wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Let the complex numbers z1,z2,z3 represent the vertices of an equilateral triangle. Then prove that 1z1z2+1z2z3+1z3z1=0.

Open in App
Solution

Since z1,z2,z3 represent the vertices of an equilateral triangle .
Then |z1z2|=|z2z3|=|z3z1|.......(1).
Now 1z1z2+1z2z3+1z3z1

=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z1z2)|z1z2|2+¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z2z3)|z2z3|2+¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z3z1)|z3z1|2

=¯¯¯¯¯¯¯¯¯(z1)¯¯¯¯¯¯¯¯¯(z2)|z1z2|2+¯¯¯¯¯¯¯¯¯(z2)¯¯¯¯¯¯¯¯¯(z3)|z1z2|2+¯¯¯¯¯¯¯¯¯(z3)¯¯¯¯¯¯¯¯¯(z1)|z1z2|2 [Using (1) and for complex number a and b,¯¯¯¯¯¯¯¯¯¯¯¯a+b=¯¯¯a+¯¯b]

=0.

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon