Abstract：Based on the quaternion algorithm, we introduce the concepts of the conjugate and inverse of vectors and the Glassman product, which represents another multiplication of two vectors. The quaternion corresponding to a large arc on a sphere is the inverse of Glassman product of the unit vector, whose center points to the end of the large arc, and the unit vector whose center points to the beginning of the large arc. The quaternion corresponding to spherical angle is the inverse of Glassman product of the unit vector of normal plane of the final arc(toward the vertex) and the unit vector of plane normal of the initial arc (away from the vertex). It is easy to derive spherical trigonometry formulae using the quaternion algorithm, including sinusoidal and cosine formulae, and the first and second five elements formulae.
