First slide

Geometry of complex numbers

Question

If the complex numbers $z_{1}, z_{2}$ and $z_{3}$ represent the vertices of an equilateral triangle such that $|z_{1}| = |z_{2}| = |z_{3}|$ , then

Moderate

A

$z_{1} + z_{2} + z_{3} = 0$
B

$z_{1} + z_{2} - z_{3} = 0$
C

$z_{1} - z_{2} + z_{3} = 0$
D

$z_{1} + z_{2} + z_{3} \neq 0$

Solution

Let $|z_{1}| = |z_{2}| = |z_{3}| = k$ (says),
$\Rightarrow z_{1}, z_{2}, z_{3}$ lie on a circle with centre at the origin and radius $k$ . As $z_{1}, z_{2}, z_{3}$ are vertices of an equilateral triangle, the circumcentre and the centroid of the triangle coincide. Therefore $\frac{1}{3} (z_{1} + z_{2} + z_{3}) = 0 \Rightarrow z_{1} + z_{2} + z_{3} = 0$

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