rotation to quaternion matrix handeness

Question

I've understand that quaternions do not have handness but rotation matricies derived from unit quaternions does.

The following formula is given by wikipedia for quaternion to rotation matrix conversion :

Given the unit quaternion $q = w + xi + yj + zk$ , the equivalent left-handed (Post-Multiplied) 3×3 rotation matrix is $$ Q = \begin{bmatrix} 1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\ 2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\ 2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2 \end{bmatrix} . $$

As mentioned, this formula is relative to a left-handed coordinate frame. What's the right-handed counterpart ?

Best : ) ,

Answer 1

Hint:

Given the unit quaternion $q = w + x \vec i + y\vec j + z \vec k=w+ \vec v$, the matrix $Q$ results from the representation of a rotation of a vector $\vec p=p_x \vec i+p_y \vec j+ p_z \vec k$ as: $$ Q(\vec p)= q \vec p q^{-1} $$ The resulting rotation is a rotation of angle $\theta$ around an axis oriented by a versor $\vec u$ such that:

$$ \cos \frac{\theta}{2}=\frac{w}{|q|} \qquad \sin \frac{\theta}{2}=\frac{|\vec v|}{|q|} $$ and $$ \vec u=\frac{\vec v}{|\vec v|} $$

This is a counter-clockwise rotation $R_{\vec u, \theta}$ around the axis $\vec u$.

You can find the clockwise rotation around the same axis changing the angle to $-\theta$, or inverting the orientation of the versor $\vec u$ (note that if you perform all the two transformation the rotation remain the same).

What does this means for the quaternion $q$ ?