1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# Given: z1+z2+z3=A;z1+z2w+z3w2=B;z1+z2w2+z3w=C where w is cube rott of unityProve: |A|2+|B|2+|C|2=3(|z1|2+|z2|2+|z3|2)

Open in App
Solution

## We|A|2+|B|2+|C|2=A¯¯¯¯A+B¯¯¯¯B+C¯¯¯¯C....(1)A¯¯¯¯A=(z1+z2+z3)(¯¯¯¯¯z1+¯¯¯¯¯z2+¯¯¯¯¯z3)=z1z1+z2z2+z3z3+¯¯¯¯¯z1(z2+z3)+¯¯¯¯¯z2(z3+z1)+¯¯¯¯¯z3(z1+z2)....B¯¯¯¯B=(z1+z2ω+z3ω2)(¯¯¯¯¯z1+¯¯¯¯¯¯¯¯z2ω+¯¯¯¯¯¯¯¯¯¯z3ω2)=(z1+z2ω+z3ω2)(¯¯¯¯¯z1+¯¯¯¯¯z2ω2+¯¯¯¯¯z3ω)[∵¯¯¯ω=ω2;(¯¯¯¯¯¯ω2)=ω]=z1¯¯¯¯¯z1+z2¯¯¯¯¯z2ω3+z3¯¯¯¯¯z3ω3+¯¯¯¯¯z1(z2ω+z3ω2)+¯¯¯¯¯z2(z3ω4+z1ω2)+¯¯¯¯¯z3(z1ω+z2ω2)=|z1|2+|z2|2+|z3|2+¯¯¯¯¯z1(z2ω+z3ω2)+¯¯¯¯¯z2(z3ω4+z1ω2)+¯¯¯¯¯z3(z1ω+z2ω2)....(2)SimilarlyC¯¯¯¯C=|z1|2+|z2|2+|z3|2+¯¯¯¯¯z1(z2ω2+z3ω)+¯¯¯¯¯z2(z3ω2+z1ω)+¯¯¯¯¯z3(z1ω2+z2ω)....(3)Adding (1), (2) and (3), we getA¯¯¯¯A+B¯¯¯¯B+C¯¯¯¯C=3[|z1|2+|z2|2+|z3|2]+¯¯¯¯¯z1[z2(1+ω+ω2)+z3(1+ω2+ω)]+¯¯¯¯¯z2[z2(1+ω+ω2)+z1(1+ω2+ω)]+¯¯¯¯¯z3[z2(1+ω+ω2)+z2(1+ω2+ω)]From (1) and (2) we conclude|A|2+|B|2+|C|2=3[|z1|2+|z2|2+|z3|2]

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Properties
MATHEMATICS
Watch in App
Join BYJU'S Learning Program