vektorstørrelsen

Vektorstørrelsen, often translated as vector magnitude or vector length, is a fundamental concept in vector calculus and linear algebra. It represents the scalar value of a vector, essentially its "size" or "length" without regard to its direction. For a vector v, its vektorstørrelse is typically denoted as ||v|| or sometimes |v|.

In a two-dimensional Cartesian coordinate system, if a vector v has components (x, y), its vektorstørrelse is calculated using the Pythagorean theorem: ||v|| = sqrt(x^2 + y^2). Similarly, in three dimensions, for a vector v with components (x, y, z), the vektorstørrelse is ||v|| = sqrt(x^2 + y^2 + z^2). This generalizes to n-dimensional Euclidean space, where the vektorstørrelse of a vector v = (v1, v2, ..., vn) is ||v|| = sqrt(v1^2 + v2^2 + ... + vn^2).

The vektorstørrelse is always a non-negative real number. A vector with a vektorstørrelse of zero is known as the zero vector, which has no direction and is located at the origin. The concept of vektorstørrelse is crucial for operations such as normalization, where a vector is scaled to have a unit vektorstørrelse (a vektorstørrelse of 1), and for calculating distances between points in geometric contexts. It is a scalar quantity and therefore can be directly compared with other scalar values.