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

Parallel plane segments with the same orientation and area corresponding to the same bivector a ∧ b.^[2]

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 asplanes, 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 theattitudeof the plane (or alternately therotation,geometric orientationorgradientof the plane), is associated with the ratio of the bivector components. In particular the three basis bivectors, e23, e31and e12, or scalar multiples of them, are associated with the yz-plane, xz-plane and xy-plane respectively.

• Themagnitudeof the bivector is associated with theareaof the plane segment. The area does not have a particular shape so any shape can be used. It can even be represented in other ways, such as by an angular measure. But if the vectors are interpreted as lengths the bivector is usually interpreted as an area with the same units, as follows.

• Like the direction of avectora plane associated with a bivector has a direction, a circulation or a sense of rota-tion in the plane, which takes two values seen asclockwise and counterclockwisewhen viewed from viewpoint not in the plane. This is associated with a change of sign in the bivector, that is if the direction is reversed the bivector is negated. Alternately if two bivectors have the same attitude and magnitude but opposite directions then one is the negative of the other.

• If imagined as a 2d parallelogram, with vector’s origin at 0, then signed area is thedeterminantof the vectors’

Cartesian coordinates ( axb_y− bxa_y).^[20]

b

a b a

b a

The cross product a × b isorthogonalto the bivector a ∧ b.

In three dimensions all bivectors can be generated by the exterior product of two vectors. If the bivector B = a ∧ b then the magnitude of B is

|B| = |a| |b| sin θ,

where θ is the angle between the vectors. This is the area of theparallelogramwith edges a and b, as shown in the diagram. One interpretation is that the area is swept out by b as it moves along a. The exterior product is

antisymmetric, so reversing the order of a and b to make a move along b results in a bivector with the opposite direction that is the negative of the first. The plane of bivector a ∧ b contains both a and b so they are both parallel to the plane.

Bivectors and axial vectors are related byHodge dual. In a real vector space the Hodge dual relates a subspace to its orthogonal complement, so if a bivector is represented by a plane then the axial vector associated with it is simply the plane’ssurface normal. The plane has two normals, one on each side, giving the two possibleorientationsfor the plane and bivector.

Relationship betweenforceF, torque τ,linear momentump, and angular momentum L.

This relates thecross productto theexterior product. It can also be used to represent physical quantities, liketorque andangular momentum. In vector algebra they are usually represented by vectors, perpendicular to the plane of the force,linear momentumor 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:

AB = A· B + A × B.

Like vectors these have magnitudes |A · B| = |A||B| cos θ and |A × B| = |A||B| sin θ, 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:

B = a∧ b C = a∧ c

a

b c

b + c

B C

B + C

Two bivectors, two of the non-parallel sides of a prism, being added to give a third bivector.^[12]

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 aprismwith a, b, c and b + c as edges. This corresponds to the two ways of calculating the area using thedistributivityof the exterior product:

B + C = a∧ b + a ∧ c

=a∧ (b + c).

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.