Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online course: Linear Algebra - Analytic Geometry - Orthogonal Groups

Rotation Matrix

[previous page] [next page] [table of contents][page overview]
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)
It's only the check, that $ Qu=u$ and a vector $ v$ which is orthogonal to $ u$, become rotated by the angle $ \vartheta$ about the axis $ u$. The first proposition is trivial. The image of $ v$ is

$\displaystyle Qv = \cos\vartheta v + \sin\vartheta u \times v \,. $

This is a rotation by $ \vartheta$ in the plane spanned by $ v$ and $ u\times v$.
(Authors: Höllig/Reble/Höfert)
  automatically generated 4/21/2005