Addition and Subtraction of Vectors

Purpose: To add and subtract vectors graphically and by components.

Simulation Used: Vector Addition from the PhET at the University of Colorado.
• Theory

If vector , shown on the diagram above, has a magnitude A and makes an angle with the positive x axis, then its x- and y-components are given as:

Using this fact, we can add and subtract vectors by manipulating their components. If we have two vectors, and , the x- and y-components of the vector c, , are given by:
, and      .
The magnitude C and the direction of , then, can be calculated as:
, and      .
Graphically, the sum and the difference of the two vectors are given by the diagonals of the parallelogram formed by the vectors as shown on the diagrams below:
.
• Preliminary Settings.
• Open the simulation Vector Addition.
• Select "Show Grid" from the menu on the right.
• Activity 1: Addition of Vectors
• From the bucket to the right, drag vectors with the following components A=(20,10), B=(-10,25), and C=(10,-25).
• What are their x- and y-components? What are the x- and y-components of A+B and A+C? Record your calculations and the results of the simulation in the table below:
 Calculated From Simulation x y x y Vector A Vector B Vector A+B Vector A+C
• What is the magnitude and the angle of the vectors A+B and A+C?
 Calculated From Simulation magnitude angle in degrees magnitude angle in degrees Vector A+B Vector A+C
• Assignment.On a graphing paper, draw all the vectors to scale. Use the parallelogram rule.
• Activity 2: Subraction of Vectors.
Note that in Activity 1, vector C is equal to the vector -B. Thus, vector A+C is actually the difference between A and B, that is A-B. So, in order for the simulation to give the correct result, students must set up the operation of the two vectors as A+(-B).
• From the bucket, drag two vectors, A=(10,10) and B=(19,-3)
• Repeat all the calculations from Activity 1 and record your results in the table below:
 Calculated From Simulation x y x y Vector A Vector B Vector A-B magnitude angle in degrees magnitude angle in degrees Vector A-B
• Assignment.On a graphing paper, draw all the vectors to scale. Use the parallelogram rule.
• Activity 3: A problem
Oasis B is located 25 km east from oasis A. A camel starts from oasis A and travels 19.6 km in the direction 14.7 degrees south of east. Then it walks 8 km directly to the north. How far is the camel from oasis B?

Assignment: Solve the problem, write down the solution and bring it to class.

You may use the simulation as a guide, but that is not the solution. You must do the calculations and solve the problem. Do your calculations agree with the result of the simulation?

Acknowledgements.
• The Java Applet comes from the PhET Interactive Simulations at the University of Colorado, Boulder.
• Funding for the creation of this lab manual was provided by the VCCS Paul Lee Professional Development Grant Program.
• Some activities are based on the "Laboratory Manual, Physics 231 - 232" by Walter Wimbush, Northern Virginia Community College, 2008.

Created: Tue Jun 02 14:32:17 Eastern Daylight Time 2009 Last modified: Thu Nov 05 16:40:45 Eastern Standard Time 2009