Complex Numbers Vectors Problem (1 Viewer)

Zen2613 · Feb 13, 2016

Let P,Q,R be 3 points in the complex plane such that P,Q,R (going anticlockwise) is an equilateral triangle.
Let the point P and the point R be representative of the complex numbers z2 and z1 respectively.
Find the complex number represented by the point Q in terms of z1 and z2.

InteGrand · Feb 13, 2016

Zen2613 said:
Let P,Q,R be 3 points in the complex plane such that P,Q,R (going anticlockwise) is an equilateral triangle.
Let the point P and the point R be representative of the complex numbers z2 and z1 respectively.
Find the complex number represented by the point Q in terms of z1 and z2.

$\noindent Let $\omega = \cos \frac{\pi}{3}+i\sin \frac{\pi}{3}$. Let $Q$ be the number $z_3$. Note that $\overrightarrow{RQ}$ is just $\overrightarrow{RP}$ rotated by $\frac{\pi}{3}$ counter-clockwise. Since $\overrightarrow{RP}$ represents the number $z_2 - z_1$ (since $P$ is $z_2$ and $R$ is $z_1$) and $\overrightarrow{RQ}$ represents $z_3 - z_1$, we have $z_3 - z_1 = \omega \left(z_2 - z_1\right)$. Solving for $z_3$ yields $z_3 = z_1 + \omega \left(z_2 - z_1\right)$, which is the answer.$

Complex Numbers Vectors Problem (1 Viewer)

Zen2613

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InteGrand

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