Vectors began as carriers before they became arrows
A Euclidean vector has length and direction—an arrow that "carries" one point to another. Astronomers coined the Latin word for carrier; nineteenth-century fights over quaternions then forged the modern toolkit of dots, crosses, and free arrows.
Graphically the object runs from an initial point to a terminal point; its magnitude is that distance and its direction the sense of the displacement. Addition, subtraction, scaling, and negation obey the familiar commutative, associative, and distributive laws, making Euclidean vectors a concrete vector space. Physics uses them for velocity, acceleration, and force even when the quantity is not a literal distance: the arrow still encodes magnitude and direction, though coordinates matter for the numerical representation. Bound vectors remember a point of application; free vectors care only about magnitude and direction—crucial in mechanics when forces act at contact points.
Development took more than two centuries and about a dozen major contributors. In 1835 Bellavitis introduced equipollence: parallel segments of equal length and orientation represent the same vector. Hamilton embedded vectors as the imaginary part of quaternions and popularized the word vector. Grassmann's 1840 tidal theory already held seeds of cross and scalar products but was neglected until the 1870s. Clifford's 1878 Elements of Dynamic peeled the quaternion product into separate dot and cross products that engineers could use in three dimensions without a fourth. Gibbs, reading Maxwell, published Elements of Vector Analysis in 1881; Wilson's 1901 Vector Analysis, based on Gibbs's lectures, dropped quaternions from the pedagogical story altogether.
Pure mathematics now calls any element of a vector space a vector; Euclidean vectors are the geometric special case inside a Euclidean space. Synthetic geometry builds them as equipollence classes of ordered point pairs—parallelograms witness sameness. Modern geometry often starts from an inner-product space of translations acting freely and transitively on points; the two Euclidean-space definitions are equivalent. Whether drawn as an arrow or written as coordinates with a dot product, the carrier idea remains: something that moves A to B without forgetting which way it pointed.
Source: Euclidean vector
