## Sums of Vectors and Their Properties

This example illustrates how using a dynamic geometrical representation can help students develop an understanding of vectors and their properties, as described in the Number and Operations Standard. Students manipulate a two vectors to control the movement of a plane in a game-like setting. Students extend their knowledge from the first lesson to further investigate the system of vectors.

High school students should not only explore the properties of number systems that they have already encountered in the lower grades but also experience how those properties extend into new systems, such as vectors. (See the Number and Operations Standard.) Building on the experiences students had with a single vector in the previous lesson, this lesson introduces a second vector, representing the velocity and direction of the wind, and asks them to consider how the two vectors combine to affect the movement of a plane.

Students are allowed a more formal look at vector addition by showing the vector representing the sum of the two velocity vectors. Students can see how changes in one of the summands affect the sum; for example, as the angle of the red vector is increased, the angle of the sum vector also increases. However, the relationship between the summands and the sum is not easily seen unless the summands have the same angle measure (in which case the lengths are added) or inverse measures (in which case the lengths are subtracted).

Students will also be able to notice that vectors with opposite directions and the same magnitude cancel each other out. In other words, when added, they result in the identity, implying that they are inverses of each other. Likewise, students should note that if the values of the two vectors are interchanged, the same sum results. Thus, vector addition is commutative.

To begin the lesson, say to students, "Today, you will be directing an airplane, much as you directed the car in the previous lesson. You will use the Dual Vector Investigation Tool from Illuminations. In this applet, you will notice a red vector representing wind on the screen. Use the blue vector to direct the airplane to catch the hurricane. How does having a wind vector change the game?" [The wind pushes the plane off-course, so that its speed and direction are not determined by the blue velocity vector alone.] Allow students some time to play the game a few times.

Vector Investigation: Dual Vector

Then, have them turn off the **Show Hurricane** feature before
exploring the questions below. (You can display these questions on the
overhead projector; or, you can simply read the questions aloud to
students one at a time.)

- Turn on the
**Show Vector Sum**option. A black sum vector will appear that cannot be directly controlled. Start the plane and move it around the screen using the red and blue vectors. - Look at various lengths and angles of the three vectors.
- Using their midpoints, arrange the three vectors so that they form a triangle. Adjust the length of one of the vectors and again form a triangle. What does the triangle that is formed tell you about the relationships among the three vectors?
- Adjust the red and blue vectors so that the plane is stationary. What do you notice about their directions and magnitudes? Find other values for the two vectors that keep the plane stationary.

To develop a conceptual understanding of vector addition, students can attempt the following investigations. During these explorations, students should notice that vector addition is commutative—that is, if the magnitude and direction of the red and blue vectors are interchanged, the magnitude and direction of the black vector stays the same; and, that if the red and blue vectors have equal magnitude but opposite direction, the plane is stationary.

- Adjust the red vector so that its magnitude is 5 and its direction is 45°. Adjust the blue vector so that its magnitude is 3 and its direction is 90°. (Note that clicking the mouse inside each box will highlight the number, and students can enter a specific value for the magnitude or direction. Alternatively, students can approximate the values by dragging the vectors.) What are the magnitude and direction of the sum, displayed at the lower right of the screen?
- Reverse the values so that the blue vector has magnitude 5 and direction 45°, and the red vector has magnitude 3 and direction 90°. Now what are the magnitude and direction of the sum?
- Try interchanging other values for the red and blue vectors and make an observation. What do you observe? How does it relate to another property you've seen before?

- Computer and Internet connection

**Questions for Students**

1. Pretend that you are able to control the wind. By adjusting the red wind vector, "blow" the airplane to catch the hurricane. Make one or more observations.

[The magnitude and direction of the red wind vector affect the path of the plane. If the wind is in the same direction as the blue vector, the plane travels in the same direction with increased speed. If the red and blue vectors point in different directions, the plane's speed is reduced and the direction of the plane is a result of some combination of the directions of the red and blue vectors.]

2. What relationship does the sum vector have to the plane? How does adjusting the red and blue vectors affect the sum vector?

[The magnitude and direction of the sum vector is a combination of the magnitude and direction of the red and blue vectors.]

3. Can you find a pattern? What happens when you increase the length of one of the vectors? Increase its angle? In what cases can you exactly predict the values for the sum vector from the values for the red and blue vectors?

[If the length of either the red or blue vector increases, the length of the sum vector will also increase. Likewise, if the angle of either the red or blue vector increases, the angle of the sum vector will also increase. If the direction of the red and blue vectors is the same, then the magnitude of the sum vector will be equal to the sum of the magnitudes of the red and blue vectors; and if the direction of the red and blue vectors is exactly opposite, the magnitude of the sum vector will be 0.]

4. In what way does the sum vector represent a sum?

[It is equal to the combination of the red and blue vectors. That is, the horizontal dimension of the black vector is equal to the sum of the horizontal dimensions of the red and blue arrows. Likewise, the vertical dimension of the black vector is equal to the sum of the vertical dimensions.]

**Teacher Reflection**

- What additional questions could students explore about vectors or vector addition if the grid is present?
- Arrange the vectors tail-to-tail by placing the endpoint of each vector at the same point. How could this arrangement be used to represent vector addition?
- Arrange the red and blue vectors head-to-tail by placing a velocity vector so its initial point is at the terminal point of the other velocity vector. What is the relationship between this arrangement and the sum vector? How could this arrangement be used to represent vector addition?
- Which arrangement do you prefer: tail-to-tail or head-to-tail? How do the two diagrams differ in the conceptual understanding of vectors and vector addition that a student might perceive?
- How might the diagram of each arrangement contribute to students' development of force?
- Which diagram emphasizes instantaneous rates of change? Discrete rate of change? Forces acting on an object?

### Components of a Vector

### Learning Objectives

Students will:

- Use a dynamic geometric representation to develop an understanding of vectors and their properties.

### NCTM Standards and Expectations

- Develop a deeper understanding of very large and very small numbers and of various representations of them.

- Understand vectors and matrices as systems that have some of the properties of the real-number system.

- Use number-theory arguments to justify relationships involving whole numbers.