Simple Vector Space Rotation

Question

Given Data in the problem & notation convenstions

1. We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation

2. We also have a vector as our initial direction $\vec{v}$

3. $^$ represents unit vector. For example $\hat{\theta_1} $, $\hat{\theta_2} $, $\hat{\theta_2} $, $\hat{v},\hat{\Omega} $, represents unit vectors. And $ | \hspace{.1cm} | $ represents the magnitude of the vector

4. Let us define a sequence of rotations as follows(call it "OPERATION #1")

   1. Rotate $\hat{v}$ around vector $\hat{\theta_1} $ with an angle $|\vec{\theta_1}|$ to get new orientation for $\hat{v}$ called $ V_1$.
   2. Rotate $ V_1 $ around vector $\hat{\theta_2} $ with an angle $|\vec{\theta_2}|$ to get new orientation for $ V_1 $ called $V_2$
   3. Rotate $V_2$ around vector $\hat{\theta_3} $ with an angle $|\vec{\theta_3}|$ to get new orientation for $ V_2 $ called $V_3$

5. Let us define another sequence of rotations as follows(call it "OPERATION #2")

   1. Find the resultant vector called $\vec{\Omega}$ from $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$
   2. Rotate $\hat{v}$ around vector $\hat{\Omega} $ with an angle $|\vec{\Omega}|$ to get new orientation for $\hat{v}$ called $ \psi$.

Question

1. Can we say "OPERATION #1" and "OPERATION #2" does the same final rotation to $\hat{v}$? In other words can we say $ \psi=V_3$ ? If so how do we prove it mathematically?

Answer 1

The answer is affirmative. There are infinitely many solutions. The solution set of $\hat\Omega$ is the unit circle bisecting $\hat v$ and unit vector $\psi$. A unit vector $\hat x$ on unit circle is described by $$(\hat v-\psi)\cdot \hat x = 0.$$ The residue of the orthogonal projection of $\hat v$ and $\psi$ on $\hat x$ are respectively $$\vec a = \hat v-\hat x (\hat x\cdot\hat v),$$ and $$\vec b = \psi-\hat x (\hat x\cdot\psi).$$ The cosine of the angle $\Omega$ of rotation is $$\cos(\Omega) = \hat a\cdot\hat b.$$