Spherical and cylindrical unit vectors - probably really short and trivial?

[jeebs]

hi,
I have this problem I've been stuck with for a long time and i can't figure out what to do.

if spherical coordinates are denoted (r,θ,ϕ) and cylindrical coordinates are denoted (ρ,ϕ,z), how do i express the radial unit vector in cylindrical coordinates, e(ρ), in terms of the spherical unit vectors (e(r), e(θ), e(ϕ)) corresponding to the same point?

what i have tried is rewriting the spherical and cylindrical coordinates in terms of cartesians (x,y,z) and equating them, like this (the list goes cartesian, spherical, cylindrical):

x = rsin(θ)cos(ϕ) = ρcos(ϕ)
y = rsin(θ)sin(ϕ) = ρsin(θ)
z = rcos(θ) = z

which gives the result ρ = rsin(θ),
from which i said e(ρ) = e(r)sin(θ). I am not even sure this is right.

then i hit a dead end. can anybody help me out, this is really getting on my nerves now.
thanks.

 

[Defennder]

You need to project e(p) onto each spherical unit vector. So the radial (I reserve the term planar radial for p-vector) component is (e(p) (dot) e(r))e(r). Same goes for the rest.