SOLUTION: If 1, Omega, omega^2 are cube roots of unity, prove that 1, Omega, omega^2 are vertices of an equilateral triangle.

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Question 1117926: If 1, Omega, omega^2 are cube roots of unity, prove that 1, Omega, omega^2 are vertices of an equilateral triangle.
Found 2 solutions by greenestamps, math_helper:
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


What kind of "proof" are you looking for?

DeMoivre's Theorem says that the three cube roots of unity have equal moduli and are spaced at equal angles around the origin. That makes the three roots the vertices of an equilateral triangle.

Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
Let the roots be +w%5Bk%5D+ where k=0,1,2.

The three roots can be found by +w%5E3+=+e%5E%282%28pi%29ki%29+ for k=0,1,2

Using this property:


for k=0,1,2

k=0: +w%5B0%5D+=+e%5E0+=+1+
k=1:
k=2:

Now if you plot the real values on the x-axis and imaginary parts along the y-axis, and you connect each point to form a triangle, you will find the distance between each of the three points (vertices) is +sqrt%283%29+ units. Thus completing the proof. There may be a more efficient method, but this method does work.