Series: Mathematics Colloquium
Location: Math 501
Presenter: Bill Faris
This is an elementary talk about vectors. Consider Euclidean space E of dimension n. A free vector is free to move in E. A bound vector consists of a point in E together with a vector attached to this point. A sliding vector consists of a line in E and a vector that is free to slide within the line. Every bound vector defines a sliding vector, and every sliding vector defines a free vector.
It is sometimes possible to add sliding vectors. Suppose two sliding vectors have their two lines in the same plane, and the two lines intersect at one point. Then there is a natural definition of the sum of the two sliding vectors: slide the vectors to the common point, and add the bound vectors at this point.
In the mechanics of a rigid body, each sliding vector represents a force with a line of application. The condition for equilibrium is that the sum of these sliding vectors is zero. In the simplest case there are three lines that intersect in one point, and the three corresponding vectors sum to zero.
In general the condition for equilibrium seems problematic, since there may be partial sums of sliding vectors that do not define a sliding vector. What saves the theory is that the set of sliding vectors is naturally embedded in a vector space. The talk will build on ideas of Grassmann (1809--1877) to give a beautiful geometrical description of this vector space.
(Tea served in First Floor Commons Room at 3:30pm.)