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

You’re just making the new i and j the hypotenuse (of the right triangle with that angle) and scaling back to unit magnitude.

Best bet is to draw the triangle/circle. In the original coordinate system, i points along the x axis, and is one unit long. j points along the y axis and is one unit long.

Rotating the coordinate system by phi is a fancy way of saying “just draw 2 different 1 unit long vectors, where the i’ is phi degrees above i and j’ is phi degrees to the left of j.

Ignore j’ for the moment. For i’:

1. Draw i’, the vector that is one unit long and makes an angle phi with the x axis.

2. Draw a line straight down from the end of i’ to hit the x-axis.

3. Look at the right triangle, use sohcahtoa

The horizontal part of that right triangle along the x axis is length cos(phi). The vertical part is length sin(phi). But horizontal means i and vertical means j. Hence i cos(phi) + j sin(phi).

Similar with j’.

Following this with just text may be tricky. Seriously, draw the picture.

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).