The unit quaternion \(q=(x,y,z,w)\) represents the rotation given by the matrix
\[R= \left( \begin{array}{ccc} 1-2(y ^ 2+z ^ 2) & 2 (x y - w z) & 2 (x z + w y) \\ 2 (x y + w z) & 1 - 2 (x ^ 2 + z ^ 2) & 2 (y z - w x) \\ 2 (x z - w
y) & 2 (y z + w x) & 1- 2 (x ^ 2 + y ^ 2) \end{array} \right) \]
Demonstration
\(v'=\mathcal{R}_q(v) = q \, q_v \, q^{\star} = ((w ^ 2-s ^ 2)\, v + 2 (s \cdot v)\,s + 2 w\,(s\times v),0)\;\;\) with \(s=(x,y,z)\)
\(v' = (w^2-x^2-y^2-z^2) v + 2 \left(\begin{array}{lll}x & y & z\end{array}\right) v \left(\begin{array}{lll} x & y & z\end{array}\right)^T + 2 w
\left(\begin{array}{l}x & y & z\end{array}\right) \times v \)
\(v' = \left((w^2-x^2-y^2-z^2)\,\mathrm{I} + 2 \left(\begin{array}{ccc}x^2 & xy & xz \\ xy & y^2 & yz \\ xz & yz & z^2\end{array}\right) + 2
w\;\left(\begin{array}{ccc} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{array}\right)\right)v\)