Proyeksi Ortogonal Vektor

Introduction

This tutorial will guide you through the concept of orthogonal projection of vectors, a fundamental topic in mathematics, particularly useful in various fields such as physics and engineering. Understanding orthogonal projections helps in visualizing and solving real-world problems involving vector components.

Step 1: Understanding Vectors

• Definition of Vectors: A vector is a mathematical object characterized by both magnitude and direction.
• Components of a Vector: Vectors can be represented in a coordinate system, typically using components along the x, y, and z axes.
• Example: A vector A can be expressed as A = (Ax, Ay, Az), where Ax, Ay, and Az are its components along the respective axes.

Step 2: Introduction to Orthogonal Projections

• Definition: Orthogonal projection involves dropping a perpendicular from one vector onto another.
• Visualization: Imagine a vector A being projected onto vector B; the projection point is where the perpendicular line intersects vector B.

Step 3: Calculating the Orthogonal Projection

• Formula: The orthogonal projection of vector A onto vector B can be calculated using the formula:

  [ \text{proj}_{\mathbf{B}} \mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{\mathbf{B} \cdot \mathbf{B}} \mathbf{B} ]

• Breakdown of the Formula:

  • Dot Product: The dot product (\mathbf{A} \cdot \mathbf{B}) gives a scalar that indicates how much of A goes in the direction of B.
  • Normalization: Dividing by (\mathbf{B} \cdot \mathbf{B}) ensures that the projection is scaled correctly along B.

Step 4: Example Calculation

• Given Vectors: Let A = (3, 4) and B = (1, 2).

• Calculate the Dot Products:

  • (\mathbf{A} \cdot \mathbf{B} = 31 + 42 = 3 + 8 = 11)
  • (\mathbf{B} \cdot \mathbf{B} = 11 + 22 = 1 + 4 = 5)

• Substituting into the Formula:

  [ \text{proj}_{\mathbf{B}} \mathbf{A} = \frac{11}{5} \mathbf{B} = \frac{11}{5} (1, 2) = \left(\frac{11}{5}, \frac{22}{5}\right) ]

Step 5: Practical Application of Orthogonal Projection

• Use in Physics: Orthogonal projections are commonly used to resolve forces into components, making it easier to analyze motion.
• Application in Computer Graphics: Understanding projections is crucial for rendering 3D environments onto 2D screens.

Conclusion

In this tutorial, you learned about vectors, orthogonal projections, and how to calculate the projection of one vector onto another. Mastering these concepts will enhance your problem-solving skills in mathematics and its applications in various fields. For further practice, try calculating projections with different vectors or explore their applications in physics and engineering problems.