Trivectors

A trivector is a mathematical object that can be visualized as a "directed area" in three-dimensional space. It is a type of multivector, a generalization of vectors in geometric algebra. While a vector has magnitude and direction, a trivector possesses magnitude and orientation in a plane. Think of it as a flat surface with a sense of which way it's "facing".

In a standard three-dimensional Euclidean space, a trivector is closely related to the concept of a plane. However, it is more than just a plane; it carries an additional sense of direction. This direction is perpendicular to the plane it defines. For instance, a unit trivector can be thought of as a unit area element with a specified normal vector. The magnitude of the trivector indicates the area it represents, and its orientation determines the plane and the direction of its normal.

Trivectors find applications in various fields, including electromagnetism and fluid dynamics, where they can elegantly represent quantities like magnetic flux or vorticities that inherently involve planar orientations. They are also fundamental in the study of geometric algebra, providing a consistent framework for geometric operations and transformations. The algebraic operations involving trivectors follow specific rules within the geometric algebra system.