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Mathematics-Online course: Linear Algebra - Analytic Geometry - Orthogonal Groups

Rotation Matrix

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A rotation in $ \mathbb{R}^3$ with normed rotation axis vector $ u$ and rotation angle $ \theta$, which is oriented like a right-handed screw, maps a vector $ x$ onto
$\displaystyle Qx = \cos\theta x + (1-\cos\theta) u u^{\operatorname t}x + \sin\theta u \times x \,.$      

The corresponding rotation matrix is defined by
$\displaystyle Q: q_{ik}$ $\displaystyle =$ $\displaystyle \cos\theta \;\delta_{ik} + (1-\cos\theta)\;u_iu_k + \sin\theta \sum\limits_j \varepsilon_{ijk}u_j \,,$  

with Kornecker symbol $ \delta_{ik}$ and $ \varepsilon$-tensor $ \varepsilon_{ijk}$.

(Authors: Höllig/Reble/Höfert)

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  automatically generated 4/21/2005