Veektory

Veektory are mathematical objects that express both magnitude and direction. In linear algebra, a veektor is an element of a vector space, and in Euclidean space, veektory can be represented as ordered tuples of numbers. The simplest case is R^n, where a veektor is an n-tuple (v1, v2, ..., vn).

Geometric interpretation: as arrows in n-dimensional space, with length |v| and direction. In many contexts veektory are stored as column vectors, with components relative to a chosen basis. The same object can be described in different coordinates via a linear transformation.

Operations: addition u+v combines vectors componentwise; scalar multiplication αu scales length. The dot product (u·v) yields a scalar that measures projection, angle, or orthogonality. The cross product in three dimensions produces a vector orthogonal to both operands. Norms and unit vectors describe length and direction with magnitude 1.

Vector spaces and bases: A veektor space is a set closed under vector addition and scalar multiplication. A basis is a minimal generating set; its size is the dimension of the space. Any veektor can be expressed uniquely as a linear combination of basis vectors.

Applications: veektory describe physical quantities like velocity and force, represent coordinates in graphics, machine learning data, and transformations in computer graphics and engineering. Vector concepts extend to abstract spaces and linear transformations, providing a fundamental language across disciplines.