Question

ABCD is a square with the letters in the anticlockwise order.The points A and B represents 2+3i and 6+i,respectively.How to find the complex numbers represented by C and D?

Answer 1

The point `C` is `=8+5i` and the point `D` is `=4+7i`

Explanation:

The point `A` is `z_A=2+3i`

The point `B` is `z_B=6+i`

The point `C` is `z_C`

The point `D` is `z_D`

The point `C` is obtained by the rotation of point `A` by `pi/2` clockwise around the center `z_B`

Therefore,

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

`theta=-pi/2`

`z_C-(6+i)=e^(-ipi/2)(2+3i-(6+i))`

`z_C-6-i=-i(-4+2i)=4i+2`

`z_C=4i+2+i+6=8+5i`

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

Therefore,

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

`theta=pi/2`

`z_D-(2+3i)=e^(ipi/2)(6+i-(2+3i))`

`z_D-2-3i=i(4-2i)=4i+2`

`z_D=4i+2+2+3i=4+7i`