Complex Number (Angle) The complex number $z$ is given by $z=-2+2i$


*

*Find the modulus and argument of $z$

*Write down the modulus and argument of $\dfrac{1}{z}$

*Show on an Argand diagram the points A,B and C representing the complex numbers $z$, $\dfrac{1}{z}$ and $z+\dfrac{1}{z}$ respectively.

*State the value of $\angle ACB$.
For 1. I got the modulus of $z=2\sqrt{2}$ and $\arg(z)=2.356\,\text{rad}$
For 2. I got the modulus of $\dfrac{1}{z}=\dfrac{1}{2\sqrt{2}}$ and $\arg\left(\dfrac{1}{z}\right)=-2.356\,\text{rad}$.
How to solve question 4.?
 A: Here is a geometric approach, which is suitable for the Argand plane.


*

*The argument of $z$ is $\frac 34\pi$, which equals the value you gave as a decimal. That means the angle from the origin to point A representing $z$ is $45°$ up from the negative $x$-axis.





*The argument of $\frac 1z$ is $-\frac 34\pi$, which equals the value you gave as a decimal. That means the angle from the origin to point B representing $\frac 1z$ is $45°$ down from the negative $x$-axis. Note that angle between the vectors from the origin to $A$ and $B$ is $45°+45°=90°$. Those vectors are perpendicular.

*By vector addition in the plane, the vector from $A$ ($z$) to $C$ ($z+\frac 1z$) is parallel to the vector from the origin to $B$ ($\frac 1z$) and has the same length. Hence, the polygon $OACB$ is a rectangle.

*The rectangle means that $\angle ACB=90°=\frac{\pi}2$.

A numeric approach could be done by using the dot product on the relevant vectors. But the geometric approach above should be sufficient.
