Equilateral triangle on the argand diagram

Question

Let $P=3+2i$ be a point in the plane.

Find points $Q$ and $R$ such that $PQR$ form an equilateral triangle with the center (of the triangle) at the origin.

Does anyone know what to do?

Answer 1

$ \alpha= tan ^{-1} \frac23 $

For $ Q$ and $R, $ $ \theta = (\alpha + 2 \pi/3, \alpha + 4 \pi/3) $

With $P$ as center all rotations of $ PQ_1,Q_1R_1,R_1P$ are admissible.

You can also start with the conjugate and negative vector.

Answer 2

Just multiply by $$e^{\pm\frac{2i\pi}{3}}$$ to get the other vertices of the triangle