Vektorlengde

Vektorlengde, often referred to as the magnitude of a vector, is a fundamental concept in linear algebra and physics. It represents the scalar size or length of a vector in a given vector space. For a vector v = (v1, v2, ..., vn) in n-dimensional Euclidean space, its length, typically denoted as ||v||, is calculated using the Pythagorean theorem. The formula for the vector length is the square root of the sum of the squares of its components: ||v|| = sqrt(v1^2 + v2^2 + ... + vn^2).

In two dimensions, for a vector v = (x, y), the length is ||v|| = sqrt(x^2 + y^2). In three dimensions, for a vector v = (x, y, z), the length is ||v|| = sqrt(x^2 + y^2 + z^2). The vector length is always a non-negative scalar value. A vector with a length of zero is called the zero vector. The concept of vector length is crucial for defining distances between points, normalizing vectors (creating unit vectors), and understanding concepts like work and displacement in physics. It provides a way to quantify the "amount" of a vector quantity.