Activity: Curvilinear Basis Vectors

Static Fields 2022 (9 years)
Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
What students learn
  • The basis vector \(\widehat{\hbox{coordinate}}\) points in the direction in which \(\hbox{coordinate}\) is increasing.
  • Basis vectors are (straight) unit vectors even when they are basis vectors for a coordinate which is an angle, e.g. \(\hat{\phi}\)
  • Some basis vectors in cylindrical and spherical coordinates, e.g. \(\hat{\phi}\), vary in direction as you move from point to point in space.
  • Basis vectors for a single coordinate are a simple iconic example of a vector space.

You will probably be doing this activity in-class, from directions given by the instructor. If you are doing it on your own, then choose a point in the room that you are in to be the origin. Imagine that your right shoulder is a point in space, relative to that origin. Point your right arm in succession in each of the directions of the basis vectors adapted the various coordinate systems:

  • \(\left\{\hat{x},\hat{y},\hat{z}\right\}\) in rectangular coordinates.
  • \(\left\{\hat{s},\hat{\phi},\hat{z}\right\}\) in cylindrical coordinates.
  • \(\left\{\hat{r},\hat{\theta},\hat{\phi}\right\}\) in spherical coordinates.

Author Information
Corinne Manogue
Learning Outcomes