Thursday, January 17, 2019

The Dot Product and Angles Between Vectors

Last year I taught the dot product for the first time but was unhappy with the presentation. I gave the algebraic and geometric definitions of the dot product and we verified their equivalence for one example. Then we started using the dot product to find angles between vectors.

This year I hoped we could arrive at the dot product somehow instead of pulling it from a hat. We started with the problem of finding angles between vectors and ended up deriving the two definitions of the dot product simultaneously.

Implementing Dan Meyer's "headache-aspirin" philosophy, we started with an example that I thought my students might answer by mere inspection: <2, 2> and <-3, 0>. I thought a headache would be created by the next problem where the angle could not be easily seen, and I could offer the Law of Cosines as aspirin. But my intentions backfired in a nice way. Although some students could see the 135 degree angle and explained how they broke it down into 90 and 45 degree angles, other students already had a headache and demanded their aspirin! So we used the Law of Cosines right away on this example.


Students solved a few more problems this way:


Then I told everyone that this method could be simplified into a real nice equation and asked if they wanted to try to get it. Some students would've been happy to continue with the now familiar method, but others asked for the equation, so we worked through it on the board together:


I was nervous that I would lose students right at the beginning writing the vector c as a - b because unfortunately we hadn't done much work visualizing vector subtraction. But students agreed that b + c = a and then we solved for c. On the side (not shown above) we calculated the components of vector c as a - b = <a1,a2> - <b1,b2> = <a1-b1, a2-b2> and then substituted these into our Pythagorean equation for magnitude. Students seemed to enjoy running through the "box method" to expand the left side, but of course the most fun part is all the cancellation that leads to the final equation!

To me this process was quite satisfying, but I'm curious about other pathways. I just checked seven precalculus textbooks from a little over a decade ago and saw that they all introduce the dot product by its algebraic definition and then either simply state an equation for the angle between vectors or derive it using the Law of Cosines. A few of the textbooks first went through properties of the dot product and then used some of these in the derivation (e.g. that the dot product of a vector with itself is the square of its magnitude), which seems too laborious. How do you teach the dot product? Any ideas will be much appreciated.