Let us consider a random variable $Z$ with a probability density function $f$ with respect to the Haar measure on SO(3). Next, we consider two fixed normal vectors $u,v$ in $\mathbb{R}^3$. Define a real random variable

$$ Y = \langle Zu, v \rangle = \| Zu \| \| v \| \cos \theta = \cos \theta , $$

where $\theta$ is the angle between $Zu$ and $v$. Then $Y$ lines almost surely in $[-1,1]$. I would like to compute

$$ \mathbb{P} ( Y \in [a,b] ). $$

In other words, I would like to know what is the probability that the cosine of an angle between $v$ and $u$ rotated by $Z$ lies in some interval. However, I do not know how to transfer integration from SO(3) to $\mathbb{R}$.

Let us consider a random variable $Z$ with a probability density function $f$ with respect to the Haar measure on $\mathrm{SO}(3)$. Next, we consider two fixed normal vectors $u,v$ in $\mathbb{R}^3$. Define a real random variable $$ Y = \langle Zu, v \rangle = \| Zu \| \| v \| \cos \theta = \cos \theta\,, $$

where $\theta$ is the angle between $Zu$ and $v$. Then $Y$ lines almost surely in $[-1,1]$. I would like to compute

$$ \mathbb{P} ( Y \in [a,b] ). $$

In other words, I would like to know what is the probability that the cosine of an angle between $v$ and $u$ rotated by $Z$ lies in some interval. However, I do not know how to transfer integration from $\mathrm{SO}(3)$ to $\mathbb{R}$.