threenumbers

Threenumbers is a term used to denote an ordered triple of numbers, written as (a, b, c). They are commonly real numbers but may come from any field. As a geometric notion, a threenumber can be viewed either as a point in three-dimensional space or as a three-component vector in three-dimensional space. The set of all threenumbers with real components is denoted R^3; if restricted to integers, Z^3; more generally, F^3 for any field F.

Operations on threenumbers are typically componentwise. Addition and scalar multiplication follow standard vector rules: (a, b, c) + (x, y, z) = (a+x, b+y, c+z); t(a, b, c) = (ta, tb, tc). The magnitude is ||(a, b, c)|| = sqrt(a^2 + b^2 + c^2), and the distance between (a, b, c) and (x, y, z) is sqrt((a-x)^2 + (b-y)^2 + (c-z)^2). The dot product is (a, b, c) · (x, y, z) = a x + b y + c z. The cross product (a, b, c) × (x, y, z) yields a third threenumber perpendicular to both, with magnitude equal to the area of the parallelogram spanned by the two vectors.

Applications of threenumbers include modeling points and vectors in 3D space for computer graphics, physics, and geometry. They can also encode a three-feature data point, such as an RGB color triple in digital images.

Related concepts include coordinate triples, 3D vectors, and n-tuples. The term threenumber overlaps with standard terminology such as triple or vector in appropriate contexts.