My text is covering projector matrices while building up to Householder triangularization. The main topic of discussion is orthogonal projector matrices that satisfy 

\begin{align} P &= P^2 \tag{1} \\ P &= P^\* \tag{2} \end{align}

It turns out that we can form a rank one orthogonal projector with any orthonormal vector $q \in \mathbb{C}^m$ 

\begin{align} P\_q = qq^\* \end{align} 

which is easily verified to satisfy (1) and (2). $P\_q$'s rank $m-1$ complement is then found by $P\_{\perp q} = I - P\_q$. So far these facts and definitions all make sense to me. But I'm having a little trouble with the following: my book goes on to say that analogous projector matrices for arbitrary nonzero vectors $a$ can be written 

\begin{align}
P\_a &= \frac{aa^\*}{a^\*a} \\ \\
P\_{\perp a} &= I - P\_a 
\end{align}

It's easy to verify that these formulas satisfy (1) and (2) but what is the motivation for the scaling factor of $a^\*a$? For example 

\begin{align}
P\_av = \left(\frac{a^\*v}{a^\*a}\right)a \implies \| P\_av\|\_2 
 =\frac{\left|a^\*v\right|}{\|a\|\_2^2} \|a\|\_2\
 = \frac{\left|a^\*v\right|}{\|a\|\_2} \tag{\*}
\end{align}

Why the scaling factor? Is it true that $a^\*v = \|a\|\_2\|v\|\_2\cos(a,v)$ for higher dimensional complex vectors? I that case (\*) becomes 

\begin{align}
\|v\|\_2 \left|\cos(a,v)\right|
\end{align} 

In hindsight that seems to make sense. $P\_av$ is just putting $v$ on $a$ and scaling it to be an orthogonal projection. I guess my true question is: is the 2-norm function equivalent to the modulus function in complex spaces? In particular, for $x,y \in \mathbb{C}^m$ prove

$$x^\*y = \|x\|\_2\|y\|\_2 \cos \alpha $$