||Components of a Vector (applet)
Students develop an understanding
that vectors are composed of both magnitude and direction.
Your task is to explore how characteristics of the vector affect the movement of the car as you use the vector to "drive" the car around without crashing into the walls. Adjust the vector by dragging either endpoint, or move it by dragging the dot on the vector. How do your adjustments of the vector affect the numbers at the bottom of the screen? Now start the car by clicking on the "Start Car" button. Try to drive the car around the box without crashing. As you do this, consider the following questions:
- How do the numbers for direction and magnitude correspond to the appearance of the vector?
- How do those numbers correspond to the movement of the car?
- What happens when you move the vector into a new position using its midpoint?
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?
Now click the box to "Show Cyclone." Your goal is to chase after and attempt to "catch" the cyclone without crashing into the walls. Try to catch the cyclone by controlling the car's movement with the vector. Then reset the game and try to catch the cyclone using only the sliders at the bottom of the screen, without directly manipulating the vector.
How to Use the Interactive Figure
To make the car begin to move, click on Start Car. Click on Stop Car to stop the car, and click on Reset Car to reset the game. To reveal the cyclone, click on Show Cyclone, and click again to hide the cyclone. To display the vector within a grid, click on Show Grid. Click again to hide the grid.
To move the entire vector, drag it from its midpoint. To adjust the vector's direction (angle) and magnitude (length) directly, drag it from its initial (starting) point or terminal (arrow) point. Adjustments to the vector's direction and magnitude can also be made by dragging the sliders found at the bottom of the figure.
Vectors are used in numerous applications and are very important in the sciences 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 components.
First, they 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).
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.
Take Time to Reflect
- 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 instructionally
appropriate ways in which this applet might be used with some inappropriate
ways. What might a teacher do to focus students' attention on the mathematics
embedded in the situation?
Special thanks to Brian Keller for timely work in creating the vector applets and to Gerd Kortemeyer for assistance in developing the activity.
- 7.1 Learning about Properties of Vectors and Vector Sums Using Dynamic Software
- 7.1.2 Sums of Vectors and their Properties