Sliding vectors, line bivectors, and torque

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.