This paper introduces quaternions and their primary application in quaternion rotation operators. Quaternions, defined as 4-tuples, are algebraic structures that extend complex numbers and are particularly useful in representing rotations in 3D space. The author, Jack B. Kuipers, provides a brief introduction to quaternion properties and algebra, emphasizing their computational advantages over traditional matrix rotation operators. Key topics include: 1. **Hamilton's Quaternions**: The historical development of quaternions, including Hamilton's discovery of the quaternion multiplication rules. 2. **Quaternion Notation**: Definitions and representations of quaternions, including scalar and vector parts. 3. **Quaternion Properties**: Complex conjugate, norm, unit quaternion, and quaternion inverse. 4. **Quaternion Algebra**: Operations and properties of quaternions, including non-commutative multiplication. 5. **Special Quaternion Triple-Product**: The quaternion rotation operator, which geometrically represents a rotation of a vector through an angle about a specified axis. 6. **Rotation Sequences**: Both open and closed rotation sequences, and their applications in aerospace and orbit analysis. 7. **Matrix vs. Quaternion Representation**: Comparing the representation of rotation sequences using matrices and quaternions. 8. **Aerospace and Orbit Sequences**: Specific sequences of rotations used in aerospace and orbit dynamics. 9. **Great Circle Paths**: Using quaternions to describe great circle paths on a sphere. 10. **Spherical Triangles and Equilateral n-Gons**: Applications of quaternions in solving spherical trigonometry problems and determining side-angle relationships in regular spherical polygons. 11. **Unmanned Weather Satellite**: A practical application of quaternions in designing an unmanned weather satellite with a near-polar orbit. The paper concludes by highlighting the computational and operational advantages of quaternions over traditional rotation matrices, making them a valuable tool in various fields such as aerospace and virtual reality.This paper introduces quaternions and their primary application in quaternion rotation operators. Quaternions, defined as 4-tuples, are algebraic structures that extend complex numbers and are particularly useful in representing rotations in 3D space. The author, Jack B. Kuipers, provides a brief introduction to quaternion properties and algebra, emphasizing their computational advantages over traditional matrix rotation operators. Key topics include: 1. **Hamilton's Quaternions**: The historical development of quaternions, including Hamilton's discovery of the quaternion multiplication rules. 2. **Quaternion Notation**: Definitions and representations of quaternions, including scalar and vector parts. 3. **Quaternion Properties**: Complex conjugate, norm, unit quaternion, and quaternion inverse. 4. **Quaternion Algebra**: Operations and properties of quaternions, including non-commutative multiplication. 5. **Special Quaternion Triple-Product**: The quaternion rotation operator, which geometrically represents a rotation of a vector through an angle about a specified axis. 6. **Rotation Sequences**: Both open and closed rotation sequences, and their applications in aerospace and orbit analysis. 7. **Matrix vs. Quaternion Representation**: Comparing the representation of rotation sequences using matrices and quaternions. 8. **Aerospace and Orbit Sequences**: Specific sequences of rotations used in aerospace and orbit dynamics. 9. **Great Circle Paths**: Using quaternions to describe great circle paths on a sphere. 10. **Spherical Triangles and Equilateral n-Gons**: Applications of quaternions in solving spherical trigonometry problems and determining side-angle relationships in regular spherical polygons. 11. **Unmanned Weather Satellite**: A practical application of quaternions in designing an unmanned weather satellite with a near-polar orbit. The paper concludes by highlighting the computational and operational advantages of quaternions over traditional rotation matrices, making them a valuable tool in various fields such as aerospace and virtual reality.