1. Using the dot product to compute angles.
2. The relationship between the sign of the dot product and the nature of the angle.

(This activity makes a good homework problem so long as it is then discussed in class.)

Both the algebraic and geometric expressions for the dot product.

• A handout for each student

The wrap-up is an essential component of this activity.

Most students are able to do the first problem, by combining the algebraic and geometric expressions for the dot product. However, most students will then use the same technique for the second problem, computing the actual angles between all pairs of vectors, rather than simply determining the sign of the dot product, which suffices to answer almost all of the questions. (Parallel vectors are most easily determined by inspection, since corresponding components must be multiples of each other.) It is important to let this happen — then discuss whether it was necessary.