What is the trick in finding the coordinates for the normal of a directional vector?

In 2D space, normals to the direction vector (a, b) are just (-b, a) or (b, -a), or in fact any non-zero multiple of this. Why? The dot product works out to zero. Restricting yourself to unit vectors, there are still two different normals.

In 3D space, normals are harder to nail down since they're much more plentiful. Restricting yourself to unit vectors, there are still infinitely many normals. You can pick one by solving the equation (a, b, c) dot (n1, n2, n3) = 0 where (a, b, c) is your direction vector and n1, n2, and n3 are unknowns, but you'll have to make some arbitrary choices.