Editing Rodrigues' rotation formula (section)

== Derivation ==

[[Image:Rodrigues-formula.svg|300px|right|thumb|Rodrigues' rotation formula rotates {{math|'''v'''}} by an angle {{math|''θ''}} around vector {{math|''k''}} by decomposing it into its components parallel and perpendicular to {{math|''k''}}, and rotating only the perpendicular component.]]
[[File:Orthogonal decomposition unit vector rodrigues rotation formula.svg|350px|thumb|Vector geometry of Rodrigues' rotation formula, as well as the decomposition into parallel and perpendicular components.]]

Let {{math|'''k'''}} be a [[unit vector]] defining a rotation axis, and let {{math|'''v'''}} be any vector to rotate about {{math|'''k'''}} by angle {{math|''θ''}} ([[right hand rule]], anticlockwise in the figure), producing the rotated vector <math>\mathbb{v}_{\text{rot}}</math>.

Using the [[dot product|dot]] and [[cross product]]s, the vector {{math|'''v'''}} can be decomposed into components parallel and perpendicular to the axis {{math|'''k'''}},

:<math> \mathbf{v} = \mathbf{v}_\parallel + \mathbf{v}_\perp \,, </math>

where the component parallel to {{math|'''k'''}} is called the [[vector projection]] of {{math|'''v'''}} on {{math|'''k'''}},

:<math> \mathbf{v}_\parallel = (\mathbf{v} \cdot \mathbf{k}) \mathbf{k} </math>,

and the component perpendicular to {{math|'''k'''}} is called the [[vector rejection]] of {{math|'''v'''}} from {{math|'''k'''}}:

:<math>\mathbf{v}_{\perp} = \mathbf{v} - \mathbf{v}_{\parallel} = \mathbf{v} - (\mathbf{k} \cdot \mathbf{v}) \mathbf{k} = - \mathbf{k}\times(\mathbf{k}\times\mathbf{v})</math>,

where the last equality follows from the [[vector triple product]] formula:  <math display="inline">\mathbf{a}\times (\mathbf{b} \times \mathbf{c})
= (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}</math>.  Finally, the vector <math>\mathbf{k} \times \mathbf{v}_{\perp} = \mathbf{k} \times \mathbf{v}</math> is a copy of <math>\mathbf{v}_{\perp}</math> rotated 90° around <math>\mathbf{k}</math>. Thus the three vectors <math display="block">\mathbf{k}\,,\  
\mathbf{v}_{\perp}\,,\, 
\mathbf{k} \times \mathbf{v}</math> form a [[Right-hand rule|right-handed]] orthogonal basis of <math>\mathbb{R}^3</math>, with the last two vectors of equal length.

Under the rotation, the component <math>\mathbf{v}_{\parallel}</math> parallel to the axis will not change magnitude nor direction:

:<math>\mathbf{v}_{\parallel\mathrm{rot}} = \mathbf{v}_\parallel \,;</math>

while the perpendicular component will retain its magnitude but rotate its direction in the perpendicular plane spanned by <math>\mathbf{v}_{\perp}</math> and <math>\mathbf{k} \times \mathbf{v}</math>, according to

:<math> \mathbf{v}_{\perp\mathrm{rot}} 
= \cos(\theta) \mathbf{v}_\perp + \sin(\theta) \mathbf{k}\times\mathbf{v}_\perp 
= \cos(\theta) \mathbf{v}_\perp + \sin(\theta) \mathbf{k}\times\mathbf{v} \,,</math>

in analogy with the planar [[polar coordinates]] {{math|(''r'', ''θ'')}} in the [[Cartesian coordinates|Cartesian basis]] {{math|'''e'''<sub>''x''</sub>}}, {{math|'''e'''<sub>''y''</sub>}}:

:<math>\mathbf{r} = r\cos(\theta) \mathbf{e}_x + r\sin(\theta) \mathbf{e}_y \,. </math>

Now the full rotated vector is:

:<math>      \mathbf{v}_{\mathrm{rot}}
   = \mathbf{v}_{\parallel\mathrm{rot}} + \mathbf{v}_{\perp\mathrm{rot}} 
 = \mathbf{v}_\parallel + \cos(\theta) \, \mathbf{v}_\perp + \sin(\theta) \mathbf{k}\times\mathbf{v} .
  </math>
Substituting <math>\mathbf{v}_{\perp } = \mathbf{v} - \mathbf{v}_{\|} </math> or <math>\mathbf{v}_{\| } = \mathbf{v} - \mathbf{v}_{\perp}</math> in the last expression gives respectively:
:<math display="block">\mathbf{v}_{\text{rot}}
 = \cos(\theta) \, \mathbf{v}
 + \sin(\theta) \mathbf{k}\times\mathbf{v}
 + (1 - \cos\theta)(\mathbf{k} \cdot \mathbf{v})\mathbf{k}</math> 
:<math display="block">\phantom{\mathbf{v}_{\text{rot}}}
 = \mathbf{v} 
 + \sin(\theta) \mathbf{k}\times\mathbf{v}
 + (1-\cos\theta)\mathbf{k}\times(\mathbf{k}\times\mathbf{v}).</math>