Given
- I have a $n$ dimensional $\vec{a}$.
- All elements of $\vec{a}$ are between 0 and a positive number $K$.
- $n$ is about 15 to 20.
Problem
I want to randomly and unbiasedly choose a vector $\vec{b}$ such that:
- Elements of $\vec{b}$ are between 0 and 1.
- $\vec{a} \cdot \vec{b} = 1$
How to do that?
Attempt
I thought about the following:
- Choose $\vec{c}$, a random direction perpendicular to $\vec{b}$.
- Find range of $\theta$ such that $\vec{b} = \frac{\vec{a}}{\lVert \vec{a} \rVert} + \theta\vec{c}$ is within bounds.
- Randoly choose a $\theta$
The problem is that this is not unbiased. Some directions have a larger range of legal $\theta$, and those directions should be chosen more often.
