Lesson 7 of 12
Objective: SWBAT find the components, magnitude, and direction of resultant vectors.
Launch and Explore
The purpose of today’s lesson is to give students practice working with resultant vectors in the hope that they will make some generalizations and will be able to streamline the process. I want students to be able to find the resultant vector if they are given the two vectors in component form or if they are given the magnitudes and directions. The connection to right triangle trigonometry is strong – so I want students to see the connection between the components and the magnitude and direction. Furthermore, today’s lesson will have implications as we start to put complex numbers into trigonometric form and as we convert rectangular coordinates to polar coordinates later on in the year.
To begin class, I prompt students to recall what a resultant vector is by giving a quick example:
If u = <5, 4> and v = <7, -6>, what is u + v?
My students have had enough experiences over the last few days to know that we just add the horizontal and vertical components to get <12, -2>. I keep this problem on the board so that students can refer back to it when they get stuck on another problem.
Then, I give students the resultant vector assignment. I ask them to work on it with their table groups. I try to let them work without instruction. I want them to persevere through the problems. If a group gets stuck, I give them a few general suggestions, but I shy away from giving them a step-by-step approach to solve the problem – I want them to develop this on their own!
Suggestions for ways to respond to students who have questions:
- Draw a diagram to represent the situation. Use the diagram to find the horizontal and vertical components.
- How do right triangles relate to this situation? Could you draw in a right triangle to allow you to find a missing value?
- How did we find the components of u + v in the example we did together?
- When we are given a magnitude and direction, how do we find the horizontal and vertical components?
I will give students the answers to this worksheet so they can check their work as they go. That way, if they have a question they can get instant feedback and check their work right away.
Share and Summarize
After students have worked for at least 30-40 minutes, I see if they found any generalizations about the process of finding resultant vectors. The big ideas that are important are in the list below. I listen carefully for evidence of these in the students explanations. I usually develop a plan for drawing these ideas out while the students are working. I take notes about which students discussed them as they were working. Then, I selectively pick students to share their generalizations with the class.
- If a vector has a magnitude of r and a direction of θ, then the component form of the vector is <r*cosθ, r*sinθ>.
- The magnitude r of a vector will always equal sqrt(x2 + y2) where x and y are the horizontal and vertical components, respectively.
- To find the direction θ of vector, you could always use arctan(y/x) and then decide what quadrant θ should be in.
- Adding the horizontal and vertical components together will always give the components of the resultant vector, so simply find the horizontal and vertical components and add them together. This will work with adding any number of vectors, not just two.
The last two questions on the worksheet are still about resultant vectors, but they are presented in a context. Question #2 is similar to the problem we started with when I introduced vectors. I find it useful to compare and contrast the solution strategy we used on that first day with the strategy we used today. I discuss this issue in the video below.