Applying a rotation \((n,\theta)\) to a vector \(v\)
- \(v = v_{\parallel} + v_{\perp}\)
- \(v' = v' _ {\parallel} + v'_{\perp}\)
- \(\Rightarrow v' = v_{\parallel} + (\cos(\theta) \, v _ {\perp} + \sin(\theta) \, n\times v _ {\perp}) \)
- \(v_{\parallel} = (v\cdot n)\,n\)
- \(v_{\perp} = v-(v\cdot n)\,n\)
- \(v' = (v\cdot n)\,n + \cos(\theta)(v-(v\cdot n)\,n) + \sin(\theta) \, n\times(v-(v\cdot n)\,n)\)
- \(\Rightarrow v' = \cos(\theta)\,v + \sin(\theta)\,n\times v + (1-\cos(\theta))\,(v\cdot n)\,n\)
- Rodrigues' rotation Formula
