In 3D, rotations have three degrees of freedom, a degree for each linearly independent plane ( bivector ) the rotation can take place in.
42.
Loosely speaking the length of a vector is a length, length of a bivector is area, and the length of a trivector is volume.
43.
In general ( and not necessarily for Euclidean vectors ) the rotation of a vector space equipped with a quadratic form can be expressed as a bivector.
44.
The exterior product of two vectors is antisymmetric, so is the negation of the bivector, producing the opposite orientation, and is the zero bivector.
45.
The exterior product of two vectors is antisymmetric, so is the negation of the bivector, producing the opposite orientation, and is the zero bivector.
46.
A bivector has a real square if and only if it is simple, and only simple bivectors can be represented geometrically by an oriented plane area.
47.
This is only true in two dimensions, so properties of the bivector in two dimensions that depend on commutativity do not usually generalise to higher dimensions.
48.
The bivector " F " " ab " yields a skew-symmetric linear operator defined by lowering one index with the metric.
49.
Under the geometric product, bivectors have negative square : the bivector \ scriptstyle \ hat x \ hat y describes the \ scriptstyle xy-plane.
50.
The derivatives that appear in Maxwell's equations are vectors and electromagnetic fields are represented by the Faraday bivector "'F " '.
