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### ACCELERATION

#### Essentials

##### Main ideas

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

##### Prerequisites

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

##### Warmup

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

##### Props

• whiteboards and pens

##### Wrapup

• 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.

#### Details

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.

##### Homework

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

(none yet)

##### Enrichment

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

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