I am trying to get a solid and intuitive handle on polar and spherical coordinates, and I'm getting stuck with what I think should be simple geometry:
To find the unit vector in Cartesian coordinates $\hat{\rho}$. It's very intuitive, I draw a y-axis and x-axis, and I see that $\hat{\rho} = \cos(\phi)\hat{x} + \sin(\phi)\hat{y}$
But I'm sitting here trying to reach the known representation of unit vector $\hat{\phi} = -\sin(\phi)\hat{x} + \cos(\phi)\hat{y}$ and I'm not getting it with simple geometry.
I often reach $-\cos(\phi)\hat{x} + \sin(\phi)\hat{y}$ which is just the negative of the $x$ unit vector.
I know how to verify this using the known relationship between $\hat{\phi}$ unit vector and $\hat{\rho}$ unit vector being perpendicular, and therefore their scalar product must equal $0$. And yet why is it not as simple and intuitive finding the $\hat{\phi}$ unit vector as it was $\hat{\rho}$? Could be I've just reached a mental block here.
I don't want to reach this by "guessing" coefficients for $\hat{x}$ and $\hat{y}$ and then verifying with scalar product equaling zero, I want to see this geometrically.
Thanks!