Vektori

Vektori are geometric or physical quantities that possess both magnitude and direction, and they form a vector space. In Euclidean space, a vector is commonly represented by an ordered tuple of numbers relative to a basis: in two dimensions as (x, y) and in three dimensions as (x, y, z). The magnitude, or length, of a vector is given by the Euclidean norm. The zero vector has zero magnitude and no defined direction.

Vectors can be added together and scaled by real numbers. The sum of vectors a and b is a + b, and scalar multiplication yields c a, where c is a scalar. The dot product a · b is a scalar defined as a1 b1 + a2 b2 (+ a3 b3). It encodes the angle between vectors via cos theta = (a · b) / (||a|| ||b||). In three dimensions, the cross product a × b is another vector perpendicular to both, with magnitude ||a|| ||b|| sin theta and direction given by the right-hand rule.

Vector spaces are sets of vectors closed under addition and scalar multiplication. A basis is a set of vectors that spans the space, and the number of vectors in a basis is the space's dimension. Every vector can be expressed uniquely as a linear combination of basis vectors, yielding its coordinates in that basis.

Applications of vectors span many fields. In physics, they represent forces, velocities, and accelerations; in computer graphics, they describe directions, normals, and lighting computations; in engineering and navigation, they model displacements and trajectories. In data science, vectors often represent feature sets for algorithms.

Related concepts include covectors (dual vectors) and vector fields, which assign a vector to each point in a space. Notation varies, with vectors often denoted by bold letters or arrows.