Because the original vector is not pointing in the same
direction as the final vector (it is at an angle of 60 degrees to it). If you do a 360 degree rotation you get back to where you started. So any number of successive
full rotations will have eigenvectors, each with eigenvalue
1. If you do a 180 degree rotation, the resulting vector
will be pointing in the opposite direction, and will have
eigenvalue -1.
You are doing a rotation in 3 dimensions. Surely the original example meant two dimensions. In 3 dimensions, the axis of rotation is indeed an eigenvector.
