METODE PENJUMLAHAN VEKTOR (Metode jajar genjang, segitiga, poligon, uraian)-CONTOH SOAL & PEMBAHASAN

Introduction

This tutorial will guide you through the methods of vector addition, specifically focusing on both graphical and analytical approaches. Understanding these methods is essential for solving problems related to vectors in physics and engineering. We will cover the triangle method, parallelogram method, polygon method, and analytical techniques involving trigonometry.

Step 1: Understanding Vector Addition

• Definition of a Vector: A vector has both magnitude and direction. It can represent physical quantities like force or velocity.
• Importance of Vector Addition: Adding vectors is essential for determining the resultant vector, which combines multiple vectors into a single vector representation.

Step 2: Graphical Methods of Vector Addition

Triangle Method

1. Draw the First Vector: Start by drawing the first vector (A) from a point.
2. Draw the Second Vector: From the tip of the first vector, draw the second vector (B) in the appropriate direction.
3. Complete the Triangle: Draw a line from the tail of the first vector to the tip of the second vector. This line represents the resultant vector (R).
4. Measure the Resultant: Use a ruler to measure the length of R, and a protractor to find its direction.

Parallelogram Method

1. Draw Both Vectors: Draw both vectors A and B starting from the same point.
2. Complete the Parallelogram: Draw parallel lines to each vector to form a parallelogram.
3. Draw the Resultant Vector: The diagonal of the parallelogram from the common point represents the resultant vector R.
4. Measure and Calculate: As in the triangle method, measure the length and direction of R.

Polygon Method

1. Sequentially Draw Vectors: Draw the first vector, then the second from the tip of the first, and continue this for all vectors in a sequence.
2. Close the Polygon: Connect the start point to the endpoint of the last vector to form a closed shape.
3. Determine the Resultant: The line from the starting point to the endpoint of the last vector is the resultant.

Step 3: Analytical Methods of Vector Addition

Decomposing Vectors

1. Identifying Components: Each vector can be broken down into its horizontal (x) and vertical (y) components using trigonometric functions.
   • For a vector A at an angle θ:
     • Ax = A * cos(θ)
     • Ay = A * sin(θ)
2. Calculate Components for All Vectors: Repeat this for all vectors involved.

Summing Components

1. Sum the x Components: Add all x components together to get the total x component (Rx).
2. Sum the y Components: Add all y components together to get the total y component (Ry).
3. Resultant Vector Calculation:
   • Calculate the magnitude of the resultant vector using: [ R = \sqrt{Rx^2 + Ry^2} ]
   • Determine the direction using: [ \theta = \tan^{-1}\left(\frac{Ry}{Rx}\right) ]

Conclusion

In this tutorial, we explored various methods of vector addition, both graphical and analytical. The graphical methods include the triangle, parallelogram, and polygon methods, each useful for visual representation. The analytical approach involves decomposing vectors into components, summing them, and calculating the resultant vector using trigonometric formulas. Understanding these methods provides a solid foundation for tackling more complex vector problems in physics and engineering. Next, practice these methods with real-world examples to enhance your skills.