Iloczynowy

Iloczynowy is an adjective derived from iloczyn, the Polish noun for product. In mathematical language it denotes a relation to a product or a multiplicative combination. The term occurs mainly in phrases naming particular product operations, such as iloczyn skalarny (dot product) and iloczyn wektorowy (cross product). It may refer to either the operation itself or its result when the context involves products of objects like vectors or polynomials.

In vector spaces, the iloczyn skalarny of two n-dimensional vectors a and b with components a1, ..., an and b1, ..., bn is a1b1 + a2b2 + ... + anbn, a scalar. It is bilinear, commutative, and invariant under orthogonal transformations. This operation often appears in geometry and physics as a measure of similarity or projection between vectors.

In three-dimensional space, the iloczyn wektorowy a × b is a vector perpendicular to a and b; its magnitude equals |a||b|sin θ, and its direction is given by the right-hand rule. The components are (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1). It is anti-commutative: a × b = −(b × a). The cross product is used to describe orientation, torque, and rotational effects in physics and engineering.

In abstract algebra, iloczyn describes the product of elements, such as polynomials or matrices; the adjective iloczynowy is less common and context-dependent, often serving as a descriptive term rather than naming a distinct object.

See also: iloczyn skalarny, iloczyn wektorowy, iloczyn algebraiczny.