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 ?
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