Describing vectors in a different coordinate system

[Mr Davis 97]

The problem I am having is a problem in my textbook. It says that if we have xy Cartesian coordinate system, and if we then have a rotated coordinate system x'y', then to get the vector in the x'y' in terms of the xy system, we use the following arguments for the unit vectors:

i' = icos\Phi + jsin\Phi

j' = jcos\Phi - isin\Phi

I don't understand how this was derived, or where it came from. I try to use the right-angle definition for trig ratios, but I keep getting different numbers, and don't see how this relation is true. I would really appreciate it if somebody could provide a simple explanation.

 

[SteamKing]

The problem I am having is a problem in my textbook. It says that if we have xy Cartesian coordinate system, and if we then have a rotated coordinate system x'y', then to get the vector in the x'y' in terms of the xy system, we use the following arguments for the unit vectors:

i' = icos\Phi + jsin\Phi

j' = jcos\Phi - isin\Phi

I don't understand how this was derived, or where it came from. I try to use the right-angle definition for trig ratios, but I keep getting different numbers, and don't see how this relation is true. I would really appreciate it if somebody could provide a simple explanation.

The derivation is mostly a matter of geometry. Perhaps this figure can clear things up:

 

[jedishrfu]

Wikipedia has an article on coordinate rotation

http://en.wikipedia.org/wiki/Coordinate_rotation

midway down in the "two Dimensions" topic they show a matrix that transforms a vector from xy to x'y'

In your case, I think you have the signs mixed up ie

i' = i cos phi - j sin phi

and

j' = i sin phi + j cos phi