Question

A and C are two points in,an argand diagram representing the complex numbers -5+2i and 1+i respectively. Given that the AC the diagonal of the square ABCD, How to find the complex numbers represented by the points B and D?

Answer 1

The point `B` is `=-3/2+9/2i` and the point `D` is `=-5/2-3/2i`

Explanation:

The points are

`z_A=-5+2i`

`z_C=1+i`

The midpoint of `AC` is `Omega` and is

`z_Omega=(z_A+z_C)/2=(-4+3i)/2=-2+3/2i`

The point `B` is obtained by the rotation of point `C` by `pi/2` anticlockwise around the center `z_Omega`

Therefore,

`z_B-z_Omega=e^(itheta)(z_C-z_Omega)`

`theta=pi/2`

So,

`e^(itheta)=e^(ipi/2)=cos(pi/2)+isin(pi/2)=0+1*i=i`

And `i^2=-1`

Therefore,

`z_B-(-2+3/2i)=e^(ipi/2)((1+i)-(-2+3/2i))`

`z_B-(-2+3/2i)=i(3-i/2)=3i+1/2=1/2+3i`

`z_B=1/2+3i-2+3/2i=-3/2+9/2i`

Similarly,

The point `D` is obtained by the rotation of point `A` by `pi/2`anticlockwise around the center `z_Omega`

`z_D-z_Omega=e^(itheta)(z_A-z_Omega)`

`z_D-(-2+3/2i)=e^(ipi/2)((-5+2i)-(-2+3/2i))`

`z_D-(-2+3/2i)=i((-5+2i)-(-2+3/2i))`

`z_D-(-2+3/2i)=i(-3+1/2i)=-3i-1/2`

`z_D=-3i-1/2-2+3/2i=-5/2-3/2i`