Geometric and Algebraic Vectors:
An Independent Learning Activity

Explore:  Work through the Group/Lab problems.  You do not have to record your answers to this part.  This is for your own learning and exploration.

Answer: Individual Homework Problems 1 - 4.

You are 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.  Respond to the following questions as you explore:

1. How do the numbers for direction and magnitude correspond to the appearance of the vector?
   
2. How do those numbers correspond to the movement of the car?
   
3. What happens when you move the vector into a new position using its midpoint?
   
4. How can you make the car stop?  What are the values of the vector's characteristics when this happens?

Now, check 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. 

5.  Which method of controlling the car do you find easier?

In this applet, you are directing an airplane, similar to the previous example, but you will also have wind to factor into the problem.  How does having a wind blowing change the game?  Play Mother Nature and control the wind to blow the airplane to catch the hurricane.  Make one or more observations about having the wind in this situation.

Vector Sums and Their Properties, Part II

Turn off the "Show Hurricane" feature.  Respond to the following questions as you explore:

1. Turn on the "Show Vector Sum" option. A black vector appears 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?
   
2. 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?
   
3. 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 relationship among the three vectors?
   
4. Adjust the red and blue vectors so that the plane is stationary.  What do you notice about their directions and magnitudes?
   
5. Adjust the red vector so that its magnitude is about 5 and its direction is close to 45^o.  Adjust the blue vector so that its magnitude is about 3 and its direction is close to 90^o.  What are the magnitude and direction of the sum?
   
6. Now, reverse the values so that the blue vector has magnitude 5 and direction 45^o and the red vector has magnitude 3 and direction 90^o.  What are the magnitude and direction of the sum?
   
7. Try interchanging other values for the red and blue vectors and make and observation.  What do you observe? How does it relate to another property you've seen before?

2-Dimensional Vector Addition:

1. Define vectors A(-150,25) and B(65, 50).  What is the resultant?
2. Make up your own set of vectors to add and show the resultant.  Copy the diagram in your work or print a screen shot of the applet.

3-Dimensional Vector Addition:

1. Devine vectors A(10.3, 7.2, 11.0) and B(15.0, -14.0, 4.7).  What is the resultant?
2. Make up your own set of vectors to add and show the resultant.  Copy the diagram in your work or print a screen shot of the applet.

The Dot (or Scalar) Product: 
http://www.netcomuk.co.uk/~jenolive/vect6.html
The Dot Product: 
http://tutorial.math.lamar.edu/classes/calcII/dotproduct.aspx
Record (i.e. work through) examples 1, 2, and 3 as notes to refer back to in future discussions.

1. A third name for the dot product or the scalar product is also the___________________ product.

2. The dot product gives us a very nice method for determining 
   if two vectors are ________________________________.

3. The dot product will give another method for determining when two vectors are ________________________________.  

4. Note as well that often we will use the term___________________________
   in place of perpendicular.

5. Now, if two vectors are orthogonal then we know that the angle between them is__________ degrees.  

6. If two vectors are orthogonal then, a.b= _____

     NOTE:  Stop once you get to the Projections topic.

The Cross (or Vector) Product:
http://www.netcomuk.co.uk/~jenolive/vect8.html
The Cross Product: http://tutorial.math.lamar.edu/Classes/CalcII/CrossProduct.aspx
Record (i.e. work through) examples 1, 2, and 3 as notes to refer back to in future discussions.

1. The cross product requires both of the vectors to be ___________________ vectors. 
2. The result of a dot product is ________________________and the result of a cross product is ____________________________.
3. The cross product is really the _________________________________ of a _____________________ matrix.
4. Switching the order of the vectors in the cross product simply changes _____________________________ in the result.  This means that the two cross products will ____________________________________________since they only differ by a sign.
5. The cross product is ________________________ to both of the original vectors.
6. What is the "right hand rule"?

http://www.frontiernet.net/~imaging/vector_calculator.html

Use the applet to explore vector addition, vertical and horizontal components, and resultants.  Respond to the following questions:

1. Create vectors  a=<7,6> and b=<-2,6>.  Use the applet to produce their sum and difference and then sketch their resultants accurately in your work, being sure to record the components, magnitudes and directions.
2. Reset the applet and create two new vectors of your choosing.  Record all the information about your vectors, sums and differences, and their components, magnitudes, and differences.
3. Sketch your two original vectors from the problem above, and draw their horizontal and vertical components to form a right triangle.  Use triangle trigonometry to calculate the components and resultants, verifying the applet's answers.

http://www.walter-fendt.de/ph14e/resultant.htm

http://www.surendranath.org/Applets/Math/VectorAddition/VectorAdditionApplet.html

http://www.phy.syr.edu/courses/java-suite/crosspro.html

http://www.slu.edu./classes/maymk/SketchpadApplets/AddVectors.html

http://id.mind.net/~zona/mstm/physics/mechanics/vectors/components/vectorComponents.html