rotations, they do not rotate about their axis anymore. The most external matrix rotates the other two, leaving the second rotation matrix over the line of nodes, and the third one in a frame comoving with the body. There are possible combinations of three basic rotations but only of them can be used for representing arbitrary 3D rotations as Euler angles. These 12 combinations avoid consecutive rotations around the same axis (such as XXY) which would reduce the degrees of freedom that can be represented. Therefore, Euler angles are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. Other conventions (e.g., rotation matrix or quaternions) are used to avoid this problem. In aviation orientation of the aircraft is usually expressed as intrinsic Tait-Bryan angles following the convention, which are called heading, elevation, and bank (or synonymously, yaw, pitch, and roll). Quaternions Quaternions, which form a four-dimensional vector space, have proven very useful in representing rotations due to several advantages over the other representations mentioned in this article. A quaternion representation of rotation is written as a versor (normalized quaternion): The above definition stores the quaternion as an array following the convention used in (Wertz 1980) and (Markley 2003). An alternative definition, used for example in (Coutsias 1999) and (Schmidt 2001), defines the "scalar" term as the first quaternion element, with the other elements shifted down one position. In terms of the Euler axis and angle this versor's components are expressed as follows: Inspection shows that the quaternion parametrization obeys the following constraint: The last term (in our definition) is often called the scalar term, which has its origin in quaternions when understood as the mathematical extension of the complex numbers, written as and where are the hypercomplex numbers satisfying Quaternion multiplication, which is used to specify a composite rotation, is performed in the same manner as multiplication of complex numbers, except that the order of the elements must be taken into account, since multiplication is not commutative. In matrix notation we can write quaternion multiplication as Combining two consecutive quaternion rotations is therefore just as simple as using the rotation matrix. Just as two successive rotation matrices, followed by , are combined as we can represent this with quaternion parameters in a similarly concise way: Quaternions are a very popular parametrization due to the following properties: More compact