According to Wikipedia, the angle between two complex vectors $u$ and $v$ in the Vector Space $\mathbb C^n$ over $\mathbb C$ is given by $$\theta=\cos^{-1}\left(\frac{\operatorname{Re}(u\cdot v)}{\|u\|\|v\|}\right)$$
I was just wondering:
- Where is the formula derived from? (For example, the real counterpart is derived from the Law of Cosines). So, I'm just wondering how was the formula obtained, or was it just defined to be that way?
- If the latter case above was true, then why can't the angle be defined this way: $$\theta=\cos^{-1}\left(\frac{|(u\cdot v)|}{\|u\|\|v\|}\right)$$
It is just defined that way.
To elaborate on an aspect of dezdichado's comment, note that if $n=1$ and $u=a+bi$ and $v=c+di$ (with $a,b,c,d$ real) then - noting that Wikipedia's definition of $\cdot$ puts a complex conjugate on the entries of the second argument - $\operatorname{Re}(u \cdot v) = ac+bd$ is the usual real inner product of $(a,b)$ and $(c,d)$ in $\mathbb{R}^2$ while $|u \cdot v| = \sqrt{(ac+bd)^2 + (bc-ad)^2}$ is not.
So, for example, with Wikipedia's definition, the angle between $1$ and $i$ in $\mathbb{C}$ is $\arccos(0) = \pi/2$, as one might expect (and not $\arccos(1) = 0$, as you would get with the proposed formula with $|u \cdot v|$ in it).