If `z_1,z_2 in C`, show that `(z_1+ z_2)^2= z_1^2+2 z_1 z_2+ z_2^2`.

For any two complex numbers z_1 and z_2 , prove that Re (z_1 z_2) = Re z_1 Re z_2 - Imz_1 Imz_2

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : R_e (z_1+ z_2)=R_e (z_1)+R_e (z_2) .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : I_m (z_1+ z_2)=I_m (z_1)+I_m(z_2) .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : R_e (z_1 z_2)=R_e (z_1)R_e (z_2)- I_m (z_1)I_m(z_2) .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : I_m (z_1 z_2)=R_e (z_1)I_m (z_2)- I_m (z_1)R_e(z_2) .

Complex numbers z_(1),z_(2)andz_(3) are the vertices A,B,C respectivelt of an isosceles right angled triangle with right angle at C. show that (z_(1)-z_(2))^(2)=2(z_1-z_(3))(z_(3)-z_(2)).

If z_1 =-3i, z_2= 3 +4i and z_3=2- 3i , verify that z_1(z_2+z_3)=z_1z_2+z_1z_3 .

If a x_1^2+by_1^2+c z_1^2=a x_2^2+b y_2^2+c z_2^2=a x_3 ^2+b y_3 ^2+c z_3^ 2=d a x_2x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1= a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that | x_1 y_1 z_1 x_2 y_2 z_2 x_3 y_3 z_3 |= (d-f){(d+2f)/(a b c)}^(1//2)