Vector notation

Vector notation is the use of symbols or ordered components to represent vectors. Common forms include bold letters, arrows over letters, and coordinate tuples.

Vector: A quantity with magnitude and direction, represented mathematically by components or geometric objects. Vector notation is the symbolic language used to write these quantities.
Ordered pair: A pair of objects in which the first and second positions have distinct meanings. In two dimensions, a vector's components are often written as an ordered pair.
Geometric vector: A directed geometric quantity independent of any particular coordinate system. Geometric vectors exist independently of the component symbols used to write them.
Vector calculus: The study of derivatives and integrals of scalar and vector fields. Its formulas rely on consistent notation for vectors, fields, and differential operators.
Scalar: A quantity described by a single value that is unchanged by coordinate rotations. Scalars lack the directional structure that vector notation must express.
Column vector: A matrix with one column, whose entries can represent a vector's components. This format makes vectors compatible with matrix multiplication.
Free vector: A vector defined by magnitude and direction, with no fixed point of application. Its coordinate entries change with the frame even though the vector itself does not.
Classical mechanics: The study of motion and forces using Newtonian and related laws. Position, velocity, and force are commonly represented as vectors.
Coordinate system: A scheme that assigns numerical coordinates to points or geometric objects. Component notation depends on a chosen coordinate system.
Einstein notation: A convention that implies summation over repeated indices in tensor expressions. Index notation writes vector components compactly and exposes coordinate relationships.

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Oliver HeavisideBroader topic: Heaviside’s streamlined use of vectors displaced much of the quaternion formalism in electromagnetism.