# Combining Two 3D Rotations

Combining Two 3D Rotations

The concentric rotations and can be combined either by doing first and then , written as , or first and then , written as . This right-to-left notation comes from the algebra of dot products, and , where is a vector and and are the matrices representing the rotations and .

A

B

A

B

B∘A

B

A

A∘B

M·M·v

B

A

M·M·v

A

B

v

M

A

M

B

A

B

In general, .

A∘B≠B∘A

You can vary the axes of rotation by changing the spherical coordinates of their corresponding vectors.