Let's assume we talk about the 3D case.
A rotation matrix is an ortho-normal 3x3 matrix. R.' * R is the unity matrix, or in other words: rotating in one direction at first and then backwards in the opposite direction does not change the data.
A general rotation in 3D is specified by a unit vector (called u here) to rotate around and the angle of rotation (called alpha). Then we get this general rotation matrix (see FEX: RotationMatrix):
s = sin(alpha);
c = cos(alpha);
% 3D rotation matrix:
x = u(1);
y = u(2);
z = u(3);
mc = 1 - c;
R = [c + x * x * mc, x * y * mc - z * s, x * z * mc + y * s; ...
x * y * mc + z * s, c + y * y * mc, y * z * mc - x * s; ...
x * z * mc - y * s, y * z * mc + x * s, c + z * z .* mc];
There are different conventions depending on the point of view: Do you rotate the coordinate system or the object. So maybe R.' is matching in your case.
This is called "direction cosine matrix" also, because it contains the projections of a vector into the coordinate system. You can interpret this matrix as rotated coordinate system also and its 3 vectors (rows or columns) build an orthonormal tripod.
While this is the general matrix, which performs the operation of rotating around a vector, there are a number of different methods, to split this matrix into 3 different rotations around axes fixed to the coordinate system or to the body. The Euler angles or Euler-Cardan (also called Tait–Bryan) angles: The matrix R is split into 3 matrices of the style:
R1 = [ c1, -s1, 0;
s1, c1, 0;
0, 0, 1];
R2 = [ c2, 0, s2;
0, 1, 0;
-s2, 0, c2];
R3 = [ 1, 0, 0;
0, c3, -s3;
0, s3, c3];
Here "s1" means: sin(alpha1) etc.
Now you can decide for a certain convention, e.g. R3*R2*R1, or R3*R2*R3 (yes, the index can be repeated), etc. The angles alpha1, alpha2, alpha3 are called the Euler angles. Their definition requires a decision for a specific order. Different fields of science use typical conventions for order, e.g. analysis of human motion and aerospace engineering.
In your case neither a,b,c nor a1,b1,c1 are Euler angles. See https://en.wikipedia.org/wiki/Euler_angles for details.
if you have the two vectors a and b and want to get the rotation matrix needed to transform one into the other, you can follow these steps:
- The axis of rotation is the normalized cross-product:
u = cross(a, b) / norm(cross(a, b))
- The rotational angle is defined by:
alpha = atan2(norm(cross(a, b)), dot(a, b))
- Use the above formula to create R.
An equivalent solution:
na = a / norm(a);
nb = b / norm(b);
v = cross(na, nb);
skew = [0, -v(3), v(2); v(3), 0, -v(1); -v(2), v(1), 0];
R = eye(3) + skew + skew ^ 2 * (1 - dot(na, nb)) / (norm(v))^2;
