Bivector

A bivector is a mathematical object that can be thought of as a directed plane segment. It is the simplest type of multivector in geometric algebra, a framework that unifies vector algebra and complex numbers. Geometrically, a bivector represents an oriented area in a plane, with both magnitude and direction. The magnitude of a bivector corresponds to the area of the parallelogram formed by two vectors, and its orientation is determined by the plane in which this parallelogram lies and the sense of rotation from one vector to the other.

In two-dimensional Euclidean space, a bivector can be represented as the wedge product of two linearly independent vectors, denoted as $u \wedge v$. This product is antisymmetric, meaning $u \wedge v = -v \wedge u$, and it is zero if the vectors are linearly dependent. In three-dimensional space, bivectors represent oriented planes. The addition of bivectors follows the standard rules of vector addition, and their product with scalars is also defined component-wise. Bivectors have applications in various fields, including physics, where they can represent angular momentum or magnetic fields, and in computer graphics for representing rotations. They are fundamental building blocks in the study of geometric algebra and its extensions.