# Vectors (online)

## Addition and Subtraction of Vectors

#### Purpose

To add and subtract vectors graphically and by components.

#### 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 $\vec{A}$ and the angle it makes with the positive x-axis $\theta$, its components are equal to:

Ax = A cos(θ)            and         Ay = A sin(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.

$A=\sqrt{A_x^2+A_y^2}$                and         $\theta = \arctan \frac{A_y}{A_x}$

If a vector $\vec{C}=\vec{A}\pm\vec{B}$, then its components can be calculated as:

Cx = Ax  ±  Bx               and        Cy = Ay  ±  By

#### 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.

VectorMagnitudeAngleMagnitudeAngle
CalculatedFrom 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 $\vec{A}=(10,25)$, $\vec{B}=(-3,7)$, and $\vec{C}=(-4,-10)$, calculate the components, the magnitude, and the direction of each vector given in the table.

x-componenty-componentMagnitudeAngle
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?