Vector Operations and Calculations
This page provides a comprehensive overview of vector operations and calculations, essential for understanding vector mathematics in both two and three-dimensional spaces.
Vector Addition and Subtraction
The document begins by illustrating vector addition. It shows how to add two vectors component-wise, which is a fundamental operation in vector mathematics.
Example: The addition of two vectors a and b is represented as:
a + b = (a₁ + b₁, a₂ + b₂, a₃ + b₃)
While not explicitly shown, vector subtraction follows a similar principle, where you subtract corresponding components.
Vector Length Calculation
The page then moves on to explain how to calculate the length of a vector, also known as the magnitude or Betrag eines Vektors.
Definition: The length of a vector in 2D space is calculated using the formula:
d = √(x₁² + y₁²)
For three-dimensional vectors, the formula extends to include the z-component:
Definition: The length of a vector in 3D space is calculated as:
d = √(x₁² + y₁² + z₁²)
Position Vectors and Direction Vectors
The concept of Ortsvektor (position vector) is introduced, which is a vector that extends from the origin to a specific point in space.
Highlight: An Ortsvektor (position vector) originates from the origin and extends to a specific point in space.
The document also touches on the concept of direction vectors, though it doesn't provide an explicit definition.
Distance Between Points
The page includes a formula for calculating the distance between two points in three-dimensional space:
Definition: The distance between two points A(a₁, a₂, a₃) and B(b₁, b₂, b₃) is given by:
|AB| = √[(b₁ - a₁)² + (b₂ - a₂)² + (b₃ - a₃)²]
Scalar Multiplication
The document briefly mentions multiplication of a vector by a scalar (a real number). This operation scales the vector while maintaining its direction (or reversing it for negative scalars).
Vocabulary: Skalar - A scalar in this context refers to a real number used to multiply a vector.
Additional Concepts
The page also includes formulas and notations for other vector operations, such as calculating angles between vectors and finding vectors between two points, though these are not elaborated upon in detail.
Overall, this page serves as a comprehensive reference for basic vector operations and calculations, providing essential formulas and concepts for working with vectors in both two and three-dimensional spaces.
