diff --git "a/Chapitre 9 - Produits scalaires et espaces euclidens/9.1 - G\303\251om\303\251trie dans le plan et l'espace.ipynb" "b/Chapitre 9 - Produits scalaires et espaces euclidens/9.1 - G\303\251om\303\251trie dans le plan et l'espace.ipynb"
index 7fec515..7eaa204 100644
--- "a/Chapitre 9 - Produits scalaires et espaces euclidens/9.1 - G\303\251om\303\251trie dans le plan et l'espace.ipynb"
+++ "b/Chapitre 9 - Produits scalaires et espaces euclidens/9.1 - G\303\251om\303\251trie dans le plan et l'espace.ipynb"
@@ -1,6 +1,247 @@
{
- "cells": [],
- "metadata": {},
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# **Concept(s)-clé(s) et théorie**\n",
+ "\n",
+ "## Définition 1\n",
+ "Le **produit scalaire** sur $\\mathbb{R}^2$ est l'application $\\cdot: \\mathbb{R}^2 \\times \\mathbb{R}^2$ définie par $$ u \\cdot v = u_1v_1 + u_2v_2$$ ceci pour tout $u = (u_1, u_2), \\ v = (v_1, v_2) \\in \\mathbb{R}^2$.\n",
+ "\n",
+ "## Propriétés 1:\n",
+ "Pour $u,v,w \\in \\mathbb{R}^2$ et $\\lambda \\in \\mathbb{R}$, on a:\n",
+ "\n",
+ "- *Symmetrie*: $u \\cdot v = v \\cdot u$;\n",
+ "- *Additivitè*: $(u + v) \\cdot w = u \\cdot w + v \\cdot w$\n",
+ "- *Bilinéarité (combiné avec 2)*: $(\\lambda u) \\cdot v = u \\cdot (\\lambda v) = \\lambda u \\cdot v$\n",
+ "- *Définie positivité*: $u \\cdot u \\geq 0$ et si $u \\cdot u = 0$ alors $u = 0$.\n",
+ "\n",
+ "## Définition 2\n",
+ "La **longeur** (ou **norme**) d'un vecteur $u \\in \\mathbb{R}^2$ est définie par $||u|| = \\sqrt{u \\cdot u}$.\n",
+ "\n",
+ "## Défintion 3\n",
+ "Soient $u,v \\in \\mathbb{R}^2$ deux vecteurs non-nuls. Alors l'**angle** entre les droites de vecteurs directeurs $u,v$ est défini comme étant l'angle $0 \\leq \\Theta \\leq \\pi$ tel que $$ \\cos \\Theta = \\dfrac{u \\cdot v}{||u|| \\ ||v||}$$.\n",
+ "\n",
+ "## Remarque 1\n",
+ "Toutes les définitions et propriétés susmentionnées se généralisent trivialement dans $\\mathbb{R}^n$ et, en particulier, dans l'espace $\\mathbb{R}^3$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Exercises et Examples"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ " \n",
+ " "
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import Librairie.AL_Fct as al\n",
+ "import Corrections.corrections as corrections"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Exercice 1\n",
+ "\n",
+ "Let $u_1, \\dots, u_m \\in \\mathbb{R}^2$. Mark those of the following statements which are correct."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "application/vnd.jupyter.widget-view+json": {
+ "model_id": "c0e95dcc8ca14cedad8fab04a3035259",
+ "version_major": 2,
+ "version_minor": 0
+ },
+ "text/plain": [
+ "interactive(children=(Checkbox(value=False, description='$$\\\\qquad u_1 \\\\cdot u_2 \\\\cdot [\\\\dots] \\\\cdot u_n =…"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "corrections.Ex1Chapitre9_1()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Exercice 2\n",
+ "\n",
+ "Give some couples of vectors and compute inner product, angle, norms --> display vectors and notation in solutions"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "$||v|| = 10.0499; \\qquad \\theta = -84.2894 °$"
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
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