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
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.
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