{{{z_1} + {z_2} + {z_3}}}{3}) where z1,z2,z3{z_1},{z_2},{z_3} are the coordinates in the complex plane z-axis.