relates the cross product to the exterior product. It can also be used to represent physical quantities, like torque and angular momentum. In vector algebra they are usually represented by vectors, perpendicular to the plane of the force, linear momentum or displacement that they are calculated from. But if a bivector is used instead the plane is the plane of the bivector, so is a more natural way to represent the quantities and the way they act. It also unlike the vector representation generalises into other dimensions. The product of two bivectors has a geometric interpretation. For non-zero bivectors A and B the product can be split into symmetric and antisymmetric parts as follows: Like vectors these have magnitudes and , where θ is the angle between the planes. In three dimensions it is the same as the angle between the normal vectors dual to the planes, and it generalises to some extent in higher dimensions. Bivectors can be added together as areas. Given two non-zero bivectors B and C in three dimensions it is always possible to find a vector that is contained in both, a say, so the bivectors can be written as exterior products involving a: This can be interpreted geometrically as seen in the diagram: the two areas sum to give a third, with the three areas forming faces of a prism with a, b, c and as edges. This corresponds to the two ways of calculating the area using the distributivity of the exterior product: This only works in three dimensions as it is the only dimension where a vector parallel to both bivectors must exist. In higher dimensions bivectors generally are not associated with a single plane, or if they are (simple bivectors) two bivectors may have no vector in common, and so sum to a non-simple bivector. Four dimensions In four dimensions the basis elements for the space ⋀2R4 of bivectors are (e12, e13, e14, e23, e24, e34), so a general bivector is of the form Orthogonality In four dimensions the Hodge dual of a bivector is a bivector, and the space ⋀2R4 is dual to itself. Normal vectors are not unique, instead every plane is orthogonal to all the vectors in its Hodge dual space. This can be used to partition the bivectors into two 'halves', in the following way. We have three pairs of orthogonal bivectors: (e12, e34), (e13, e24) and