Rotate normal vector onto axis plane

Question

I have a set of data points in 3D space which apparently all fall onto a specific plane. I use PCA to compute the plane parameters. The 3rd component of PCA gives me the normal

Answer 1

Although there were other interesting responses, this is the solution we figured out while waiting for answers:

function roti = magic_cosini(n)
    b = acos(n(2) / sqrt(n(1)*n(1) + n(2)*n(2)));
    bwinkel = b * 360 / 2 / pi;
    if (n(1) >= 0)
        rotb = [cos(-b) -sin(-b) 0; sin(-b) cos(-b) 0; 0 0 1];
    else
        rotb = [cos(-b) sin(-b) 0; -sin(-b) cos(-b) 0; 0 0 1];
    end
    n2 = n * rotb;
    a = acos(n2(3) / sqrt(n2(2)*n2(2) + n2(3)*n2(3)));
    awinkel = a * 360 / 2 / pi;
    rota = [1 0 0; 0 cos(-a) -sin(-a); 0 sin(-a) cos(-a)];
    roti = rotb * rota;

(It's returning a hopefully correct double rotation matrix)

The flaw we had before and fixed here was to esp. deal with the sign of the X component, which was not covered in the cosine computations. This made us rotate in the wrong direction once (rotating with 180° - angle).

I hope I will also find time to try Nosredna's solution! It's always good to avoid trigonometry.