## 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

Calculate the magnitude and the angle with respect 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 , calculate 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

Submit your results to receive credit for the lab.