Angle between vectors
25 minutes in class
Summary of Activity
In this personalized task, we first ask students a series of 8 entertaining questions about themselves, that are numerical in nature. (How old do you want to live? What’s the furthest distance you ever ran? If you had a time-machine, which century would you want to visit?…). After that, we consider their list of answers as a single vector in 8 dimensions. In order to determine how much they resemble their neighbor at the table, we calculate the cosine of the angle between the two vectors obtained. This in fact yields the correlation coefficient r between the two sets of data !
Notes and Insights
The activity is all the more striking for students who have studied the correlation coefficient in other classes, which seems not to be the case very much at that point in the year for me (even in AP Stats).
One thing to keep in mind is to keep if we want to write more questions is to make sure we keep them within a range of reasonably equivalent value-ranges. If some questions require single digit answers but the last one yields an answer in the millions, the angle between vectors will be tiny.
Correlation is usually very high because few questions allow for negative numbers. Given that the purpose of this activity isn’t the actual numbers but to demonstrate the bridge between statistics and linear algebra, it doesn’t matter.
I have a Keynote presentation that I will share if you contact me.