3space

3space, commonly denoted R^3 or E^3, is the three-dimensional Euclidean space that serves as the standard setting for classical geometry and physical space. It consists of all ordered triples (x, y, z) of real numbers.

As a real vector space, 3space has dimension 3 with the standard basis (1,0,0), (0,1,0), (0,0,1). The Euclidean distance between two points p = (x1, y1, z1) and q = (x2, y2, z2) is sqrt((x1−x2)^2 + (y1−y2)^2 + (z1−z2)^2), and the inner product is x1x2 + y1y2 + z1z2. The length of a vector and the angle between vectors follow from this inner product, yielding the familiar Euclidean geometry.

Geometric objects in 3space include lines, defined as p + t v for a point p and direction v, and planes, defined by ax + by + cz = d. One can describe curves, surfaces, and more complex shapes within this framework. While Cartesian coordinates are standard, cylindrical and spherical coordinates are also used to describe points in 3space.

Transformations that preserve distances are isometries, forming the Euclidean group in three dimensions, including rotations, translations, and reflections. More general affine transformations combine linear maps with translations and may alter distances while preserving parallelism. 3space can also be extended or related to other geometric settings, such as Minkowski space in relativity or projective 3-space in projective geometry.

Applications of 3space span physics (spatial description in classical mechanics and as the spatial part of spacetime models), computer graphics and vision (3D modeling and rendering), engineering, and the natural sciences. It provides the standard model for visualizing and analyzing three-dimensional phenomena.