Rigid body orientation using Euler angles confusion

[Tar]

Hello,

Homework Statement

I'm given the following exercise:

"A rod with neglected thickness exists. What is the relation between the α,β angles to Euler angles of orientation?
α is defined as the angle between the rod and its projection on the XY plane.
β is defined as the angle between the X-axis and the rod's projection on the XY plane.
(Note: We are using ZXZ Euler angles convention, having the following order of angles: ϕ,θ,ψ.)"

The answer is: α=θ+0.5π, β=ϕ
I don't understand why.

The Attempt at a Solution

As far as I know, usually the body points in the e3 direction in the body frame. Initially the body frame points towards the original Z direction, after rotation of ϕ around the Z axis, and then rotation of θ around the x′ axis, we will receive the following:

Note that the new z′ axis doesn't point to where it should be (according to the answer) and ϕ doesn't represent the angle between z' axis' projection on the XY plane and X-axis.

What am I missing?

Thank you.

 

[kuruman]

As far as I know, usually the body points in the e3 direction in the body frame.

This is not clear from the statement of the problem, but let's assume that it is true. If e3 is the Z direction, a rotation about that axis by any angle will not change the orientation of the rod, but will change the orientation of the x and y axes to x' and y'. Now to get the rod below the x'y' plane, you will need angle θ greater than π/2. The solution says that's not the case. So either your assumption about the original orientation of the rod is incorrect or the solution is incorrect.

On edit: Welcome to PF.

 

[Tar]

This is not clear from the statement of the problem, but let's assume that it is true. If e3 is the Z direction, a rotation about that axis by any angle will not change the orientation of the rod, but will change the orientation of the x and y axes to x' and y'. Now to get the rod below the x'y' plane, you will need angle θ greater than π/2. The solution says that's not the case. So either your assumption about the original orientation of the rod is incorrect or the solution is incorrect.

On edit: Welcome to PF.

You're right. the lecturer have verified again (for the 3rd time) and there's a mistake that he hadn't noticed on his first verification.

I think this thread should be deleted as it doesn't help anyone,

Thanks for your help.