Composition of Rotations

Rotations can be composed of multiple rotations. This page describes the relationship between composition of rotations and order of operation using examples. Suppose we have frame A and we first rotate it by rotation matrix R₁ to make frame B, and then by another rotation matrix R₂ to make frame C:

The rotation from frame A to C (that is, the composition of R₁ and R₂) can be calculated by multiplying the two matrices. The order of multiplication changes depending on how R₂ applies the rotation on frame B:

• Post-multiply (R₁R₂)
  R₂ is a rotation of frame B with respect to frame B, the intermediate frame. This is called intrinsic rotation.
• Pre-multiply (R₂R₁)
  R₂ is a rotation of frame B with respect to frame A, the original frame. This is called extrinsic rotation.

The order of multiplication and its interpretation applies equally with quaternions.

Example

• R₁R₂
• R₂R₁

Example

• q₁q₂
• q₂q₁