Lösung 3.2:3

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If we mark the three complex numbers in the plane, we see that the fourth corner will have \displaystyle \text{3}+\text{2}i and \displaystyle \text{3}i as neighbouring corners.


In order to find the fourth corner, we use the fact that in a square opposite sides are parallel and all sides have the same length. This means that the vector from \displaystyle \text{1}+i to \displaystyle \text{3}i is equal to the vector from \displaystyle \text{3}+\text{2}i to the fourth corner.


If we interpret the complex numbers as vectors, this means that the vector from \displaystyle \text{1}+i to \displaystyle \text{3}i is


\displaystyle 3i-\left( 1+i \right)=-1+2i


And we obtain the fourth corner if we add this vector to the corner \displaystyle \text{3}+\text{2}i,


\displaystyle \text{3}+\text{2}i+\left( -1+2i \right)=2+4i