Express a vector in a rotated coordinate system

[Niles]

Homework Statement

Hi

I have a coordinate system (x', y') and a vector v'=(1, 0) here. There is a different coordinate system (x, y), which is rotated about the y-axis relative to (x', y') by an angle Ω. I am trying to express v' in the system (x, y).

At first what I tried to do was to rotate v' by an angle Ω around the y-axis by a rotation matrix, but then it occurred to me that this only rotates the vector, it does not express it in the system (x, y). Can I get a hint to how to achieve this?

Thanks in advance.

 

[HallsofIvy]

Expressing a vector in a coordinate system that has been rotated by angle $\theta$ is the same as rotating the vector, in the original coordinate system by $-\theta$.

For example, if I rotate coordinate system x'y' by 90 degrees, counterclock wise, then the new positive x-axis is the old y'-axis and the new positive y-axis is the old negative x'-axis. (1, 0) becomes (0, -1), exactly the same as rotating the vector itself 90 degrees clockwise.

 

[Niles]

I see, that makes good sense. Thanks for helping me out these past days.