Vector (mathematics)

A vector is a mathematical object that captures both magnitude and direction—unlike a mere number, which has only size. Vectors are fundamental to describing the physical world: they represent forces, velocities, and spatial relationships with elegant precision.

In its most common form, a vector is an ordered list of numbers (called components). In 2D space, it might be (3, 4); in 3D, perhaps (1, 2, 5). These components encode direction and length. Vectors can be visualized as arrows pointing from one location to another, making them intuitive for physics and engineering.

Vectors obey special arithmetic rules. You can add them (tip-to-tail), stretch them by scalars, and measure angles between them using the dot product. This algebraic structure—called a vector space—is so powerful that it generalizes far beyond geometry: functions, probability distributions, and digital signals all behave like vectors.

The toolkit for working with vectors includes matrices, calculus, and linear algebra. Applications span data analysis, computer graphics, machine learning, and theoretical physics. Understanding vectors unlocks the language in which much of modern science and technology speaks.

Related

Vector space, Linear algebra, Physics, Magnitude, Dot product, Calculus