Vectors (mathematics)

A vector is a mathematical object that combines magnitude (size) with direction, representing everything from velocity and force to displacement and abstract data. Unlike scalars, which have only size, vectors capture the full story of how things move, pull, or change through space.

Vectors appear everywhere. In classical mechanics, they describe how objects accelerate under forces. In Commutative algebra, they form the backbone of linear algebra. Computer scientists use vectors to represent high-dimensional data—think of each word in a document as a dimension. Machine learning models encode meaning as vectors in abstract space, allowing neural networks to reason about language and images.

The elegance of vectors lies in their algebra: you can add them, scale them, and measure angles between them using the dot product. This simple toolkit powers everything from computer graphics to signal processing.

Vectors live in spaces of any dimension. A 2D vector points across a plane; a 1000-dimensional vector exists in a space our intuition cannot visualize, yet mathematics handles it seamlessly. This dimensionless abstraction is why vectors unify physics, data science, and pure mathematics.

Related

Linear algebra, Matrix (mathematics), Euclidean space, Magnitude, Direction (geometry)