Right triangle with complex vertices

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SUMMARY

The discussion focuses on determining the geometric representation of the complex number z, where z, z², and z³ serve as the vertices of a right triangle in the complex plane. The participants emphasize the application of the Pythagorean theorem and vector analysis to solve the problem. The key insight is to identify the hypotenuse and ensure that the angle between the other two sides equals π/2. This approach is essential for visualizing and solving the geometric configuration of complex numbers.

PREREQUISITES
  • Understanding of complex numbers and their geometric representation
  • Familiarity with the Pythagorean theorem
  • Basic knowledge of vector addition and subtraction
  • Ability to work with angles in radians
NEXT STEPS
  • Study the properties of complex numbers in the complex plane
  • Learn about vector operations and their applications in geometry
  • Explore the concept of angles and their measurement in radians
  • Investigate the relationship between complex numbers and geometric shapes
USEFUL FOR

Students studying complex analysis, mathematicians interested in geometric interpretations, and educators teaching advanced geometry concepts.

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Homework Statement


Find the geometric image of the complex number z, if z, z^2, z^3 are the vertices of a right triangle.


Homework Equations





The Attempt at a Solution



I tried expanding z^2, z^3, and than using both the pythagoras theorem, and vectors (in separate attempts), but failed to get at a solution.

All help is very appreciated :)
 
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These are the vertices of a right triangle in the complex plain right?
So this is basically a vector addition/subtraction problem.
Each side is the difference between two position vectors.

Figure out which is the hypotenuse and then you need only set the angle between the other two sides to \pi/2.
 

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