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Given that $z_1=1+2i$ and $z_2=\frac{3}{5}+\frac{4}{5}i$, write $z_1z_2$ and $\frac{z_1}{z_2}$ in the form $p+iq$, where $p$ and $q \in R$. In an Argand diagram, the origin O and the points representing $z_1z_2$, $\frac{z_1}{z_2}$,$z_3$ are the vertices of a rhombus. Find $z_3$ and sketch the rhombus on this Argand diagram. Show that $\left | z_3 \right |=\frac{6\sqrt{5}}{5}$.

My attempt,

I found $z_1z_2=-1+2i$ and $\frac{z_1}{z_2}=\frac{11}{5}+\frac{2}{5}i$.

How to find $z_3$? And can anyone let me know what program can I use to plot this kind of complex number? Thanks a lot

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  • $\begingroup$ Are you sure about $|z_3|$? $\endgroup$ – Demosthene May 24 '15 at 14:35
  • $\begingroup$ Yeap. It's $|z_3|$. This is the question extracted from Edexcel exam board $\endgroup$ – Mathxx May 24 '15 at 14:36
  • $\begingroup$ I mean, are you sure that it's the correct value for $|z_3|$? I find something slightly different. $\endgroup$ – Demosthene May 24 '15 at 14:38
  • $\begingroup$ Oh. I spotted it. I'm going to edit it. $\endgroup$ – Mathxx May 24 '15 at 14:38
  • $\begingroup$ Yeah, got it :) $\endgroup$ – Demosthene May 24 '15 at 14:39
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Your calculations of $z_1z_2$ and $\frac{z_1}{z_2}$ are correct. Now, you have to build a rhombus using these two points and the origin. You can plot them in an Argand diagram and you will see that we're looking for a $z_3$ that is in the upper right quadrant of the complex plane.

For simplicity, let me call $O$ the origin, $A$ the point $z_1z_2$, $B$ the point $\frac{z_1}{z_2}$ and $C$, $z_3$. Using the fact that a rhombus has four equal sides, pairwise parallel, we have $\overrightarrow{OA}=\overrightarrow{BC}$ and $\overrightarrow{OB}=\overrightarrow{AC}$. But since $O$ is the origin, the complex representation of $\overrightarrow{OM}$ is simply $z_M$, for some point $M$. Thus: $$z_3=\overrightarrow{OC}=\overrightarrow{OA}+\overrightarrow{AC}=\overrightarrow{OA}+\overrightarrow{OB}=z_1z_2+\dfrac{z_1}{z_2}$$ And you should find: $$z_3=\dfrac{6}{5}\left(1+2i\right)$$ $$|z_3|=\dfrac{6}{\sqrt{5}}$$

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An Argand diagram is just a graph in the Complex Plane. You have three points, $0$, $-1+2i$, and $\frac{11}{5}+\frac{2}{5}i$. So these correspond to the ordered pairs $(0,0)$, $(-1,2)$, $(\frac{11}{5},\frac{2}{5})$. Plot these in this plane and using a little geometry, find the fourth point.

It may be helpful to notice that if $z=x+iy$, then $z=\left | z \right |e^{i\theta }$ where $\theta $ is $\tan ^{1}\frac{y}{x}$.

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  • $\begingroup$ I've forgotten the geometry skill. Can you give me more ideas of it? $\endgroup$ – Mathxx May 24 '15 at 14:45
  • $\begingroup$ Plot the three points in the plane as ordered pairs. Then, observe that a rhombus has four parallel sides of equal length, so you can find the fourth point by using distances. $\endgroup$ – Matematleta May 24 '15 at 14:55

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