threevectors

Three-vectors are mathematical objects representing quantities with three components in ordinary three-dimensional space. They are elements of the real vector space R^3 and can be written as (x, y, z) relative to a chosen basis, or as a column vector [x; y; z] in matrix notation.

Operations on three-vectors follow standard vector algebra. Addition is componentwise: (x1, y1, z1) + (x2, y2, z2) = (x1+x2, y1+y2, z1+z2). Scalar multiplication scales each component: c(x, y, z) = (cx, cy, cz). The dot product of two three-vectors v = (x1, y1, z1) and w = (x2, y2, z2) is v·w = x1x2 + y1y2 + z1z2, yielding a scalar. The Euclidean norm is |v| = sqrt(v·v). The cross product v × w, defined in R^3, yields a vector perpendicular to both inputs with magnitude equal to the area of the parallelogram spanned by v and w, given by (y1 z2 − z1 y2, z1 x2 − x1 z2, x1 y2 − y1 x2).

Three-vectors form a real 3D vector space and transform under coordinate changes by 3×3 matrices. If the basis is rotated by an orthogonal matrix Q, coordinates transform as v′ = Qv, preserving dot products and norms.

Applications of three-vectors span physics and engineering, including velocity, force, and displacement in spatial problems, as well as computer graphics and 3D modeling. In physics, three-vectors describe spatial components separate from time; this contrasts with four-vectors used in spacetime formalisms.