Euclidean Space
Introduction
Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid, who first presented its foundational elements in his work "Elements". Euclidean spaces are central to mathematical analysis and they are used in many areas of mathematics, physics, and engineering.
Definition
In mathematics, Euclidean space is the fundamental space of geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the dimension zero. These spaces are only a few of the many examples of Euclidean spaces.
Mathematical Description
The Euclidean space of dimension n, often denoted as ℝn, is an n-dimensional space with a Euclidean metric. This metric is determined by a positive-definite symmetric bilinear form, and the set ℝn equipped with this form is a real inner product space. Points in this space are usually represented as n-tuples of real numbers.
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.
Euclidean Distance
In Euclidean space, the distance between two points is defined as the square root of the sum of the squares of the differences of their corresponding coordinates. This is known as the Euclidean distance, and it is the most common way of measuring distance in Euclidean space.
Euclidean Vector
In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points to.
Properties of Euclidean Space
Euclidean space has a number of important properties that distinguish it from other spaces. These include the properties of homogeneity and isotropy, which state that the space appears the same at every point and in every direction, respectively.
Applications
Euclidean spaces have many applications in various fields of study, including physics, engineering, computer science, and economics. For instance, in physics, the three-dimensional Euclidean space is commonly used to describe the physical universe.