Bivectoren

Bivectoren are mathematical objects that can be thought of as directed areas. They are a type of Clifford algebra element, specifically within the geometric algebra of a 3D Euclidean space. A bivector can be visualized as a parallelogram with a direction and magnitude, where the magnitude is the area of the parallelogram and the direction is perpendicular to the plane containing it, determined by the right-hand rule applied to the two vectors that define it.

In geometric algebra, bivectors arise from the geometric product of two vectors. For instance, if you have two vectors u and v, their geometric product uv can be decomposed into a scalar part (their dot product u.v) and a bivector part (their wedge product u ^ v). This wedge product represents the oriented area spanned by u and v. Bivectors are fundamental to understanding rotations in 3D space. A rotation can be represented by an exponential of a bivector, known as a rotor.

The algebra of bivectors in 3D is isomorphic to the algebra of 3D vectors, which is why rotations in 3D are often described using vector cross products. However, bivectors offer a more unified and elegant framework within geometric algebra, providing a consistent way to handle geometric operations like rotations, reflections, and transformations across different dimensions. They are not limited to 3D and play a crucial role in higher-dimensional geometric algebras as well.