Let w _ 1 be the point obtained by the rotation of z _ 1 = 5 + 4i...

Given,

Two complex numbers ${w}_{1} \text{ and } {w}_{2}$

${w}_{1} = 3 + 5i$ 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 $\pi - \tan^{- 1}\dfrac{8}{9}$ {as complex number is in second quadrant so principal argument is given by $\pi - \tan^{- 1}\left|\dfrac{y}{x}\right|$}