This must be very elementary, but I am really stuck after much reading.
Given a vector v and a basis A, to find v‘s coordinates in a new basis A’ which is a rotation of A by angle θ, I understand that you must proceed as follows:
a) Find the coordinates in the original basis A of the unit vectors of the new basis (say e1‘ and e2‘), which happen to be as shown in this picture:
change of basis
b) To get coordinate v1‘ in the new basis A’, make the dot product between v (coordinates in A) and e1‘ (also as per coordinates in A); whereas for v2‘, you would need dot product with again v but e2‘ in this case.
My problem is with a), second part. I see that a projection of e1‘ over X axis gives out cosθ as x coordinate and sinθ as y coordinate. But if we repeated the same projection with e2 over X axis, we would get -cosθ and sinθ, which is not the right answer.
I tend to think that my initial mistake, in the second problematic case, was projecting over X. After all, here the rotation starts from e2, not from e1, even if it is the same number of degrees… So I hypothesized that I should project e2‘ over e2 and then somehow convert the result into the language of the initial angle for the sake of homogeneity. Thus for the first coordinate I got a cos, which would be equivalent of a sine in terms of the first angle, but why then the negative sign, so I am lost…