# I Angles between complex vectors

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1. Mar 31, 2016

### Physgeek64

So I was trying to learn how to find the angle between two complex 4-dimentional vectors. I came across this paper, http://arxiv.org/pdf/math/9904077.pdf which I found to be a little confusing and as a result not overly helpful. I was wondering if anyone could help at all?

2. Mar 31, 2016

### Ssnow

Assuming that you have a $4$ dimensional complex vector space $V_{\mathbb{C}}$ you have essentially two possibility, You can consider the complex space isometric to the real space $\mathbb{R}^{8}$ so you have the relation for the Euclidean angle $\Theta$:

$$\cos{\Theta(A,B)}=\frac{(A,B)}{|A||B|}$$

where $(,)$ is the product in $\mathbb{R}^{8}$ or you can consider your complex space isometric to $\mathbb{C}^{4}$ and have the relation for the Hermitian angle $\Theta_{c}$:

$$\cos{\Theta_{c}(A,B)}=\frac{(A,B)_{\mathbb{C}}}{|A||B|}$$

where now $(,)_{\mathbb{C}}$ is the hermitian product on $\mathbb{C}^{4}$. Defining and almost complex structure you can have other kind of angles (Kahler) but depends what you need ... the last part of the article works in order to find relations between these kind of angles ...