In mechanics, for example, the angular velocity can be interpreted either as a pseudovector formula_178 or as an anti-symmetric matrix or skew-symmetric tensor formula_179. In the latter case, the velocity law for a rigid body looks:  where Ω is formally defined from the rotation matrix formula_181 associated to body's frame: formula_182 In three-dimensions holds:  In quantum mechanics the angular momentum formula_184 is often represented as an anti-symmetric matrix or tensor operator. More precisely, it is the result of cross product involving position formula_185 and linear momentum formula_111:  Since both formula_185 and formula_111 can have an arbitrary number formula_190 of components, that kind of cross product can be extended to any dimension, holding the "physical" interpretation of the operation. See for numerical details. In 1773, Joseph-Louis Lagrange introduced the component form of both the dot and cross products in order to study the tetrahedron in three dimensions. In 1843, William Rowan Hamilton introduced the quaternion product, and with it the terms "vector" and "scalar". Given two quaternions and , where u and v are vectors in R3, their quaternion product can be summarized as . James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education. In 1878 William Kingdon Clifford published his Elements of Dynamic which was an advanced text for its time. He defined the product of two vectors to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane. Oliver Heaviside and Josiah Willard Gibbs also felt that quaternion methods were too cumbersome, often requiring the scalar or vector part of a result to be extracted. Thus, about forty years after the quaternion product, the dot product and cross product were introduced — to heated opposition. Pivotal to (eventual) acceptance was the efficiency of the new approach, allowing Heaviside to reduce the equations of electromagnetism from Maxwell's original 20 to the four commonly seen today. Largely independent of this development, and largely unappreciated at the time, Hermann Grassmann created a geometric algebra not tied to dimension two or three, with the exterior product playing a central role. In 1853 Augustin-Louis Cauchy, a contemporary of Grassmann, published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product.