Guies: Izwi = 1 and erg (2), - Ong (W) = IIs show that Zw=-j

1.	Guies: Izwi = 1 and erg (2), - Ong (W) = IIs show that Zw=-j
Answer» \|zw\| = 1 \|z\| = 1/\|w\| arg(z) - arg(w) = pie/2 arg( z/w) = pie/2 According to rotation theorem z and w are perpendicular So e^i pie/2 will rotate it by 90 z/\|z\| = e^ i pie/2 . w/\|w\| e^i pie/2 = i z/\|z\| = i w \|z\| z = i w \|z\|^2 As z z' = \|z\|^2 ( z' is conjugate) z= \|z\|^2/z' So \|z\|^2/z' = iw \|z\|^2 z' iw = 1 z' w = 1/i z' w = i/ i^2 = - i Proved✌✌✌✌✌✌

Answer»

|zw| = 1

|z| = 1/|w|

arg(z) - arg(w) = pie/2

arg( z/w) = pie/2

According to rotation theorem

z and w are perpendicular

So e^i pie/2 will rotate it by 90

z/|z| = e^ i pie/2 . w/|w|

e^i pie/2 = i

z/|z| = i w |z|

z = i w |z|^2

As z z' = |z|^2 ( z' is conjugate)

z= |z|^2/z'

|z|^2/z' = iw |z|^2

z' iw = 1

z' w = 1/i

z' w = i/ i^2 = - i

Proved✌✌✌✌✌✌

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