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You can find if two rea-valued vectors $x$ and $y$ in $\Re^n$ are colinear by calculating.

$\cos(\theta) = \frac{\langle x, y \rangle}{\lvert x\rvert\,\lvert y\rvert} $

For complex vectors ($z \in C^n$), this definition doesn't work. In this post, they comment on hyperbolic geometry and how to find the angle between two vectors. Why do we use cosh to define the angle between two vectors in hyperbolic geometry?

However, what would this hyperbolic angle mean in terms of colinearity?

MvG
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user3284182
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  • For what it's worth, that expression still detects whether two complex vectors are colinear: $x$ and $y$ in $\mathbb{C}^n$ are colinear if and only if $\langle x, y \rangle / |x| |y|$ is a complex number of absolute value $1$. This is the equality case in the Cauchy-Schwarz inequality, and essentially the same as for real vectors. – Gunnar Þór Magnússon Jan 17 '24 at 10:10

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