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I am looking into performing a rotation of a 3D vector by a quaternion. I understand this entails pre-multiplying the vector $v$ (made into a "pure" quaternion) with $q$ and then post-multiplying the result by the quaternion conjugate $q*$, hence $p=qvq*.$

However, I am having a hard time understanding how the quaternion product is used to expand this equation in this document I found. Could someone please explain to me how to get the matrix (6) from Equation (5) in this link?

Thanks in advance!

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You just do it by hand first: $$ v' = qvq* = (q_0,q_1,q_2,q_3)(0,v_1,v_2,v_3)(q_0,-q_1,-q_2,-q_3) = (w,x,y,z) $$

If you have done everything correctly, you should get $w=0$ and $x,y,z$ as linear functions of $v_1,v_2,v_3$. Then you just write the coefficients in a matrix form, so that $$ \begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}R_{11}&R_{12}&R_{13}\\R_{21}&R_{22}&R_{23}\\R_{31}&R_{32}&R_{33}\end{pmatrix}\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix} $$

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