A vector is a quantity defined by magnitude, direction, and sense, geometrically represented by an arrow (a directed line segment), analogous to how a scalar is represented by a length. Vector equality is defined by equal magnitude, parallel direction, and identical sense, and this definition motivates the operations of vector arithmetic: addition (defined via the parallelogram/resultant rule, equivalently head-to-tail placement), scalar multiplication (scaling magnitude by |c| and reversing sense when c is negative), the zero vector (the unique additive identity, forced to have zero magnitude), and the additive inverse (a vector of equal magnitude and direction but opposite sense). This belongs to the domain of vector algebra as a foundation for multivariable calculus, where definitions are chosen axiomatically to preserve structural analogies with ordinary numerical arithmetic (commutativity, associativity, identity, and inverse properties) rather than derived from any inherent "truth."
A vector is a quantity defined by magnitude, direction, and sense, geometrically represented by an arrow (a directed line segment), analogous to how a scalar is represented by a length. Vector equali…