Contextualization

Theoretical Introduction:

Vectors are mathematical objects that have a magnitude and a direction, being extensively used in physics and engineering to represent physical quantities such as force, velocity, and acceleration. However, among the many operations that can be performed with vectors, addition is one of the most fundamental and essential.

Vector addition is a vector operation that results in a vector, which is obtained from the parallelogram rule or from its components in the Cartesian plane. To add two vectors, we use the parallelogram rule, where we draw the two vectors in a way that their initial points coincide, thus forming a parallelogram. The diagonal vector of this parallelogram will represent the sum vector.

On the other hand, vector addition can also be done in the Cartesian plane, where each vector is represented as a point in space, and the sum of the vectors is obtained by adding their respective components. This way of adding vectors is particularly useful when dealing with vectors in more than two dimensions, as it allows us to treat each dimension independently.

Importance of the Theme and its Application in the Real World:

Vectors and the operation of vector addition are essential tools for understanding and modeling a variety of phenomena in the real world. From describing the motion of an object in free fall to simulating complex dynamic systems, the proper use of vectors allows scientists and engineers to understand and predict the behavior of physical systems.

For example, in programming video games, vectors are used to represent the position, velocity, and acceleration of characters and objects. In robotics, vectors are used to describe the direction and movement of robots. In earth sciences, they are used to describe the direction and magnitude of tectonic forces, wind speed and direction, among other things.

Suggestions for study resources:

• Book: Halliday, D., Resnick, R., & Walker, J. (2018). Fundamentals of physics (11th ed.). Hoboken, NJ: Wiley.

• Video: Vectors - What it is, addition and properties

• Webpage: Vectors at Brasil Escola

Practical Activity

Activity Title:

"Building Bridges: A Vector Approach"

Project Objective:

To understand the importance of vectors and their addition through the construction of a bridge model, considering the forces involved in different parts of the structure.

Detailed Project Description:

Students will be divided into groups of 3 to 5, and each group will receive the challenge of designing and building a bridge model using popsicle sticks, glue, and string, in order to apply the concepts of vectors and their addition.

Each group must make a paper project of the bridge before starting the construction. In this project, they should indicate the direction and magnitude of the forces acting on different parts of the bridge, representing them through vectors. They should also calculate the sum of the forces at each joint, ensuring that the resultant is zero (equilibrium condition). Here, an important Physics concept applies: the principle of transmissibility, according to which the effect of a force on a body is the same at any point of its line of action, as long as the direction and magnitude do not change.

When building the bridge, students should follow the project, ensuring that the forces represented by vectors are balanced. The bridge will be tested at the end by placing progressively heavier weights until it fails. The idea is to verify if the vector representation of forces helps to build a more resistant structure.

Required Materials:

• Popsicle sticks
• Glue
• String
• Weights to test the bridge's resistance (can be rice bags, beans, etc.)
• Paper and colored pens to draw the vectors in the project

Step by Step:

1. Study the concept of vectors, their representation, and addition operation.
2. Based on the acquired knowledge, make a paper bridge project, representing the forces acting on each part of the structure by vectors and calculating the sum of the vectors at each joint.
3. After finishing the project, start building the bridge with popsicle sticks, glue, and strings, always ensuring that the forces are balanced at all points, as designed.
4. After construction, perform the bridge's resistance test, noting the maximum weight it supported before failing.
5. Discuss the results with the group and possible improvements in the structure based on the study of vectors.

Project Delivery:

Groups must deliver the following:

1. The original paper project of the bridge, with vectors indicating the forces and their sums.

2. The built bridge.

3. A written report of the activity, including:

   • Introduction: Explanation of the concepts of vectors and vector addition, their relevance and real-world application, as well as the objective of this project.
   • Development: Detailing of the applied theory, the description of the activity, the methodology used, and the results obtained.
   • Conclusion: Reflection on the experience, what was learned, and suggestions for improvement for the bridge based on the study of vectors.
   • Bibliography: Listing of the sources consulted in the project development.

The project should consider the organization and distribution of tasks within the group, the quality of the original project and the built bridge, the ability to apply the concepts of vectors and vector addition in the analysis of forces, and the quality of the written report.

The work will allow students to apply theoretical concepts of vectors in practice, as well as develop socio-emotional skills such as teamwork, time management, and problem-solving.