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A regular hexagon has its centre at the complex number z = i. If one vertex is located at 2 + i, then the two vertices next to it are at which points?

1. 1 ± 2i
2. 1 ± i√3
3. 2 + i(1 ± √3)
4. 1 + i(1 ± √3)

Correct answer: 1 + i(1 ± √3)

Solution

Vector from centre i to vertex (2+i) is 2. Adjacent vertices: 2*e^(+-i*pi/3)=2(1/2 +- i*sqrt(3)/2)=1 +- i*sqrt(3). Add centre i: 1 + i(1 +- sqrt(3)).

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