What is Euclidean Geometry

Introduction:

         A Euclidean geometry in a linear plane X on the real’s is furnished by a real-valued function of two vector arguments labeled a scalar product and denotes as (x, y), which the learn euclidean geometry definition has the following properties:

(a)     (x, y) is a bi linear function; that is, it is a linear function of each argument when the other is kept fixed.

(b)    It is symmetric:

                                                     (x, y) = (y, x).

(c)     It is positive:

                                              (x, x) > 0 except for x = 0.

       The distance of two vectors x and y in a linear space with Euclidean norm is defined as || x - y ||.

Definition:

       Let X be a finite-dimensional linear space with Euclidean structure, Y a subspace of  X. The orthogonal complement of Y,  denoted as Y ^`_|_` , consists of all vectors z in X that are orthogonal to every y in Y :

                                        z in  Y ^`_|_` if (y, z)  = 0       for all y  in Y.

 Euclidean Geometry:

          In learn euclidean geometry definition , we define the Euclidean length (also called norm) of x by   

                                                | x || = (x, x)^1/2.

         A scalar product is also called an inner product, or a dot product.

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