Formula for 3D Rotation

r

α

1.05

OD = OB + BC + CD

OB = (r.ω) ω

BC = (r-(r.ω) ω) cos(α)

CD = ω×r sin(α)

This Demonstration explains a formula for the rotation of the vector

r

around the axis given by the unit vector

ω

through the angle

α

.

The formula is

(r.ω)ω+(r-(r.ω)ω)cos(α)+(ω×r)sin(α)

, using the dot and cross product of vectors.

The resultant vector is

OD=OB+BC+CD

.

The vector

OB=(r.ω)ω

is the orthogonal projection of the vector

r=OA

onto the vector

ω

.

The vector

BD

is the result of the rotation of the vector

BA=r-(r.ω)ω

around

ω

through the angle

α

.

The vector

BC=(r-(r.ω)ω)cos(α)

is the orthogonal projection of

BD

onto

BA

.

CD

is the orthogonal projection of

BD

onto

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)