## Components of a Vector

• Lesson
• 1
• 2
9-12
1

In this lesson, students manipulate a velocity vector to control the movement of a car in a game setting. Students learn that vectors are composed of two components: magnitude and direction.

Vectors are used in numerous applications and are very important in the science and engineering. Vectors extend students' thinking about rates of change and should receive concentrated attention in schools. They are useful in representing various situations; in this example a vector is used to represent the velocity and direction of a moving object. Through experiences with the applet, students should make a number of observations about vectors and their properties.

First, students should see that vectors have two components—magnitude and direction. In this case, the magnitude of the vector controls the speed of the car, and the direction of the vector controls the direction of the car. Vectors can be represented graphically, in the form of an arrow, or numerically, as length and angle measurements. By dynamically linking the graphical and numerical representations, this applet enhances students' ability to connect algebra and geometry

Students should further come to see that the position of the vector on the screen is of no importance; dragging it around by its midpoint does not change the speed or direction of the car. The relevant features of a vector are its length (magnitude) and direction (angle), not its position.

Finally, students might observe that adjusting the length of the vector to 0 causes the car to be stationary. They may note that this state could be called the identity element for vector addition. They could be challenged to think about whether the identity is unique, since if the length of the vector is 0, its angle has no effect.

At their computers, have students open the Single-Vector Investigation Tool. Explain to students that their task is to explore how the characteristics of a vector affect the movement of a car as they use the vector to "drive" the car around without crashing into the walls.

Students can adjust the vector by dragging either endpoint, and they can move it by dragging the dot on the middle of the vector. Ask students, "How do your adjustments of the vector affect the numbers at the bottom of the screen?" [The magnitude increases (or decreases) as the vector is made longer (or shorter). The direction changes as the vector points in different directions.]

Now, have students start the car by clicking on the Start Car button. Have them drive the car around the box without crashing into the walls. (If they do crash, they can use the Reset Car button to start over.)

Then, have students click the box to Show Cyclone. Their goal is to chase after and attempt to intercept the cyclone without crashing into the walls. They can try to catch the cyclone by controlling the car's movement with the vector. Then, have students reset the game and try to catch the cyclone using only the sliders at the bottom of the screen, without directly manipulating the vector.

Special thanks to Brian Keller for creating the vector applets and to Gerd Kortemeyer for assistance in developing this activity.

• Computers with Internet connection

Extension

Move on to the next lesson, Sums of Vectors and Their Properties.

Questions for Students

1. How do the numbers for direction and magnitude correspond to the appearance of the vector?

[The magnitude indicates the length of the vector; the direction indicates which way the arrow points.]

2. How do those numbers correspond to the movement of the car?

[The magnitude is equivalent to the car's speed. The direction dictates the path that the car will take.]

3. What happens when you move the vector into a new position using its midpoint?

[Nothing. The location of the vector on the screen has no effect on the motion of the car. Only the magnitude and direction matter.]

4.How can you make the car stop? What are the values of the vector's characteristics when this happens? What might this situation be called?

[The car will stop when the magnitude is 0. This makes sense, since the magnitude of the vector represents the speed of the car. The direction of the vector will be irrelevant; if the car is not moving, the direction is unimportant.]

Teacher Reflection

• Why might you wish to begin the study of vectors with a vector that represents velocity rather than one that represents change in position?
• What are some of the advantages of using a dynamic representation? Are there disadvantages?
• Contrast the pedagogically appropriate ways in which this applet might be used with some inappropriate ways. What might you do to focus students' attention on the mathematics embedded in the situation?

### Sums of Vectors and Their Properties

9-12
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.

### Learning Objectives

Students will:

• Use a dynamic geometric representation to understand 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.