Question

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

Answer 1

Hint: First, complex numbers are the real and imaginary combined numbers in the form of $ z = x + iy $ , where $ x $ and $ y $ are the real numbers and $ i $ are the imaginary.
Imaginary $ i $ can be also represented into the real values only if, $ {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 $ {z_1},{z_2},{z_3} $

Complete step by step answer:
Since from the given that $ {z_1},{z_2},{z_3} $ is represented as the vertices of an equilateral triangle.
So, there are $ 3 $ 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 = {z_1} $ (the first vertex point in the equilateral triangle with point A)
Similarly, say the vertices as $ OB = {z_2} $ (the second vertex point in the equilateral triangle with the point B) and say the vertices as $ OC = {z_3} $ (the third vertex point in the equilateral triangle with the point C)
Also, from the given question we have $ \left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| $ vertices and apply the vertices according to the given points as $ OA = {z_1},OB = {z_2},OC = {z_3} $
Thus, we get, $ \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, $ \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 $ {z_1} + {z_2} + {z_3} = 0 $ (as O is the circumcenters of $ \vartriangle ABC $ )
Which is the centroid of the given circumcenter with the origin point O.
Thus, we get $ \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 \[[(\dfrac{{{x_1} + {x_2} + {x_3}}}{3}),(\dfrac{{{y_1} + {y_2} + {y_3}}}{3})]\]for real line, where \[{x_1},{x_2},{x_3}\]are the coordinate points in the x-axis and\[{y_1},{y_2},{y_3}\]are the coordinate points in the y-axis.
In complex numbers, the centroid of the triangle can be expressed as $ (\dfrac{{{z_1} + {z_2} + {z_3}}}{3}) $ where $ {z_1},{z_2},{z_3} $ are the coordinates in the complex plane z-axis.