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## 7.1.2 Sums of Vectors and their Properties

 Sums of Vectors and Their Properties  (applet) Students extend their knowledge of number systems to the system of vectors.

1. Today you will be directing an airplane, much as you directed the car in the previous part. 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 change the game?
2.  Now 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.

How to Use the Interactive Figure

To make the plane begin to move, click on Start Plane. Click on Stop Plane to stop the plane, and click on Reset Plane to reset the game. To reveal the hurricane, click on Show Hurricane, and click again to hide the hurricane. To display the vector within a grid, click on Show Grid. Click again to hide the grid.

The blue vector and sliders adjust the plane's velocity and direction.

The red vector and sliders adjust the wind's velocity and direction.

To move an entire vector, drag it from its midpoint. To adjust a vector's direction (angle) and magnitude (length) directly, drag it from its initial (starting) point or terminal (arrow) point. Adjustments to a vector's direction and magnitude can also be made by dragging the sliders found at the bottom of the figure.

To show the sum of the red and the blue vectors, click on Show Vector Sum. The magnitude and direction of the sum will appear below the Show Vector Sum box. Click it again to hide the vector sum.

1. Now turn off the Show Hurricane feature. Start the plane and explore the following:
2.  Turn on the Show Vector Sum option. A black, "sum vector" will appear that you cannot directly control. Start the plane and begin moving it around the screen using the red or the blue vectors. What relationship does the sum vector have to the plane? How does adjusting the red and blue vectors affect the sum vector?
3.  Look at various lengths and angles of the three vectors. 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?
4.  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?
5.  In what way is the sum vector a sum?
6.  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.

Try the following:

• Adjust the red vector so that its magnitude is about 5 and its direction is close to 45º. Adjust the blue vector so that its magnitude is about 3 and its direction is close to 90º. What are the magnitude and direction of the sum, displayed at the lower right of the screen.
• Now reverse the values so that the blue vector has magnitude 5 and direction 45º and the red vector has magnitude 3 and direction 90º. 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?

Discussion

High school students should not only explore more formally 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 first part, this part gives them 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 the plane.

The third task allows 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).

In the fourth task, students can 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 in the fifth task that if the values of the two vectors are interchanged, the same sum results. Thus, vector addition is commutative.

Take Time to Reflect

• Turn on the grid. What additional questions could students explore about vectors or vector addition if the grid is present?
•  Arrange the vectors tail to tail by placing all three vectors so they share a common initial point. How could this arrangement be used to represent vector addition?
•  Arrange the vectors head to tail by placing a velocity vector and the sum vector so they share a common initial point and then place the other velocity vector so its initial point is at the terminal point of the other velocity 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?

Also see:

• 7.1  Learning about Properties of Vectors and Vector Sums Using Dynamic Software

• 7.1.1  Components of a Vector