SOLUTION: Show that ǀz1 + z2ǀ˄2 + ǀz1 - z2ǀ˄2= 2ǀz1ǀ˄2 + 2ǀz2ǀ˄2

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Show that ǀz1 + z2ǀ˄2 + ǀz1 - z2ǀ˄2= 2ǀz1ǀ˄2 + 2ǀz2ǀ˄2      Log On


   

Question 1123289: Show that ǀz1 + z2ǀ˄2 + ǀz1 - z2ǀ˄2= 2ǀz1ǀ˄2 + 2ǀz2ǀ˄2
Found 2 solutions by math_helper, ikleyn:
Answer by math_helper(2461) About Me  (Show Source):
Answer by ikleyn(52788) About Me  (Show Source):
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Let Z1 be the complex conjugate to z1, ans Let Z2 be the complex conjugate to z2.


Then


|z1 + z2|^2 + |z1 - z2|^2 = (z1 + z2)*(Z1 + Z2) + (z1 - z2)*(Z1 - Z2) = 


                          = z1*Z1 + z2*Z1 + z1*Z2 + z2*Z2 + z1*Z1 - z2*Z1 - z1*Z2 + z2*Z2 = 


                          = 2*z1*Z1 + 2*z2*Z2  = 2*|z1|^2 + 2*|z2|^2.


QED.

Simply another proof.