which is itself a three-dimensional linear space. So if a general bivector is: they can be added like vectors while when multiplied they produce the following which can be split into symmetric scalar and antisymmetric bivector parts as follows The exterior product of two bivectors in three dimensions is zero. A bivector B can be written as the product of its magnitude and a unit bivector, so writing β for |B| and using the Taylor series for the exponential map it can be shown that This is another version of Euler's formula, but with a general bivector in three dimensions. Unlike in two dimensions bivectors are not commutative so properties that depend on commutativity do not apply in three dimensions. For example, in general in three (or more) dimensions. The full geometric algebra in three dimensions, Cl3(R), has basis (1, e1, e2, e3, e23, e31, e12, e123). The element e123 is a trivector and the pseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified with pseudovectors to which they are related, as discussed below. Quaternions Bivectors are not closed under the geometric product, but the even subalgebra is. In three dimensions it consists of all scalar and bivector elements of the geometric algebra, so a general element can be written for example a + A, where a is the scalar part and A is the bivector part. It is written Cl and has basis (1, e23, e31, e12). The product of two general elements of the even subalgebra is The even subalgebra, that is the algebra consisting of scalars and bivectors, is isomorphic to the quaternions, H. This can be seen by comparing the basis to the quaternion basis, or from the above product which is identical to the quaternion product, except for a change of sign which relates to the negative products in the bivector scalar product . Other quaternion properties can be similarly related to or derived from geometric algebra. This suggests that the usual split of a quaternion into scalar and vector parts would be better represented as a split into scalar and bivector parts; if this is done the quaternion product is merely the geometric product. It also relates quaternions in three dimensions to complex numbers in two, as each is isomorphic to the even subalgebra for the dimension, a relationship that generalises to higher dimensions. Rotation vector The rotation vector, from the axis–angle