I have troubles with this example:
Let $z_1=2i$, $z_2=1+i$, and $z_3=2+i$ be complex numbers. Represent graphically $$ \bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1 .$$
Thanks for any help.
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$\begingroup$I assume you know how to draw $z_1,z_2,z_3$ individually as a vector? One way would be to just make your hands dirty and actually calculate. You get a complex number which easily can be drawn as a vector.
A better approach would be to think what actually happens graphically when you add, multiply and take the complex conjugate of complex numbers. Divide it into a few steps.
Let $z=x+i\cdot y$, then $\overline{z}=x-i\cdot y$. Thus graphically the complex conjugate is found by just reflecting $z$ across the real axis.
Let $z=x+i\cdot y,w=u+i\cdot v$, then $z+w=(x+u)+i\cdot(y+v)$. Thus graphically the sum of two complex numbers is the sum of the vectors. This becomes very clear when identifying $z=x+i\cdot y$ with the vector $(x,y)$ and $w=u+i\cdot v$ with the vector $(u,v)$.
Let $z,w\in\mathbb C$, using polar coordinates we can write $$z=r_1\cdot e^{i\cdot\varphi},w=r_2\cdot e^{i\cdot\psi}.$$ Thus we get $$z\cdot w = r_1\cdot r_2\cdot e^{i\cdot(\varphi+\psi)}.$$ Thus graphically we can multiply complex numbers by multiplying the lengths and adding the arguments/angles $\varphi$ and $\psi$. One example picture for the multiplication:
By drawing $z_1,z_2,z_3$ individually in the complex plane you can now use this to easily draw $\overline{z_1},\overline{z_2},\overline{z_3}$, then $\overline{z_1}z_2$, $\overline{z_2}z_3$ and $\overline{z_3}z_1$ and then finally $\overline{z_1}z_2+\overline{z_2}z_3+\overline{z_3}z_1$.
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