as the symmetric part of the geometric product of another quantity, the exterior product (sometimes known as the "wedge" or "progressive" product) can be formulated as its antisymmetric part: It is antisymmetric in a and b and by addition: That is, the geometric product is the sum of the symmetric scalar product and alternating exterior product. To examine the nature of , consider the formula which using the Pythagorean trigonometric identity gives the value of With a negative square it cannot be a scalar or vector quantity, so it is a new sort of object, a bivector. It has magnitude , where θ is the angle between the vectors, and so is zero for parallel vectors. To distinguish them from vectors, bivectors are written here with bold capitals, for example: although other conventions are used, in particular as vectors and bivectors are both elements of the geometric algebra. Properties The space ⋀2Rn The algebra generated by the geometric product is the geometric algebra over the vector space. For a Euclidean vector space it is written or Cln(R), where n is the dimension of the vector space Rn. Cln(R) is both a vector space and an algebra, generated by all the products between vectors in Rn, so it contains all vectors and bivectors. More precisely as a vector space it contains the vectors and bivectors as linear subspaces, though not subalgebras (since the geometric product of two vectors is not generally another vector). The space of all bivectors is written ⋀2Rn. The even subalgebra The subalgebra generated by the bivectors is the even subalgebra of the geometric algebra, written Cl(R). This algebra results from considering all products of scalars and bivectors generated by the geometric product. It has dimension , and contains ⋀2Rn as a linear subspace with dimension (a triangular number). In two and three dimensions the even subalgebra contains only scalars and bivectors, and each is of particular interest. In two dimensions the even subalgebra is isomorphic to the complex numbers, C, while in three it is isomorphic to the quaternions, H. More generally the even subalgebra can be used to generate rotations in any dimension, and can be generated by bivectors in the algebra. Magnitude As noted in the previous section the magnitude of a simple bivector, that is one that is the exterior product of two vectors a and b, is , where θ is the angle