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rachneet kaur Grade:
        

if ABC is an equilateral triangle then if A and B are denoted by complex nos z1 and z2 then prove that C is denoted by z3 =z1 e^-i(pi\3)+z2 e^i(pi\3).

6 years ago

Answers : (1)

Priyansh Bajaj AskiitiansExpert-IITD
30 Points
										


Dear Rachneet Kaur,


Solution:- By rotation theorem of complex no.'s, ((z3 - z1) / (z2 - z1)) = ( |z3 - z1| / |z2 - z1| )e^i(pi/3)


[Refer Rotation Theorem from http://www.askiitians.com/iit-jee-algebra/complex-numbers/rotation.aspx ]


Since ABC is an equilateral triangle, |z3 - z1| = |z2 - z1|


So, we get ((z3 - z1) / (z2 - z1)) = e^i(pi/3)


or, (z3 - z1) = e^i(pi/3) (z2 - z1)


or, z3 = z2 e^i(pi/3) + z1 (1-e^i(pi/3))



or, z3 = z2 e^i(pi/3) + z1 (1/2 - i (sqrt(3)/2))



or, z3 = z2 e^i(pi/3) + z1 e^-i(pi/3)



Hence Proved.


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All the best Rachneet Kaur!!!



Regards,


Askiitians Experts


Priyansh Bajaj


6 years ago
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