3D Vectors and the Dot Product

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SWBAT work with 3D vectors and define the dot product.

Big Idea

3D vectors are a natural extension of the work we did in two-dimensions.

Launch and Explore

30 minutes

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.

  1. Vectors have magnitude and direction.
  2. They have a horizontal and vertical component.
  3. Can be used to model forces, gravity, paths, etc.
  4. There are different types of notation for vectors.
  5. Have a start point (tail) and end point (head).
  6. Are represented as a directed line segment.

After we review these ideas, I will give students about 20 minutes to work on the rest of the worksheet with their table group.

Share and Summarize

20 minutes

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:

  • #4d) - A unit vector is something that students may not know how to find immediately, but the concept of scaling down a vector is important as it builds upon past knowledge of proportionality and similarity. If students can't get this, it might be good to give a really easy vector like <4, 0, 0> and start by finding a unit vector going in the same direction.
  • #6 - Finding the dot product is not too complicated, but I definitely want to make sure that students computed it correctly. This concept will be important as we continue working in this unit so you want to make sure that students can easily find the dot product.
  • #9and #10 - Again, this formula will be important as we work to find the equation of a plane, so I want to make sure that students understand. For #10, I am looking for students to tell me that is the angle is 90°, then the left side of the equation will be zero. Furthermore, the numerator of the fraction must also be zero, so the dot product of the two vectors will be zero.

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.