A 2D rotation transform by θ radians:

Rotate a vector:

Rotate around the axis:

Rotate a 2D graphic by 30° about the origin:

Rotate around the axis:

Rotation by θ radians about the point {px,py}:

Rotation by θ radians around the axis about the point {px,py,pz}:

A 2D rotation by θ in the plane:

A 3D rotation by θ in the plane:

A 4D rotation by θ in the plane:

A 3D rotation by θ in the plane parametrized by s{1,-1,1}+t{1,1,1}:

This rotates the vector {1,1,1}:

Generate the transform for a symbolic vector, assuming that all quantities are real:

Simplify the result further by assuming that {x,y,z} is a unit vector:

Transformation applied to a 2D shape:

Transformation applied to a 3D shape:

The rotation transformation is an isometric transform, i.e. preserves distances:

The linear part of a rotation transform is given by RotationMatrix:

The matrix for the linear part is OrthogonalMatrixQ for real rotations:

The inverse of RotationTransform[θ,{u,v}] is RotationTransform[-θ,{u,v}]:

The inverse of RotationTransform[θ,{u,v}] is RotationTransform[θ,{v,u}]:

If u or v is not real, the relationship is more complex:

The inverse of RotationTransform[θ] is given by RotationTransform[-θ]:

The inverse of RotationTransform[θ,w] is given by RotationTransform[-θ,w]:

The inverse of RotationTransform[θ,w] is also given by RotationTransform[θ,-w]:

If w is not real, the relationship is more complex:

The composition of rotations is a rotation:

For graphics transformation, use Rotate:

The order in which rotations are applied is important:

Compare the results of the two possible orders; the result is not zero:

Rotate a 3D object about a point p:

Rotate around the axis, in the plane:

Rotate around the axis, in the plane:

Rotate around the axis, in the plane: