The complex number $z$ is given by $$z=\frac{1+\sqrt3\,i}{1-\sqrt3\,i}.$$ Give all arguments as pr

Question

The complex number $z$ is given by $$z=\frac{1+\sqrt3\,i}{1-\sqrt3\,i}.$$ Give all arguments as principal arguments in radians. Express $z$ in the form $x+iy$, and find $|z|$ and $\arg z$.[4] The point $A$ represents $z$ on an Argand diagram. It is rotated clockwise through $\frac{\pi}{2}$ about the origin to the point $B$. Sketch $A$ and $B$, and state the complex number represented by $B$, both in terms of $z$ and in Cartesian form.[3]

Accepted Answer

(a) $z=-\frac12+\frac{\sqrt3}{2}i,\ |z|=1,\ \arg z=\frac{2\pi}{3}$ (b) $z_B=-iz=\frac{\sqrt3}{2}+\frac12i$

A quotient followed by a clockwise rotation

A quotient followed by a clockwise rotation

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