# Geometry - Object

> Geometry (Spatial, Linear Algebra)

### Table of Contents

## 1 - About

Viz - Graphic Object in Geometry

An object can be:

## 2 - Articles Related

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## 3 - Notion

Notions such as:

- The length of a vector
- The angle between two vectors.

are introduced with the inner product of two vectors

## 4 - Object

### 4.1 - Containing the origin

A geometric object (point, line, plane, …) which contains **the origin** can be representing by a subset of in two different ways :

- Span of some vectors
- Solution set of some system of linear equations with zero right-hand sides

Representation | Line | Plan |
---|---|---|

Picture | ||

Span of some vectors ( subset of ) | The Span of one “3-vectors” is a line through the origin in three dimensions Span {[1, 2, -2]} | The Span of two “3-vectors” is a plane through the origin in three dimensions Span {[4,-1,1], [0, 1, 1]} This plane contains this two lines: - Span {[4,-1,1]} - Span {[0,1,1]} |

Solution set of system of linear equations with zero right-hand sides ( subset of ) | {[x, y, z] : [1,2,-2] . [x, y, z] = 0} | {[x, y, z] : [4,-1, 1].[x, y, z] = 0, [0, 1, 1].[x, y, z] = 0} |

- Point: Zero-dimensional: Span of the empty set: The origin

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### 4.2 - Not containing the origin

A geometric object (point, line, plane, …) can be expressed as the solution set of a system of linear equations.

<math>\{x : \alpha_1 . x = \beta_1 , ... \alpha_n = \beta_n\}</math>