Belajar Bareng Aljabar Linier: Norm, Jarak, dan Hasil Kali Titik (Bagian 1)

Introduction

This tutorial focuses on understanding the concepts of vector norms, distances, and dot products as introduced in the video "Belajar Bareng Aljabar Linier: Norm, Jarak, dan Hasil Kali Titik (Bagian 1)" by Eka Resti Wulan. These foundational topics in linear algebra are crucial for students in mathematics and related fields, as they provide essential tools for analyzing geometric and algebraic relationships.

Step 1: Understanding Vector Norms

Vector norms are a way to measure the length or magnitude of a vector. Here are the key points:

• Definition of Norm: The norm of a vector v, denoted as ||v||, represents its length in a multi-dimensional space.
• Common Norms:
  • Euclidean Norm: Calculated using the formula: [ ||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2} ]
  • Properties of Norms:
    • Non-negativity: ||v|| ≥ 0
    • Definiteness: ||v|| = 0 if and only if v is the zero vector.
    • Homogeneity: ||cv|| = |c| * ||v|| for any scalar c.
    • Triangle Inequality: ||u + v|| ≤ ||u|| + ||v||.

Step 2: Calculating Distances

The distance between two points represented by vectors can be computed using norms. Follow these steps:

• Distance Formula: The distance d between two points A and B can be found using the formula: [ d(A, B) = ||B - A|| ]
• Example:
  • Let A = (x1, y1) and B = (x2, y2). The distance is: [ d(A, B) = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} ]

Step 3: Dot Product of Vectors

The dot product is a way to multiply two vectors, resulting in a scalar. Here are the essential details:

• Definition: For two vectors u and v, the dot product is calculated as: [ u \cdot v = u_1v_1 + u_2v_2 + ... + u_nv_n ]
• Geometric Interpretation: The dot product can also be interpreted as: [ u \cdot v = ||u|| \cdot ||v|| \cdot \cos(\theta) ] where θ is the angle between the two vectors.
• Applications: The dot product helps determine the angle between vectors and can indicate orthogonality (u ⋅ v = 0 means u and v are perpendicular).

Conclusion

Understanding vector norms, distances, and dot products is crucial in linear algebra and has practical applications in geometry, physics, and engineering. To further your knowledge, consider practicing with different vector examples, calculating their norms, distances, and dot products. Exploring these concepts will deepen your understanding and prepare you for more advanced topics in linear algebra.