Modeling Real Objects and Motions with Vectors

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Next sectionHigh School Geometry with Geometric Algebra.

News Bulletin: The World Health Organization has announced a world-wide epidemic of the Coordinate Virus in mathematics and physics courses at all grade levels. Students infected with the virus exhibit compulsive vector avoidance behavior, unable to conceive of a vector except as a list of numbers, and seizing every opportunity to replace vectors by coordinates. At least two thirds of physics graduate students are severely infected by the virus, and half of those may be permanently damaged so they will never recover. The most promising treatment is a strong dose of Geometric Algebra.

It may be surprising that the concept of vector is so difficult for students, since
intuitive notions of direction and distance are essential for navigating the everyday world. Surely these intuitions need to be engaged in learning the algebraic concept of vector, as they are essential for applications. The necessary engagement occurs only haphazardly in conventional instruction, and that is evidently insufficient for most students.

One barrier to developing the vector concept is the fact that the correspondence between vector and directed line segment has
many different interpretations in modeling properties of real objects and their motions:

          • Abstract depiction of vectors as manipulatable arrows has no physical interpretation, 
             though it can be intuitively helpful in developing an abstract geometric interpretation.

Vectors as points designate places in a Euclidean space or with respect to a physical reference frame.
             Requires designation of a distinguished point (the
origin) by the zero vector.

Position vector x for a particle which can move along a particle trajectory  x = x(t)
             must be distinguished from places which remain fixed.

Kinematic vectors, such as velocity  v = v(t) and acceleration are tied to particle position x(t).
             Actually, they are vector fields defined along the whole trajectory.

Dynamic vectors such as momentum and force representing particle interactions. 

Rigid bodies. It is often convenient to use a vector a as a 1d geometric model for a rigid body like a rod or a ruler.
             Its magnitude  
a = |a|  is then equal to the length of the body, and its direction  a/a  represents the body’s orientation,
             or, better, its
attitude in space. The endpoints of a correspond to ends of the rigid body,
             as expressed in the equation

\[\Large{{\bf{x}}\left( \alpha \right)\,\,\, = \,\,\,{{\bf{x}}_0}\,\,\, + \,\,\,\alpha \,{\bf{a}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {0 \le \alpha \le 1} \right\}}\]

             for the position vectors of a continuous distribution of particles in the body.
             Note the
crucial distinction between curves (and their parametric equations) that represent particle paths
curves that represent geometric features of physical bodies.

Exercises with Barycentric Coordinates

          (1)  Discuss and sketch values of the parametric equation  x
αa + βb  with the constraint  α + β = 1. 

          (2)  For the parametric equation  x
= αa + βb +γc  with  α + β + γ = 1,
                discuss values of the 
parameters that give vertices, edges and interior points of a triangle. 

          (3)  Discuss the relation of barycentric coordinates to center of mass.

     (Click here for solutions.)

Previous sectionVector Identities and Plane Trigonometry with GA.

Next section: High School Geometry with Geometric Algebra.

© David Hestenes 2005, 2014