Let $$w_1$$ be the point obtained by the rotation of $$z_1 = 5 + 4i$$ about the origin through a right angle in the anticlockwise direction, and $$w_2$$ be the point obtained by the rotation of $$z_2 = 3 + 5i$$ about the origin through a right angle in the clockwise direction. Then the principal argument of $$w_1 - w_2$$ is equal to

Now, find $$w_1$$.

Rotation of $$z_1 = 5 + 4i$$ through 90° anticlockwise is achieved by multiplying by $$i$$:

$$w_1 = i \cdot z_1 = i(5 + 4i) = 5i + 4i^2 = -4 + 5i$$

Now, find $$w_2$$.

Rotation of $$z_2 = 3 + 5i$$ through 90° clockwise is achieved by multiplying by $$-i$$:

$$w_2 = -i \cdot z_2 = -i(3 + 5i) = -3i - 5i^2 = 5 - 3i$$

Now, find $$w_1 - w_2$$.

$$w_1 - w_2 = (-4 + 5i) - (5 - 3i) = -9 + 8i$$

Now, find the principal argument.

The point $$-9 + 8i$$ lies in the second quadrant (negative real, positive imaginary).

$$\arg(w_1 - w_2) = \pi - \tan^{-1}\left(\frac{8}{9}\right)$$