On yesterday’s assignment, students were to come up with conjectures about how operations on vectors would work out. Today we are going to work with their conjectures and make generalizations about how we can represent vector operations in coordinate form and graphically.
I start class by going over Question #7 from yesterday’s assignment. Have a few students show their graphs and see if they made the connection from the Airplane Task to find out that the vectors should be placed head to tail when they are added together. Students should also know that doubling a vector is the same as adding it to itself, and the opposite of a vector is the 180° rotation of itself. I see if students made the connection to what these operations do with the coordinates.
To prepare for today’s lesson, give every student an overhead transparency that includes the four vectors on this worksheet. Students should cut their transparency in fourths so that each vector is separated. On the board,I write a vector operation (like u + v) and students will have to use their vector cutouts to place them on top of each other to show them being connected head to tail. Then, students can take a dry erase marker and they can draw in the resultant vector. I explain the process in the video below.
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The goal for this portion of the lesson is to allow them to see the graphical meaning of the vector operations. The vector cutouts will not have units or coordinates, so we will just be looking at the meaning on a visual level. It is really important for them to see these vector operations as movement – not just static entities on a worksheet!
Here are the operations that I would go through so that students can get a feel for them. After each one, I randomly select a student to instruct the class how they performed the operation.
1. u + v
2. u + v + w
4. u + w
5. u – v
6. w – u
After these have been completed, you can ask a question like whether or not vector addition is commutative, and students can use their cards to investigate the answer. I have students share out with the whole class to reach a consensus.
I find that analyzing vector operations graphically is not always ideal. When the components are given, it is often much more efficient to use those. This PowerPoint summarizes how to perform the vector operations when you are given the components. Slide #2 is a good starting point for -students to work on in groups. As they work, I will check-in with each table to see if they have any questions and to assess their understanding.
Slide #3 has a problem that is a little more challenging. Students are to find the magnitude of the sum of two vectors, but students are not given the components of the vectors or the directions – only the magnitudes and the angle between the vectors. Students will have to use our graphical approach to connect the vectors head to tail, and then they can use the Law of Cosines to find the magnitude of the resultant.
One common misconception is that students will find the distance from the tail of vector a to the tail of vector b, therefore neglecting to put the vectors head to tail. If this occurs, I remind the students about the yesterday’s problem about the airplane and how we found the resultant vector.