Addition and Subtraction of Vectors

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

• 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?