the Hodge dual then gives the isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice versa; that is where ∗ indicates the Hodge dual. Note that if the underlying orientation is reversed by inversion through the origin, both the identification of the axial vectors with the usual vectors and the Hodge dual change sign, but the bivectors don't budge. Alternately, using the unit pseudoscalar in Cl3(R), gives This is easier to use as the product is just the geometric product. But it is antisymmetric because (as in two dimensions) the unit pseudoscalar i squares to −1, so a negative is needed in one of the products. This relationship extends to operations like the vector-valued cross product and bivector-valued exterior product, as when written as determinants they are calculated in the same way: so are related by the Hodge dual: Bivectors have a number of advantages over axial vectors. They better disambiguate axial and polar vectors, that is the quantities represented by them, so it is clearer which operations are allowed and what their results are. For example, the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector. They generalise to other dimensions; in particular bivectors can be used to describe quantities like torque and angular momentum in two as well as three dimensions. Also, they closely match geometric intuition in a number of ways, as seen in the next section. Geometric interpretation As suggested by their name and that of the algebra, one of the attractions of bivectors is that they have a natural geometric interpretation. This can be described in any dimension but is best done in three where parallels can be drawn with more familiar objects, before being applied to higher dimensions. In two dimensions the geometric interpretation is trivial, as the space is two-dimensional so has only one plane, and all bivectors are associated with it differing only by a scale factor. All bivectors can be interpreted as planes, or more precisely as directed plane segments. In three dimensions there are three properties of a bivector that can be interpreted geometrically: The arrangement of the plane in space, precisely the attitude of the plane