question in helix

if r is a position, what do you mean by " r is equal to the helix"?

r is the position but the position shouyld be a function of helix right ?

Do you mean a function of the angle θ?

yes Sir

So if by "helix" you mean "the angle θ"
and I interpret your question as:
"i need to start from the position r of the particle where r is equal to the angle θ"
then that does not seem correct, unless you are trying to say "when r and θ are both 0"
and by r you mean something like "the arc-length parameter s"
If you mean "I need to find  position r of the particle as a function of the angle θ"
then that sounds like a useful thing to do, although it may be possible to answer some of the questions in problem 2.8 without need of finding the function.

any idea or any tip on how to start working on this then ?

Finding the position of the bead as a function of the angle θ seems like a reasonable way to start.

can i say that r = angle e1 and theta e2   ( e1 and e2 are unti vectors)

I'm confused.
What is "angle e1 and theta e2"
if e1 is a unit vector, what is "angle e1"
and if theta is an angle, what do you mean by "theta e2"?
Do you mean "the angle between e1 and e2"?
How are you defining the unit vectors e1 and e2?
If r is a position in 3 dimensions, don't you want 3 parameters to describe it?

i was thinking to use two reference systems. One fixed to the ground and one fixed to the helix. So the position of the particle can be described as r = cos angle ( in the e1 direction and theta in the e2 direction)

I thought everything was fixed except for the bead.
If neither the helix nor the ground are moving, then both are fixed to each other.
If by "ground" you mean "the base of the helix", are you trying to say you want the two references to be:
the position of the point on the ground closest to the bead
and, the height of the bead above the ground?

i guess we say the same things using different terms. But yes we agree.
since angle and theta are changing is my initial position function r correct ?

"since angle and theta are changing"
How does "angle" change with "theta"?
If e2 points along the axis of the helix, then the proportionality constant between theta and the projection of r along e2 would depend on R and φ
and you'd still need a third unit vector to describe the position of the bead.