Super Dot Product
Main Concepts
Inner product, Inner product spaces
Duration
40-50 minutes in class
Summary of Activity
This exploration is meant as an introduction to inner products.
The dot product is an operation that allows us to find characteristics of vectors like their magnitude, the angle between them, etc. We would like to define the equivalent of a “dot product” for linear spaces other than 
. Students receive the task to study properties of the dot product and then come up with an idea for an equivalent “super-dot product” that they have to apply to a quadratic function. They use their creation to calculate the “magnitude” of the quadratic function, and find two quadratic functions that are “perpendicular” to each other.
Notes and Insights
The notion of inner product is perplexing to some students. This exploration is supposed to help ease the process of introducing that notion to students. The openness of the task however makes many feel unsure as to what they are attempting to do. It is good to start by solliciting any idea from students as a group and show what the process is for that particular example before letting students try with their own experiment.
What I think is key is to communicate that math is a field of astonishing freedom. Mathematicians can really do what they want! In physics one wouldn’t decide that F=ma2, and in chemistry we wouldn’t suggest that water is H3O to see what happens. In math, we may study worlds like that and see what happens. This is what we suggest the students to do in this activity: try something and see where it goes.