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 **e _{1}‘** and

**e**), which happen to be as shown in this picture:

_{2}‘change of basis

b) To get coordinate v_{1}‘ in the new basis A’, make the dot product between **v** (coordinates in A) and **e _{1}‘** (also as per coordinates in A); whereas for v

_{2}‘, you would need dot product with again

**v**but

**e**in this case.

_{2}‘My problem is with a), second part. I see that a projection of **e _{1}‘** over X axis gives out cosθ as x coordinate and sinθ as y coordinate. But if we repeated the same projection with

**e**over X axis, we would get -cosθ and sinθ, which is not the right answer.

_{2}I tend to think that my initial mistake, in the second problematic case, was projecting over X. After all, here the rotation starts from **e _{2}**, not from

**e**, even if it is the same number of degrees… So I hypothesized that I should project

_{1}**e**

_{2}‘**over e**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…

_{2}