Vektore

Vektore are fundamental objects in mathematics, physics, and related fields that represent both magnitude and direction. In Euclidean space, a vector in R^n is written as an ordered n-tuple v = (v1, v2, ..., vn). The magnitude, or norm, is ||v|| = sqrt(v1^2 + v2^2 + ... + vn^2), and the direction is given by the unit vector v/||v||.

Vectors can be added and scaled according to the rules of a vector space. For vectors v and w and a scalar α, the sum v + w and the product αv satisfy properties such as commutativity, associativity, and distributivity. The dot product v · w = v1w1 + v2w2 + ... + vnw n yields a scalar and encodes the cosine of the angle between v and w. In three-dimensional space, the cross product v × w produces another vector perpendicular to both, with magnitude equal to the area of the parallelogram spanned by v and w.

Any vector can be expressed in a basis as v = ∑ vi ei, where ei are basis vectors and vi are the coordinates or components of v relative to that basis. Key types include the zero vector, unit vectors, and position vectors, which point from the origin to a given point.

Applications of vektore span many areas: in physics for velocity and force, in computer graphics for directions and transformations, and in data science for feature vectors that represent data points. Historically, the modern concept developed in the 19th century with contributions from Grassmann, and the term and calculus were popularized by Gibbs and Heaviside. In computer science, the word vector also denotes a dynamic array data structure.