Résultant

Résultant is a term used in French to denote the net effect or outcome of combining several quantities. In physics and vector analysis, the resultant of a set of vectors is their vector sum. If forces F1, F2, … act on a point, the resultant force R is R = F1 + F2 + … . The resultant represents the overall effect of the individual contributions. In statics and dynamics, the body is in equilibrium when the resultant of all external forces is zero, and in kinematics the resultant can describe net displacement or velocity when multiple motions are involved. The concept extends to any additive vector quantity, with components added component-wise along the coordinate axes.

In mathematics, the term resultant also denotes a specific scalar associated with two polynomials. The resultant of polynomials p(x) and q(x) is a quantity that vanishes precisely when p and q share a common root in an algebraic closure of the base field. It can be computed as the determinant of the Sylvester matrix or through elimination techniques. The resultant is a tool in algebraic geometry and computer algebra for testing polynomial compatibility and for eliminating variables to find common factors.

Etymology and usage: résultant comes from the verb résulter, meaning to result or to follow from something. In French, résulteur is related but unused; résul‑ is the common stem, and résultat refers to the result itself. Résultant may also function as an adjective meaning resulting, as in forces résultantes.