## equivalent vectors.pdf - Section 3: Determining equivalent vectors

# Introduction to Vectors

Lesson 1 of 11

## Objective: SWBAT find the magnitude and component forms of vectors

#### Bell Work

*10 min*

Today students are introduced to some key terms for vectors: magnitude and direction. Some of my students have been introduced to vectors through their Physics course. I use these students as experts to help other students that are struggling.

To introduce vectors, student watch a video clip from "Despicable Me" This clip shows the character Vector introducing himself. Most of the students have seen the movie so students immediately become curious about how this movie connects to the lesson. As we the unit progresses students will quote the character Vector anytime I ask "What is a vector?"

After the clip, students are given a minute or two to write a definition of a vector. I move around the room and ask students what they wrote. Most students wrote "A line with both magnitude and direction." Students share their definitions with each other.

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#### Introduction to vectors

*15 min*

Of course the students' definitions leads to the next question, "What is meant be magnitude and direction?" I give students some time to think and make a conjectures. I put students ideas on the board (I usually ask the students who are in Physics to keep their knowledge to themselves right now. I want all students to explore these terms.) After 3-4 minutes I begin putting students ideas on the board. Once most of the ideas are shared I ask the students who are in Physics to move to different corners of the room. Students are then told to go to those students and ask the meaning of magnitude and direction and to ask any other questions they may have about vectors. I let the "experts" answer student questions as I listen to the groups.

Students stay in the groups for a couple of minutes before we come back as a class. I now ask students to share what the "experts" explained. (The experts tell the students that magnitude is length and direction means it has a starting point and an end point and we know how it is going.)

Students now read page 445 from "Precalculus with Limits" by Larson. This page gives another way to define a vector and also discusses some notation and terms. After reading the text section I ask several questions:

- What is meant by "directed line segment?"
- Look at the diagram, could a vector be confused with something from geometry? (Students may think it is representing a ray)
- What is the notation for the magnitude?
- How can you find the magnitude?
- Is vector PQ equivalent to QP? (This makes students think at first some say yes, until others say but the direction is not the same so they are not equal)
- How do we identify or label vectors? (Student answer with the book notation but the physics students show the method I use on the board)

The questions above are designed to help students pull information out of the text. Many students only skim the reading and miss important information. The reading today has a lot of information in a couple of paragraphs.

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From the reading and questioning students determine vectors are equivalent if they have the same magnitude and direction. I give students a problem to determine if the vectors are equivalent. I do not discuss how to determine the magnitude or direction because I want students to try different ways. The most common methods are using the slopes to determine the direction or by finding the angle of elevation. Students may struggle with finding the magnitude again students may have different methods.

After working for a few minute, students to share some ideas on how they are finding the direction and method. I ask "How might we find the direction of the vectors?" and "How could we determine the distance?" After sharing processes, I let students work a minute or 2 more to determine if the vectors are equivalent. For students who are struggling I ask the following questions:

- What do you need to find to show the vectors are equivalent?
- How do you find the length of a line segment? (If students don't remember the distance formula I suggest making a right triangle)
- If 2 lines are going the same direction what are they called?(I am looking for parallel lines)
- What can you use to determine if 2 lines are parallel? (I may need to prompt more about slope with some students)

Once most of the students have determined the equivalence of the vectors a student asked to share their process and answer. I sometimes have 2 different students share work if the students have different methods. I have moved around the room and know what the students have done which helps me determine which students to put at the board.

If I do not have a student that has found the angle of elevation I ask "Could we use the angle of elevation to determine the direction?" (This is a good time to review the term angle of elevation.) I draw a a horizontal line and ask if we could determine the angle from horizontal (angle of elevation). We discuss how determining the angle from horizontal can be used to describe a direction.

I also discuss how some fields such as navigation use bearings to determine the direction. Many students have heard this in movies but have not understood the meaning. We will discuss how would we get from Independence to another city (such as Dallas) if we were to fly.

#### Resources

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#### Closure

*5 min*

As an exit slip I ask:** "How many different vectors have an initial point at (-3,4) and a magnitude of 5? Explain whether the vectors are equivalent?"**

I have students discuss ideas with a partner before writing the answer down. I do not expect everyone to have the same answers but discussing the ideas will help them think about the questions.

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Introduction to Vectors
- LESSON 2: Component Form of Vectors
- LESSON 3: Operation with Vectors
- LESSON 4: Solving Problems with Vectors
- LESSON 5: Review of Complex Numbers
- LESSON 6: Complex Numbers and Trigonometry
- LESSON 7: Operations of Complex Numbers in Trigonometric Form
- LESSON 8: DeMoivre's Theorem
- LESSON 9: Roots of Complex Numbers
- LESSON 10: Review
- LESSON 11: Assessment