Sliding vectors, line bivectors, and torque

TL;DR

This paper provides a modern geometric framework for understanding sliding vectors, line bivectors, and torque in Euclidean spaces, connecting classical ideas with Grassmann algebra and applications in physics and engineering.

Contribution

It introduces the concept of line bivectors as an embedding of sliding vectors in a higher-dimensional space and relates them to moment functions and dualities in physics.

Findings

Describes the structure of sliding vectors and line bivectors in Euclidean space.

Provides a Grassmann algebra-based representation of line bivectors.

Connects mathematical structures to physical concepts like force, torque, and duality.

Abstract

This paper is a modern exposition of old ideas. The setting is a Euclidian space $E$ of dimension $n$ with associated vector space $V$ of dimension $n$. A (non-zero) sliding vector is a vector in $V$ that is free to move, but only within a line $L$ of $E$. The set of sliding vectors has dimension $2n-1$. This set is naturally embedded in a vector space of dimension ${n +1 \choose 2}$. An element of this vector space will be called a line bivector. Other terms used in applications are screw and wrench. There is a nice description of line bivectors in terms of Grassmann algebra in a projective representation. It is shown that this abstract description has a concrete realization in terms of moment functions from $E$ to bivectors over $V$. The literature in physics and engineering mainly deals with the special case $n=3$. The results of the paper apply in this case and to its most common application, where the vectors in $V$ represent force and the bivectors over $V$ represent torque. It concludes with a discussion of duality, such as that of force and velocity or of torque and angular velocity.