article arose. Henry Forder used the term bivector to develop exterior algebra in 1941. In the 1890s Josiah Willard Gibbs and Oliver Heaviside developed vector calculus, which included separate cross product and dot products that were derived from quaternion multiplication. The success of vector calculus, and of the book Vector Analysis by Gibbs and Wilson, had the effect that the insights of Hamilton and Clifford were overlooked for a long time, since much of 20th century mathematics and physics was formulated in vector terms. Gibbs used vectors to fill the role of bivectors in three dimensions, and used bivector in Hamilton's sense, a use that has sometimes been copied. Today the bivector is largely studied as a topic in geometric algebra, a Clifford algebra over real or complex vector spaces with a quadratic form. Its resurgence was led by David Hestenes who, along with others, applied geometric algebra to a range of new applications in physics. Derivation For this article the bivector will be considered only in real geometric algebras. This in practice is not much of a restriction, as all useful applications are drawn from such algebras. Also unless otherwise stated, all examples have a Euclidean metric and so a positive-definite quadratic form. Geometric algebra and the geometric product The bivector arises from the definition of the geometric product over a vector space. For vectors a, b and c, the geometric product on vectors is defined as follows: Associativity Left and right distributivity Contraction Where Q is the quadratic form, |a| is the magnitude of a and ϵa is the metric signature. For a space with Euclidean metric ϵa is 1 so can be omitted, and the contraction condition becomes: The scalar product From associativity, , is a scalar times b. When b is not parallel to and hence not a scalar multiple of a, ab cannot be a scalar. But is a sum of scalars and so a scalar. From the law of cosines on the triangle formed by the vectors its value is , where θ is the angle between the vectors. It is therefore identical to the scalar product between two vectors, and is written the same way, It is symmetric, scalar-valued, and can be used to determine the angle between two vectors: in particular if a and b are orthogonal the product is zero. The exterior product Just as the scalar product can be formulated