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Given the complex numbers z1 = 1 + i and z2 = -1 - i, find a complex number z3 such that z1, z2, z3 are the vertices of an equilateral triangle.

1. z3 = sqrt(3)*(1 - i) or z3 = -sqrt(3)*(1 - i)
2. z3 = sqrt(3)*(1 + i) or z3 = -sqrt(3)*(1 + i)
3. z3 = 1 - i or z3 = -1 + i
4. z3 = sqrt(3) + i or z3 = -sqrt(3) - i

Correct answer: z3 = sqrt(3)*(1 - i) or z3 = -sqrt(3)*(1 - i)

Solution

The segment z1z2 passes through the origin with |z1 - z2| = 2*sqrt(2); the third vertex lies on the perpendicular bisector at distance (sqrt(3)/2)*2*sqrt(2) = sqrt(6), giving z3 = +/- sqrt(3)*(1 - i).

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