0
$\begingroup$

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?
$\endgroup$
1
$\begingroup$

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 $$\cos(\Omega) = \hat a\cdot\hat b.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.