**Task**

- 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?
- 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.

**Follow-up Tasks**

- Now turn off the Show Hurricane feature. Start the plane and explore the following:
- 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?
- 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?
- 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?
- In what way is the sum vector a sum?
- 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?