PERKALIAN SILANG VEKTOR (CROSS PRODUCT)

Introduction

This tutorial explores the concept of the cross product of vectors, a crucial operation in vector mathematics with applications in physics and engineering. Understanding the cross product helps in solving problems related to force, torque, and rotational dynamics. We will break down the process step-by-step, providing examples and practical applications.

Step 1: Understand the Cross Product Definition

The cross product of two vectors results in a third vector that is perpendicular to the plane formed by the original vectors.

Key Concepts

• If A and B are two vectors, their cross product is denoted as A × B.

• The magnitude of the cross product can be calculated using the formula:

  [ |A × B| = |A||B| \sin(\theta) ]

  Where:

  • |A| and |B| are the magnitudes of vectors A and B.
  • θ is the angle between the two vectors.

Practical Tip

• The right-hand rule can be used to determine the direction of the resulting vector. Point your thumb in the direction of A, your fingers in the direction of B, and your palm will face the direction of A × B.

Step 2: Calculate the Cross Product in 3D

To calculate the cross product of two vectors in a three-dimensional space, use their component forms.

Vector Representation

• Let A = (A1, A2, A3)
• Let B = (B1, B2, B3)

Formula for Cross Product

The components of the cross product C = A × B can be calculated as:

[ C_x = A_y B_z - A_z B_y ] [ C_y = A_z B_x - A_x B_z ] [ C_z = A_x B_y - A_y B_x ]

Example

Given:

• A = (2, 3, 4)
• B = (5, 6, 7)

Calculate:

• ( C_x = 3 \cdot 7 - 4 \cdot 6 = 21 - 24 = -3 )
• ( C_y = 4 \cdot 5 - 2 \cdot 7 = 20 - 14 = 6 )
• ( C_z = 2 \cdot 6 - 3 \cdot 5 = 12 - 15 = -3 )

Thus, C = (-3, 6, -3).

Step 3: Apply the Cross Product in Real-World Scenarios

Understanding how to apply the cross product can help solve various problems in physics, such as calculating torque or finding the area of parallelograms formed by vectors.

Example Application: Torque Calculation

• Torque (τ) can be calculated using the cross product of the radius vector (r) and the force vector (F):

[ \tau = r × F ]

Important Note

• Torque is also a vector that indicates the rotational effect of a force applied at a distance from a pivot point.

Conclusion

The cross product is a fundamental operation in vector mathematics with practical applications in various fields. By mastering the definition, calculation, and application of the cross product, you can enhance your problem-solving skills in physics and engineering.

Next steps could include exploring more complex vector operations or applying the cross product in different contexts, such as computer graphics or robotics.