I was reading about: https://www.switchcraft.org/learning...ol-for-dummies

I don't understand why do we write at Clarke transform to convert from 3-phase system to 2-phase orthogonal:

$\displaystyle I_{\beta} =\frac{\sqrt{3}}{2}I_b - \frac{\sqrt{3}}{2}I_c $

If you see the picture between $a$ and $b$ is an angle of 120°. Between $\alpha$ and $\beta$ is 90°. So between $\beta$ and $b$ is 30°. If we use SOH-CAH-TOA, and get that the $I_{\beta}$ is:

$\displaystyle I_{\beta} = cos(30°)b$

Why do we need then $c$? Maybe because the net vector is rotating? Can someone derive or explain?

Thanks.