**Introduction of b****asic geometry of Euclidean spaces:**

In Euclidean structure, learn euclidean geometry definition helps to take a position O as origin in real n-dimensional Euclidean space. In Euclidean space, the length of vector x represented by || x ||. The length of vector is the distance to the origin.

Let us introduce a Cartesian coordinate system and denote the Cartesian coordinates of x as x_{1}...x_{n}.

In learn euclidean geometry definition, we can use the Pythagorean Theorem frequently we can express the length of x in terms of its Cartesian coordinates.

|| x || =
x_{1}^{2}+....+x_{n}^{2} ( 1 )

In learn euclidean geometry gefinition assisits the scalar product of two vectors x and y, denoted as (x, y), is defined by

(x, y) = ∑ x_{j} y_{j}
(
2 )

We can express the length of a vector as

|| x || ^{2} = (x, x).
( 3 )

The scalar product is commutative:

(x, y) = (y, x) ( 4 )

And bilinear

(x + u, y) = (x, y) + (u, y).

(x, y + v) = (x, y) + (x, v). ( 5 )

Using the algebraic properties of scalar product we can derive the identity

(x – y, x – y) = (x, x) – 2(x, y) + (y, y).

Using (3), we can rewrite this identity as

|| x – y ||^{2} = || x ||^{2} – 2(x, y) + ||y
||^{2}. ( 6)

The term on the left is the distance of x from y, squared; the first and third terms on the right are the distance of x and y from O, squared.

Learn euclidean geometry definition:

The learn euclidean geometry definition, such as

**Definition:**

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.

**Defimlion:**

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.