Let $z_1 = x_1 +iy_1 $ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $< z_1 , z_2> = x_1x_2+y_1y_2$

Question

Let $z_1 = x_1 +iy_1$ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $\langle z_1 , z_2 \rangle = x_1x_2+y_1y_2$ For non zero $z_1$ and $z_2$ prove the following

$ \langle z_1, z_2 \rangle = |z_1||z_2|\cos\theta$ (where $0 \le \theta \le \pi$ is the angel between $z_1$ and $z_2$.)

$\langle z_1, z_2 \rangle$ = $Re(\overline{z_1}z_2)$

$\langle z_1, z_2 \rangle$ = $\frac{1}{2}(\overline{z_1}z_2 + z_1\overline{z_2})$

$z_1 ⊥ z_2 \iff \theta = \frac{\pi}{2}$

Answer 1

Hint for the second one:

$$\bar{z_1}z_2=(a_1-b_1i)(a_2+b_2i)=(a_1a_2+b_1b_2)+[a_1(+b_2)+a_2(-b_1)]i$$

Answer 2

I think all you need is to understand the definition. I'm going to write only the definitions in order for you to solve them by yourself.

Let $i=1,2$. We have two representations as the followings: $$z_i=x_i+iy_i=r_i(\cos\theta_i+i\sin\theta_i)$$ where $r_i=|z_i|=\sqrt{{x_i}^2+{y_i}^2}.$

You can prove them only by usig these.