# Relating rotation angles of a plane with angles formed by direction vectors

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.

• It’s not clear what you mean by “replace the angles” and your second illustration doesn’t help. After aligning $s$ with the $z$-axis (also ambiguous since you haven’t specified whether you want it pointing in the positive or negative direction) to what are you measuring the two angles? – amd Apr 11 '18 at 19:52
• @amd I have edited the question slightly. By "replicate the angle" I mean that I want to recreate the first illustration starting from a state in which the $\hat{s}$ vector is orthogonal to the plane. After aligning the $\hat{s}$ vector with the $\boldsymbol{z}$ axis I rotate and measure the angle $\xi$ w.r.t. $\boldsymbol{z}$ axis, this takes the plane from green outlined state to the blue outlied state. Then from that state, I rotate and measure the angle $\eta$ w.r.t $\boldsymbol{y'}$ axis, this takes the plane from blue outlined state to the black outlined state. – dykes Apr 11 '18 at 21:50
• Use spherical coordinates. – amd Apr 11 '18 at 22:03
• Yes, but I need these angles ($\xi$ and $\eta$) related to the angles $\zeta$ and $\chi$ in some equation form. I am not sure how I would do that in spherical coordinates. – dykes Apr 11 '18 at 22:09
• I am sorry, I have re-edited the image. I rotate and measure the angle $\xi$ w.r.t. $\boldsymbol{z}$ axis, this takes the plane from green outlined state to the blue outlied state. Then from that state, I rotate and measure the angle $\eta$ w.r.t $\boldsymbol{x′}$ axis, this takes the plane from blue outlined state to the black outlined state. Now maybe I can use $\xi=\zeta$ and $\eta=\chi$. – dykes Apr 11 '18 at 22:18