I’m sure it’s super simple, and I usually don’t have problems with trig, but for some reason I can’t wrap my head around converting between a vector in one coordinate system to one rotated about the origin with respect to that 1st coordinate system.
Specifically, I mean that (given the unit vectors in the original coordinate system to be **i** and **j**, and the second coordinate system to be **i’** and **j’,** and the angle to be Φ) we have:
**i’** = **i** cos(Φ) + **j** sin(Φ)
and
**j’** = -**i** cos(Φ) + **j** sin(Φ)
*not entirely sure if I have that second one correct
In: Physics
I’m not sure exactly what kind of answer you’re seeking. The “shut up and calculate” answer is, if you have a vector <i,j> and you want to rotate it by Φ radians counterclockwise, you plug it into those two equations and the first answer is the new i and the second one is the new j, giving you <i’,j’>.
To get a deeper understanding of why those are the equations you can follow a derivation. Here is [one](http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_05_Coordinate_Transformation_Vectors.pdf) and here’s a [one worded slightly differently](https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec03.pdf).
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