# Vectors in rotated coordinate systems

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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). 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.