Multivector

A multivector, in the context of geometric algebra and physics, is a mathematical object composed of a sum of different-dimensional elements called *k*-blades. Each *k*-blade represents an oriented subspace of a vector space. Multivectors offer a unified framework for describing vectors, scalars, bivectors (representing oriented areas), trivectors (representing oriented volumes), and higher-dimensional geometric elements. The algebraic operations performed on multivectors, such as the geometric product, provide powerful tools for geometric calculations and transformations.

Multivector meaning with examples

• In physics, the electromagnetic field can be elegantly represented using a multivector in spacetime algebra. This allows for compact and intuitive formulations of Maxwell's equations, unifying the electric and magnetic fields and enabling relativistic transformations in a geometrically natural way.
• Computer graphics often leverage multivectors, specifically in the context of geometric algebra, to represent and manipulate 3D objects. Transformations such as rotations and reflections are performed through the geometric product, which allows for streamlined code and more intuitive geometric manipulations.
• Using multivectors, a robotic system’s spatial relationships could be fully determined. The position and orientation of each link could be represented by a multivector. These can then be combined to determine the position and orientation of any point or the end-effector in the system.
• The development of a multivector based physics engine could greatly enhance how physical simulations are created. The orientation and positions of all objects within the simulation would be easily represented and manipulated.
• The creation of advanced AI algorithms that utilise multivectors to create and work with 3D structures and their interactions with the physical world can be enhanced by the use of multivectors.