Bivector

Article summary

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. Geometrically, a simple bivector can be interpreted as characterizing a directed plane segment (or oriented plane segment), much as vectors can be thought of as characterizing directed line segments. The bivector has an attitude (or direction) of the plane spanned by and , has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, it has a magnitude equal to the area of the parallelogram with edges and ), and has an orientation being the side of on which lies within the plane spanned by and . In layman terms, any surface defines the same bivector if it is parallel to the same plane (same attitude), has the same area, and same orientation (see figure).…

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