## 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

**Objectives**

- To learn how to resolve vectors into components.
- To learn how to add and subtract vectors by components.
- To learn how to use vectors to calculate physics problems.

#### Theory

If we know the magnitude of the vector and the angle it makes with the positive *x*-axis , its components are equal to:

*A _{x} = A *cos

*(θ)*and

*A*sin

_{y}= A*(θ)*

Conversely, if we know the components of a vector, then we can determine its magnitude and direction with respective to the positive *x*-axis.

and

If a vector , then its components can be calculated as:

*C _{x} = A_{x} ± B_{x}* and

*C*

_{y}= A_{y}± B_{y}**Preliminary Settings**

- Open the simulation Vector Addition.
- Select “Show Grid” from the menu on the right.

#### Activity 1. Switch from component- to magnitude-angle representation of a vector

Determine the magnitude and the angle with respective to the positive *x*-axis for the following vectors. Check with the simulation to confirm your result.

Vector | Magnitude | Angle | Magnitude | Angle |
---|---|---|---|---|

Calculated | From Simulation | |||

(25, 10) | ||||

(-10, 5) |

*Question: For the second vector, did you get the same angle as the simulation showed?*

#### Activity 2. Adding and Subtracting Vectors

For the vectors , , and , determine the components, the magnitude, and the direction of each vector given in the table.

x-component | y-component | Magnitude | Angle | |
---|---|---|---|---|

A + B | ||||

B - C |

*Question: Were your results confirmed by the simulation?*

#### Activity 3. Solving 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?

*Hint: Use the simulation to construct the path of the camel so that you can visualize the problem better. *

*Question: In which quadrant is the camel located after the second leg of the trip?*

#### Results

Bring your results and the answers to the questions. You can submit your lab work either in paper at the beginning of our next lab, or submit electronically via Blackboard.