3vector

3vector is a term used to describe a vector with three components in mathematics and physics, representing quantities in three-dimensional space. A typical 3vector is written as v = (v1, v2, v3) or v = [v1, v2, v3], with v1, v2, v3 real or complex numbers depending on the context. Three-vectors form the three-dimensional Euclidean space R^3 and are closed under vector addition and scalar multiplication, making them a basic object in linear algebra.

Key operations include the dot product, v · w = v1w1 + v2w2 + v3w3, which yields a scalar, and the cross product, v × w = (v2w3 − v3w2, v3w1 − v1w3, v1w2 − v2w1), which yields a vector orthogonal to both v and w. The magnitude (or norm) of v is ||v|| = sqrt(v1^2 + v2^2 + v3^2). The distance between two 3-vectors corresponds to the norm of their difference.

3-vectors are used to describe position, velocity, force, and other spatial quantities. They are commonly represented in programming and graphics as arrays or objects with x, y, and z components (for example, vec3). In transformations, 3x3 matrices perform linear changes such as rotation and scaling, while 4×4 homogeneous coordinates are used to encapsulate translation in 3D graphics.

In relativity and some areas of physics, the term 3-vector is distinguished from higher-dimensional objects like 4-vectors. See also vector, linear algebra, cross product, dot product, and three-dimensional space.