We have worked with vectors in the past (here, here, and here), but were limited to two-dimensions. Today we will be investigating vectors in three dimensions and looking at a new operation with vectors - the dot product.
The nice thing about today's lesson is that students can be very self-sufficient. The properties and operations for 3D vectors are very closely aligned with what we did in two-dimensions, so there are not many intellectual hurdles that students must go over. However, learning the dot product may be very abstract.
I begin by giving students this worksheet and have them work on question #1 with their table group. This will get them thinking about what they remember about vectors. After they brainstorm for about 5 minutes we will discuss as a class. Here is a list of what I think would be important characteristics to discuss as a class. If some of these ideas are not brought up, I will ask questions to get students there.
After we review these ideas, I will give students about 20 minutes to work on the rest of the worksheet with their table group.
When most students have wrapped up, I will discuss some of the problems from the worksheet with students. Many of the vector problems should be straightforward and we will not have to go through them together, but I will walk around while students are working and get a feel for problems that we will need to talk about. Here are a few questions that I usually feel are worth discussing as a whole class:
You may want to prove the formula for the angle of two vectors with your students. I don't necessarily go through the work, but I may show them a video like the one below that outlines the process so they get the gist of it.
At the end of this lesson, I will assign a few problems from the textbook to cement the concepts of 3D vectors and the dot product.