If z1, z2, z3 are three complex numbers, then z1 textIm( barz2 z3)+z2 textIm( barz3 z1)+z3 textIm( barz1 z2) is equal to

Question

If z1, z2, z3 are three complex numbers, then z1 textIm( barz2 z3)+z2 textIm( barz3 z1)+z3 textIm( barz1 z2) is equal to (A) 1 (B) -1 (C) 0 (D) None of these

Tardigrade Admin · Accepted Answer

Let z1=x1+i y1 z2=x2+i y2 z3=x3+i y3 barz2 z3=(x2-i y2)(x3+i y3) =(x2 x3+y2 y3)+i(x2 y3-x3 y2) ⇒ textIm( barz2 z3)=x2 y3-x3 y2 ∴ z1 textIm( barz2 z3)=(x1+i y1)(x2 y3-x3 y2) Similarly z2 textIm( barz3 z1)=(x2+i y2)(x3 y1-x1 y3) z3 textIm( barz1 z2)=(x3+i y3)(x1 y2-x2 y1) Adding we get the result = 0

Q. If z1​,z2​,z3​ are three complex numbers, then z1​Im(zˉ2​z3​)+z2​Im(zˉ3​z1​)+z3​Im(zˉ1​z2​) is equal to

1

-1

0

None of these

Q. If $z_{1}, z_{2}, z_{3}$ are three complex numbers, then $z_{1} Im (\overset{z}{ˉ}_{2} z_{3}) + z_{2} Im (\overset{z}{ˉ}_{3} z_{1}) + z_{3} Im (\overset{z}{ˉ}_{1} z_{2})$ is equal to