reflection of about the plane perpendicular to gives a geometric interpretation to the rotation operation: the rotation preserves the components that are parallel to and changes only those that are perpendicular. The terms are then computed: The result of the rotation is then A simple check on this result is the angle . Such a rotation should map to . Indeed, the rotation reduces to exactly as expected. This rotation formula is valid not only for vectors but for any multivector. In addition, when Euler angles are used, the complexity of the operation is much reduced. Compounded rotations come from multiplying the rotors, so the total rotor from Euler angles is but These rotors come back out of the exponentials like so: where refers to rotation in the original coordinates. Similarly for the rotation, Noting that and commute (rotations in the same plane must commute), and the total rotor becomes Thus, the compounded rotations of Euler angles become a series of equivalent rotations in the original fixed frame. While rotors in geometric algebra work almost identically to quaternions in three dimensions, the power of this formalism is its generality: this method is appropriate and valid in spaces with any number of dimensions. In 3D, rotations have three degrees of freedom, a degree for each linearly independent plane (bivector) the rotation can take place in. It has been known that pairs of quaternions can be used to generate rotations in 4D, yielding six degrees of freedom, and the geometric algebra approach verifies this result: in 4D, there are six linearly independent bivectors that can be used as the generators of rotations. Angle-Angle-Angle Rotations can be modeled as an axis and an angle; as illustrated with a gyroscope which has an axis through the rotor, and the amount of spin around that axis demonstrated by the rotation of the rotor; this rotation can be expressed as where axis is a unit vector specifying the direction of the rotor axis. From the origin, in any direction, is the same rotation axis, with the scale of the angle equivalent to the distance from the origin. From any other point in space, similarly the same direction vector applied relative to the orientation represented by the starting point rather than the origin applies the same change around the same axes that the unit vector specifies. The scaling each point gives a unique coordinate in Angle-Angle-Angle notation.