• Outer products yield projection operators
• Projection operators are idempotes (they square to themselves)
• A complete set of outer products of an orthonormal basis is the identity (a completeness relation)

A group of students, tethered together, are floating freely in outer space. Their task is to devise a method to reach a food cache some distance from their group.

In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.

Students practice using inner products to find the components of the cartesian basis vectors in the polar basis and vice versa. Then, students use a completeness relation to change bases or cartesian/polar bases and for different spin bases.