Let $\vec{a}$ be a random unitary vector. If $\vec{\lambda}$ is a uniformly distributed vector on $\mathbb{S}_2$ (the unitary sphere?), could we say that the result $|\vec{a}.\vec{\lambda}|$ is uniformly distributed on $[0,1]$ ?
Edit (following jkn's comment) :
And what if we do it with $\vec{\lambda}$ uniformly distributed on the unit ball instead of the unit sphere ?
PS : here is a link to the article where I found this. They assume this at III. B., page 5.
No you can't. Keep in mind the definition of the dot product.
$\vec{a} \cdot \vec{b} = (a)(b)\cos(\theta)$
Clearly your example will not be uniformly distributed