To introduce the dot product to the precalc class, I had them first find the angle between two vectors using our old trig methods. Interestingly they used right triangle trig to get the answer; I didn’t anticipate that. Lovely!

To link it in better with the future (how did they not anticipate the future !!!????!!!), I asked them to solve it with law of cosines as well:

Then I had them work in pairs on the following Desmos Activity: Vector Investigation ?????????????????? where they were answering questions based on the *dot product* but it was called *B* in the activity. There are a couple of reasons that I decided to not tell them what was being measured, and after doing the activity I think it was the right move. Still if you want to try out the activity with the labels *dot product*, here is a link to the same activity. Some great observations and guesses from the students, many went down the path of thinking about it in terms of quadrants instead of angles, but I think they were able to build some intuition for what the *dot product *was measuring.

Here are some sample responses from the first slide.

After the first three slides, the groups had a very good idea that the mysterious *B* thing was somehow related to the angle between the two vectors (and not related, directly, to the quadrants). We still have a bit of work left to nail down dot product, but it was a nice start. To finish off the day we solved the original problem, but with dot products. (Quite a bit easier). I’m still pondering if I’d like to show the proof of how the dot product is related to the law of cosines, or to bag it.