Formula for 3D Rotation
Formula for 3D Rotation
This Demonstration explains a formula for the rotation of the vector around the axis given by the unit vector through the angle .
r
ω
α
The formula is , using the dot and cross product of vectors.
(r.ω)ω+(r-(r.ω)ω)cos(α)+(ω×r)sin(α)
The resultant vector is .
OD=OB+BC+CD
The vector is the orthogonal projection of the vector onto the vector .
OB=(r.ω)ω
r=OA
ω
The vector is the result of the rotation of the vector around through the angle .
BD
BA=r-(r.ω)ω
ω
α
The vector is the orthogonal projection of onto .
BC=(r-(r.ω)ω)cos(α)
BD
BA
CD
BD
ω×r=ω×(r-(r.ω)ω)
|CD|=|BD|sin(α)=|BA|sin(α)=|r-(r.ω)ω|sin(α)=|ω×(r-(r.ω)ω)|sin(α)=|ω×r|sin(α)