If you are going to do 3d you need to understand linear algebra and matrix notation.[1] Affine 4x4 matrices are the basis of space transformations in all 3d applications I've ever seen, (Euler angles just give alternate means to describe that matrix). Even unity uses matrices, although to be able to efficiently do the Euler-Lagrange particle motion equation they prefer to have the decomposed form. A matrix encodes a entire space with 4 vectors. This is conceptually easy in this case (not the only use for matrices), the matrix encodes the directions x y z and offset vector w.
The reason matrix notation is useful is: It becomes possible to manipulate the things like normal math symbols. If you remember from school solving x from:
a * x = b
Divide both sides by a and you get
a/a * x = b /a ->
x = b / a
Now if you have 2 spaces with 3 vectors each you essentially have 2 fully formed spaces at origin. Assuming the vectors span a 3D space (in other words dont point all in one plane, its even better if they are orthogonal to each other in which case you can just use transformation functions directly). That means you have 3 spaces. So your problem is given you know 2 spaces. You need to know the space transform form space A -> space B (its customary to give matrices big letters to denote they are more complex). This is mathematically:
A * X = B
Where * is a matrix multiplications and A, X and B are transformation matrices. So then divide by A, but alas there's no matrix division, fortunately there is inverse and division is multiplication by inverse so that's what we do instead. Also one other note rotations are not commutative so we need to specify on which side we multiply so because A is before X we multiply on the left hand side with inverse of A. So we get:
A^-1 * A * X = A^-1 * B
Where ^-1 denotes matrix inverse. This simplifies to :
I * X = A^-1 * B ->
X = A^-1 * B
X is the rotation space. In unity code this looks like:
X = A.inverse * B
Where X, A and B are Matrix4x4 elements The language you use may have other conventions I'm using the java script reference here. You can covert this matrix to a quaternion and from there to Euler angles an example of this can be found here.
How to form A and B from the vectors? Just put the vector for starting space to A's columns 0-2 and destination spaces correspondingly to B columns[2].
[1] Yes its compulsory, its much simpler than it may seem at first. While you can live quite far without them they aren't any harder to use than saying rotate about x axis fro so and so. Also learn quats.
[2] I should check this, but unity seems to use column matrices so it should be right
PS: By the way if you have noisy data and more then 3 vectors per instance then you can use least squares to average the matrix t a 3 by 3 sub matrix.
