Q23a)
Use your knowledge of vectors to complete this question noting that
![](https://latex.codecogs.com/png.latex?\bg_white \overrightarrow{AB}=B-A)
b)
Since we are focusing on the difference between the arguments between the real axis which we know for an Argand diagram is the x-axis. The angle is one minus the other angle since one line could be
![](https://latex.codecogs.com/png.latex?\bg_white \arg{w}=\frac{y_{1}}{x_{1}})
with the x-axis now we want the part between these two lines so, therefore, it becomes
![](https://latex.codecogs.com/png.latex?\bg_white \arg{z}-\arg{w})
and the argument laws should finish the rest.
Q24)
Recall how vectors work. Slide w or z onto z or w respectively so using the parallelogram all will flow out naturally.
Q25a)
![](https://latex.codecogs.com/png.latex?\bg_white \Re(z))
is just the real section of the complex numbers which is x.
![](https://latex.codecogs.com/png.latex?\bg_white |z|)
is the length of the complex number from the origin which gives us
![](https://latex.codecogs.com/png.latex?\bg_white \sqrt{x^{2}+y^{2}})
if you want them to be equal then make them the same.
b)
Starting with
![](https://latex.codecogs.com/png.latex?\bg_white |z+w|^{2}=(z+w)\overline{(z+w)}=(z+w)(\bar{z}+\bar{w})=z\bar{z}+w\bar{w}+w\bar{z}+z\bar{w}=|z|^{2}+|w|^{2}+w\bar{z}+z\bar{w})
. Notice something here, do you recall what you said in Q25a if you can see that then it is clear that
![](https://latex.codecogs.com/png.latex?\bg_white |z|^{2}+|w|^{2}+w\bar{z}+z\bar{w}\leq{|z|^{2}+|z||w|+|z||w|+|w|^{2}=|z|^{2}+2|z||w|+w^{2}=(|z|+|w|)^{2})
Therefore,
![](https://latex.codecogs.com/png.latex?\bg_white |z+w|\leq{|z|+|w|})
The infamous triangular inequality an mathematical phenomenom.
Anyways amdspotter is asking a lot of Qs from this chapter because he/she is finding difficulties with the exercises and wants some valuable guidance which we as members of Bored of Studies are keen to offer.