General 2D rotation matrix for rotating a vector about the origin:

Apply rotation by to a unit vector in the direction:

Counterclockwise rotation by 30°:

Rotation that transforms the direction of {1,1} into the direction of {0,–1}:

3D rotation around the axis:

A 4D rotation matrix, rotating in the plane:

A general 3D rotation matrix, rotating in the plane given by t{1,1,1} + s{1,–2,1}:

Rotate the vector {1,0,0} to the vector {0,0,1}:

Generate the rotation matrix for symbolic vectors, assuming that all quantities are real:

Rotating {0,0,1} gives the normalized {x,y,z} vector:

Transformation applied to a 2D shape:

Transformation applied to a 3D shape:

Rotating 3D shapes:

Produce a basis for all rotations in dimension :

All rotations in 2D:

All rotations in 3D:

All rotations in 4D; in general basis elements are needed for dimension :

A rotation matrix is orthogonal, i.e. the inverse is equal to the transpose:

In the complex case, the rotation matrix is unitary:

A rotation matrix has determinant :

Multiplying by the rotation matrix preserves the norm of a vector:

The inverse of RotationMatrix [θ,{u,v}] is given by RotationMatrix [-θ,{u,v}]:

The inverse of RotationMatrix [θ,{u,v}] is also given by RotationMatrix [θ,{v,u}]:

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

In 2D the inverse of RotationMatrix [θ] is given by RotationMatrix [-θ]:

In 3D the inverse of RotationMatrix [θ,w] is given by RotationMatrix [θ,-w]:

If w is not real the relationship is more complex:

The composition of rotations is a rotation:

The order in which rotations are performed is important:

Rotating around and then is not the same as first rotating around and then :

Rotations of a circular sector: