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.
