Table of Contents

Navigate back to the activity.



Main ideas

  • Geometric introduction of $\rhat$ and $\phat$.
  • Geometric introduction of unit tangent and normal vectors.


  • The position vector $\rr$.
  • The derivative of the position vector is tangent to the curve.


See the prerequisites. It is possible to briefly introduce these ideas immediately preceding this activity.


  • whiteboards and pens


  • Emphasize that $\rhat$ and $\phat$ do not live at the origin! Encourage students to use the figure provided, which may help alleviate this confusion.
  • Point out to the students that $\rhat$ and $\phat$ are defined everywhere (except at the origin), whereas $\TT$ and $\NN$ are properties of the curve. It is only on circles that these two notions coincide; $\rhat$ and $\phat$ are adapted to round problems, and circles are round! Symmetry is important.
  • Emphasize that $\{\rhat,\phat\}$ can be used as a basis (except at the origin). Point out to the students that their answer to the last problem gives them a formula expressing $\rhat$ and $\phat$ in terms of $\ii$ and $\jj$. When comparing these basis vectors, they should all be drawn with their tails at the same point.


We have had success helping students master the idea of “direction of bending” by describing the curve as part of a pickle jar; the principal unit normal vector points at the pickles!

In the Classroom

The easiest way to find $\NN$ is to use the dot product to find vectors orthogonal to $\TT$, then normalize. Students must then use the “direction of bending” criterion to choose between the two possible orientations.

Finding $\NN$ in this way requires the student to give names to the its unknown components. This is a nontrivial skill; many students will have trouble with this. This is a good example of the general skill discussed in Section 11.1.

Subsidiary ideas

  • Dividing any vector by its length yields a unit vector.
  • Using the dot product to find vectors perpendicular to a given vector.


  • Some students will not be comfortable unless they work out the components of $\rhat$ and $\phat$ with respect to $\ii$ and $\jj$. Let them.

Essay questions

(none yet)


  • What units does a unit vector have? Do $\rhat$ and $\phat$ have the same units?

Personal Tools