I have a plane, $z=0$, as shown in the image below where $\hat{s}$ is a direction unit vector displayed by the red arrow and $\hat{n}$ is the normal unit vector to the plane.
The angle between $\hat{s}$ and $\hat{n}$ is given as:
$$\zeta = \arccos\left(\frac{\boldsymbol{\hat{s}}.\boldsymbol{\hat{n}}}{\lVert \boldsymbol{\hat{s}} \rVert \lVert \boldsymbol{\hat{n}} \rVert} \right) \tag{1}$$
The projection of $\hat{s}$ on the plane is given as:
$$\boldsymbol{s'} = \boldsymbol{\hat{s}}-\frac{\langle \boldsymbol{\hat{s}},\boldsymbol{\hat{n}} \rangle}{ \lVert \boldsymbol{\hat{s}} \rVert \lVert \boldsymbol{\hat{n}} \rVert} \boldsymbol{\hat{n}} \tag{2}$$
and the angle between $\boldsymbol{x}$ and $\boldsymbol{s'}$ is given as:
$$ \chi = \arg(s'_1 + is'_2) \tag{3}$$
I use the arrangement in the first image to measure the angles $\zeta$ and $\chi$ formed by the $\hat{s}$ direction vector and the arrangement in the second image to replicate these angles ($\zeta$ and $\chi$) by placing the $\hat{s}$ vector parallel to the $\boldsymbol{z}$ axis, as shown in the image below, and then rotating the plane.
EDIT:
I have changed the second illustration. As shown in the second image I first align the plane orthogonal to $\hat{s}$ (this is represented by the green outlined plane), next I rotate the plane by an angle $\xi$ about the $\boldsymbol{x}$ axis (this is represented by blue outlined plane) and finally by an angle $\eta$ about the $\boldsymbol{z'}$ axis.
I am finding it difficult to visualize this rotation strategy. I am looking for a consistent way to do this so that I can relate $\xi$ with $\zeta$ and $\eta$ with $\chi$ or in other words, I want to use the angles $\zeta$ and $\chi$ to rotate the plane.