Let w 1 be the point obtained by the rotation of z 1 = 5 + 4 i about the origin through a right angle in the…

Given,

Two complex numbers $w_1 \& w_2$

$w_1=3+5i$ and $w_2=5+4i$ are both rotated by ${90}^{°}$ with respect to origin anticlock-wise and clockwise respectively,

So by concept of rotation we get,

$w_3=i{w}_1=i\left(3+5i\right)=-5+3i$

And $w_4=-i{w}_2=-i\left(5+4i\right)=4-5i$

Now principal argument of $w_3-w_4=-9+8i$ will be $\mathrm{\pi }-{\tan }^{-1}\frac{8}{9}$ {as complex number is in second quadrant so principal argument is given by $\mathrm{\pi }-{\tan }^{-1}\left|\frac{\mathrm{y}}{\mathrm{x}}\right|$}