Vektorers

A vector is a mathematical object with magnitude and direction, used to represent quantities such as displacement, velocity, or force. Vectors are fundamental in physics, engineering, and geometry because they encapsulate both size and orientation in a single entity.

In n-dimensional Euclidean space, a vector is represented by an ordered n-tuple of numbers relative to a basis; it can be denoted by a bold letter v or with an arrow. The components are coordinates: v = (v1, v2, ..., vn). Vectors may be free (position-independent) or anchored at a point.

Core operations include addition and subtraction, which combine vectors componentwise, and scalar multiplication, which scales the magnitude. The dot product ⟨u, v⟩ equals the sum of the products of corresponding components and yields a scalar related to the angle between vectors and their projection. In three dimensions, the cross product u × v yields a vector orthogonal to both, with magnitude equal to the area of the parallelogram spanned by the vectors.

Norms and distances are derived from vectors' magnitudes: ||v|| = sqrt(v1^2 + ... + vn^2). The distance between vectors u and v is ||u − v||. Vectors can form linear combinations, spanning subspaces and enabling the description of more complex shapes and motions.

Vector spaces are sets of vectors closed under addition and scalar multiplication; a basis provides a minimal, spanning set, and the dimension counts how many basis vectors are needed. Any vector in the space can be expressed uniquely as a linear combination of basis vectors.

Applications span physics, computer graphics, machine learning, and navigation, where vectors model directions, forces, features, and coordinate representations in data and space. In Nordic languages, the plural form vektorer is used for vectors, with related forms appearing according to grammar.