The sqrt of three comes from the example I linked to where they used a vector length of the sqrt of three for convience. The cosine of the angle is length independent so you can change the length if you desire.
If I read your problem correctly, all the vectors must be at the same angle to each other so the shape formed by connecting the ends of the vector will be a regular polyhedron. If you find one angle using a regular polyhedron as a base you should get all of them. The link has a nice geometrical explanation.