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 41e9d6f..4950486 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" @@ -1,1066 +1,279 @@ { "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, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - " \n", - " " - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "import Librairie.AL_Fct as al\n", "import Corrections.corrections as corrections\n", "import numpy as np" ] }, { "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, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "145603d9790042d6872921fc2c8c0a05", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "interactive(children=(Checkbox(value=False, description='$$\\\\qquad u_1 \\\\cdot u_2 \\\\cdot [\\\\dots] \\\\cdot u_n =…" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "corrections.Ex1Chapitre9_1()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 2\n", "\n", "For each of the following couples of vectors $(u_i, v_i) \\in \\mathbb{R}^2 \\times \\mathbb{R}^2$, determine their norms, their inner product and the angle (in degrees) between them.\n", "\n", "$$\n", "u_1 = \n", "\\begin{pmatrix}\n", "1 \\\\ 1\n", "\\end{pmatrix}; \\quad\n", "v_1 =\n", "\\begin{pmatrix}\n", "-1 \\\\ -1\n", "\\end{pmatrix}; \\qquad \\qquad \n", "u_2 = \n", "\\begin{pmatrix}\n", "\\dfrac{1}{2}\\\\ -\\dfrac{\\sqrt{3}}{2}\n", "\\end{pmatrix}; \\quad\n", "v_2 =\n", "\\begin{pmatrix}\n", "-\\dfrac{\\sqrt{3}}{2} \\\\ \\dfrac{1}{2}\n", "\\end{pmatrix}; \\qquad \\qquad \n", "u_3 = \n", "\\begin{pmatrix}\n", "2 \\\\ 1\n", "\\end{pmatrix}; \\quad\n", "v_3 =\n", "\\begin{pmatrix}\n", "-2 \\\\ \\sqrt{2}\n", "\\end{pmatrix}; \\\\ \\quad\n", "u_4 = \n", "\\begin{pmatrix}\n", "\\dfrac{1}{2} \\\\ \\dfrac{2}{3}\n", "\\end{pmatrix}; \\quad \n", "v_4 =\n", "\\begin{pmatrix}\n", "-1 \\\\ 2\n", "\\end{pmatrix}; \\qquad \\qquad \n", "u_5 = \n", "\\begin{pmatrix}\n", "\\dfrac{\\sqrt{3}}{2} \\\\ -\\dfrac{1}{2} \n", "\\end{pmatrix}; \\quad \n", "v_5 =\n", "\\begin{pmatrix}\n", "-\\dfrac{\\sqrt{2}}{2}\\\\ \\dfrac{\\sqrt{2}}{2}\n", "\\end{pmatrix}; \\qquad \\qquad \n", "u_6 = \n", "\\begin{pmatrix}\n", "0 \\\\ 0\n", "\\end{pmatrix}; \\quad\n", "v_6 =\n", "\\begin{pmatrix}\n", "0 \\\\ 1\n", "\\end{pmatrix}; \\qquad \\quad\n", "$$" ] }, { "cell_type": "code", - "execution_count": 3, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "Insérez les valeurs des quantités listées ci-dessous. Entrez les valeurs avec 4 chiffres après la virgule! Si l'angle n'est pas défini, entrez -999!" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "e9d97e3c5b244e8195f85d22c54f320d", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||u||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "1988086a22d845e1abb0264fb9264259", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||v||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "6cacc8c39e4c412a8e3c8528fee45c51", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$u \\\\cdot v$:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "7907ec812e9245889a347b3b3976639c", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$\\\\Delta\\\\theta$', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "9180937e3229418a88e77d74b104632d", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "interactive(children=(Button(description='Run Interact', style=ButtonStyle()), Output()), _dom_classes=('widge…" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "f42ad963b59f42cfade9c2c2bdd809ef", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "HBox(children=(Button(description='Solution', disabled=True, style=ButtonStyle()),))" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "b0b4cfca00a4455cbdc0693c4870a1c6", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "Output()" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "# Case 1\n", "u_1 = [1, 1]\n", "v_1 = [-1, -1]\n", "corrections.Ex2Chapitre9_1(u_1, v_1)" ] }, { "cell_type": "code", - "execution_count": 4, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "Insérez les valeurs des quantités listées ci-dessous. Entrez les valeurs avec 4 chiffres après la virgule! Si l'angle n'est pas défini, entrez -999!" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "cb2cae64fbcf4072a787b9ea893e5843", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||u||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "fbf6d5a03ae441a79079c26e1d26216f", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||v||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "23bd18e598ce495a94b7cb13722b5024", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$u \\\\cdot v$:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "fcaa02fb1c3a44788334930da29db29c", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$\\\\Delta\\\\theta$', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "12ca9306d4f6438687903f7a7352c996", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "interactive(children=(Button(description='Run Interact', style=ButtonStyle()), Output()), _dom_classes=('widge…" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "2be90614d6c44b4b9c8fa6ed82d7e510", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "HBox(children=(Button(description='Solution', disabled=True, style=ButtonStyle()),))" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "61f1be29a5b548948ade39c84aea0830", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "Output()" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "# Case 2\n", "u_2 = [1/2, -np.sqrt(3)/2]\n", "v_2 = [-np.sqrt(3)/2, 1/2]\n", "corrections.Ex2Chapitre9_1(u_2, v_2)" ] }, { "cell_type": "code", - "execution_count": 5, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "Insérez les valeurs des quantités listées ci-dessous. Entrez les valeurs avec 4 chiffres après la virgule! Si l'angle n'est pas défini, entrez -999!" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "9b9997d07ef14feeb84804d17b2f753c", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||u||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "ecc2140a730f47fc9eaee90ca0f31300", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||v||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "a1512298a98c431ba0a97e4fbe0756e6", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$u \\\\cdot v$:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "508e7081e4b840e0b1102c3c7a13763a", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$\\\\Delta\\\\theta$', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "e78b2941257540b2be9194e55a0d19a7", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "interactive(children=(Button(description='Run Interact', style=ButtonStyle()), Output()), _dom_classes=('widge…" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "04343db159c74c189c4f57f53f1ec312", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "HBox(children=(Button(description='Solution', disabled=True, style=ButtonStyle()),))" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "27c66765315e4cfabbc198790021c070", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "Output()" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "# Case 3\n", "u_3 = [2, 1]\n", "v_3 = [-2, np.sqrt(2)]\n", "corrections.Ex2Chapitre9_1(u_3, v_3)" ] }, { "cell_type": "code", - "execution_count": 6, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "Insérez les valeurs des quantités listées ci-dessous. Entrez les valeurs avec 4 chiffres après la virgule! Si l'angle n'est pas défini, entrez -999!" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "c00bbb391c6540bba212ec0e488c6eca", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||u||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "fcbc933c5f8a493b8b3adc14b5e594b7", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||v||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "0ece33fb3ab84af19625d38493885a2c", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$u \\\\cdot v$:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "f27d18313a8042618a146a71f3ab88c0", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$\\\\Delta\\\\theta$', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "313351909ed547e8a55cfadb1642a0e3", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "interactive(children=(Button(description='Run Interact', style=ButtonStyle()), Output()), _dom_classes=('widge…" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "ff2f5c9d6d5647b9877587ad407f1e92", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "HBox(children=(Button(description='Solution', disabled=True, style=ButtonStyle()),))" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "2fc1302b48b34a98bfc6f7aafb054a10", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "Output()" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "# Case 4\n", "u_4 = [1/2, 2/3]\n", "v_4 = [-1, 2]\n", "corrections.Ex2Chapitre9_1(u_4, v_4)" ] }, { "cell_type": "code", - "execution_count": 7, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "Insérez les valeurs des quantités listées ci-dessous. Entrez les valeurs avec 4 chiffres après la virgule! Si l'angle n'est pas défini, entrez -999!" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "7d44e824c7e94784898e3445f3f871e3", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||u||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "651ef19d61d54c8ab7555202e1248745", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||v||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "0cf2e1f22db74b28ad08b69f5ad484db", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$u \\\\cdot v$:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "4c549e20109241b99118b95ee01831fb", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$\\\\Delta\\\\theta$', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "cb129d34a2324492b26610b908a5d4d9", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "interactive(children=(Button(description='Run Interact', style=ButtonStyle()), Output()), _dom_classes=('widge…" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "04da8b23e2d645b8bfb62cbbabe20e20", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "HBox(children=(Button(description='Solution', disabled=True, style=ButtonStyle()),))" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "985034506d414f7cb1c0719588f7557e", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "Output()" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "# Case 5\n", "u_5 = [np.sqrt(3)/2, -1/2]\n", "v_5 = [-np.sqrt(2)/2, np.sqrt(2)/2]\n", "corrections.Ex2Chapitre9_1(u_5, v_5)" ] }, { "cell_type": "code", - "execution_count": 8, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "Insérez les valeurs des quantités listées ci-dessous. Entrez les valeurs avec 4 chiffres après la virgule! Si l'angle n'est pas défini, entrez -999!" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "add5b99e80c0465393bc9ab57d2926b4", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||u||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "9af7cf0523cc4c13a39e2dbad574765f", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='||v||:', step=0.0001)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "806dc9a2fac34f03b72fbe2547076aee", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "FloatText(value=0.0, description='$u \\\\cdot v$:', step=0.0001)" - ] - }, 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style=ButtonStyle()),))" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "c9e1e93c05c64a828aff005d72eeb24d", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "Output()" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "# Case 6\n", "u_6 = [0, 0]\n", "v_6 = [0, -1]\n", "corrections.Ex2Chapitre9_1(u_6, v_6)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercice 3\n", "\n", "Pour chacun des tracés suivants, marquez les énoncés qui sont corrects.\n", "\n", "### Remarque 1\n", "$\\theta$ fait référence à l'angle qu'un vecteur forme avec l'axe $x$ (c'est-à-dire avec le vecteur $e_1 = [1, 0]$); supposons que cet angle soit donné en radians et appartienne à l'intervalle $I_{\\theta} = [0; 2\\pi]$.\n", "\n", "### Remarque 2 - Notation\n", "La notation suivante est adoptée:\n", "\n", "- $\\mathbf{\\lfloor x \\rfloor}$: **partie entière de $x$** c'est-à-dire l'entier inférieur à $x$ qui est le plus proche de $x$ \\\n", "*Ex*: (1) $\\lfloor 1.4 \\rfloor = 1$; (2) $\\lfloor -2.3 \\rfloor = -3$;\n", "- $\\mathbf{\\lceil x \\rceil}$: **partie entière de $ x $ plus un** c'est-à-dire l'entier supérieur à $x$ qui est le plus proche de $x$. \\\n", "*Ex*: (1) $\\lceil 1.4 \\rceil = 2$; (2) $\\lceil -2.3 \\rceil = -2$.\n", "- $\\mathbf{a \\% b}$ **(\"a modulo b\")**: nombre tel que ce qui suit détient: $$\\dfrac{a}{b} = \\Bigl\\lfloor\\frac{a}{b}\\Bigr\\rfloor + \\dfrac{a \\% b}{b}$$\n", "*Ex*: (1) $5 \\% 4 = 1$; (2) $-3 \\% 2 = 1$; (3) $\\dfrac{7}{4} \\% 1 = \\dfrac{3}{4}$\n" ] }, { "cell_type": "code", - "execution_count": 9, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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\n", 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\n", 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\n", - "text/plain": [ - "
" - ] - }, - "metadata": { - "needs_background": "light" - }, - "output_type": "display_data" - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "4318393ab68d4ca79c2821d69f5fff89", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "interactive(children=(Checkbox(value=False, description=\"L'élément de l'ensemble $S$ avec la norme maximale a …" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ "# CAS 3\n", "corrections.Ex3Chapitre9_1(case=3)" ] }, { - "cell_type": "code", - "execution_count": null, + "cell_type": "markdown", "metadata": {}, - "outputs": [], - "source": [] + "source": [ + "[Passez au notebook du chapitre 9.2: Produit scalaires, définitions, exemples](./9.2%20-%20Produit%20scalaires%2C%20définitions%2C%20exemples.ipynb)" + ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 4 } diff --git "a/Chapitre 9 - Produits scalaires et espaces euclidens/9.2 - Produit scalaires, d\303\251finitions, exemples.ipynb" "b/Chapitre 9 - Produits scalaires et espaces euclidens/9.2 - Produit scalaires, d\303\251finitions, exemples.ipynb" index 5932a62..2d6ca36 100644 --- "a/Chapitre 9 - Produits scalaires et espaces euclidens/9.2 - Produit scalaires, d\303\251finitions, exemples.ipynb" +++ "b/Chapitre 9 - Produits scalaires et espaces euclidens/9.2 - Produit scalaires, d\303\251finitions, exemples.ipynb" @@ -1,183 +1,240 @@ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# **Concept(s)-clé(s) et théorie**\n", "\n", "## Définition 1 - Produit Scalaire\n", "Soit $V$ un $\\mathbb{R}$-espace vectoriel. Un **produit scalaire** sur $V$ est une application qui fait correspondre à chaque paire ordonnée $(u,v) \\in V \\times V$ un nombre réel, noté $\\langle u, v \\rangle \\in \\mathbb{R}$, telle que les conditions suivantes soient vérifiées, pour tous $u,v,w \\in V, \\alpha \\in \\mathbb{R}$:\n", "\n", "1. *Symmétrie*: $\\langle u,v \\rangle = \\langle v, u \\rangle$\n", "2. *Additivité*: $\\langle u+v, w \\rangle = \\langle u,w \\rangle + \\langle v,w \\rangle$\n", "3. *Bilinearité (combinè avec 2)*: $\\langle \\alpha u, v \\rangle = \\alpha \\langle u,v \\rangle = \\langle u, \\alpha v \\rangle$\n", "4. *Definié Positivité*: $\\langle u,u \\rangle \\geq 0 \\ \\forall u \\in V$ et si $\\langle u,u \\rangle = 0$ alors $u=0$.\n", "\n", "## Définition 2 - Espace Euclidien\n", "Un $\\mathbb{R}$-espace vectoriel *de dimension finie* muni d'un produit scalaire s'appelle un **espace euclidien**.\n", "\n", + "## Définition 3 - Orthogonalité\n", + "Soiet $V$ un $\\mathbb{R}$-espace vectoriel muni d'un produit scalaire $\\langle \\cdot,\\cdot \\rangle$ et $u,v \\in V$. On dit que $u$ et $v$ sont **orthogonaux** si $\\langle u,v \\rangle = 0$. \n", + "\n", "## Example 1 \n", "Un example de produit scalaire dans $V = \\mathcal{M}_{n \\times n}(\\mathbb{R})$ est: $$ \\langle A,B \\rangle = Trace(A^TB)$$ ou la trace d'un matrice carée de dimension $n$ est definie comme suit: $$Trace(A) = \\sum\\limits_{i=1}^n a_{ii}$$\n", "\n", "## Example 2\n", "Un example de produit scalaire dans $V = \\mathcal{C}^0([a;b], \\mathbb{R}) =: \\{f: [a;b] \\rightarrow \\mathbb{R} : f \\ fonction \\ continue\\}$ (avec $[a;b]$ un intervalle de $\\mathbb{R}$) est: $$ \\langle f,g \\rangle = \\int_a^b f(x)g(x) \\ dx$$ où $\\int_a^b f(x) \\ dx$ désigne l'intégrale de Riemann de $f$ dans l'intervalle $[a;b]$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Exercises et Examples" + "# Exercises et Exemples" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 1, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + " \n", + " " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import Librairie.AL_Fct as al\n", "import Corrections.corrections as corrections\n", "import numpy as np\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise 1\n", "Considérez les couples de $\\mathbb{R}$-espaces vectoriels et d'opérateurs suivants et marquez ceux des déclarations suivantes qui sont corrects.\n", "\n", "1. $\\big(\\mathbf{V}, \\langle a, b \\rangle\\big) = \\big(\\mathbb{R}^3, \\ a_1b_1 - a_1b_2 - a_2b_1 + a_2b_2 - a_2b_3 - a_3b_2 + 2a_3b_3\\big)$\n", "2. $\\big(\\mathbf{V}, \\langle a, b \\rangle\\big) = \\big(\\mathbb{R}^3, \\ a^TMb\\big) \\qquad$ with $M = \\begin{pmatrix} 2 & 0 & 1\\\\ 0 & 2 & -1 \\\\ -1 & -1 & 2 \\end{pmatrix}$\n", "3. $\\big(\\mathbf{V}, \\langle a, b\\rangle\\big) = \\big(\\mathbb{P}^2(\\mathbb{R}), \\ 2c^a_0c^b_0 - c^a_0c^b_2 + c^a_1c^b_1 - c^a_2c^b_0 + c^a_2c^b_2\\big) \\qquad$ with $a(x) =: c^a_2 x^2 + c^a_1 x + c^a_0 \\ $ and $ \\ b(x) =: c^b_2x^2 + c^b_1 x + c^b_0$\n", "4. $\\big(\\mathbf{V}, \\langle a, b\\rangle\\big) = \\big(\\mathbb{P}^3(\\mathbb{R}), \\ 2c^a_0c^b_0 - c^a_0c^b_2 + c^a_1c^b_1 - c^a_2c^b_0 + c^a_2c^b_2\\big) \\qquad$ with $a(x) =: c^a_3 x^3 + c^a_2 x^2 + c^a_1 x + c^a_0 \\ $ and $ \\ b(x) =: c^a_3 x^3 + c^b_2x^2 + c^b_1 x + c^b_0$\n", "5. $\\big(\\mathbf{V}, \\langle a, b\\rangle\\big) = \\big(\\mathcal{C}^1([x_0, x_1]; \\mathbb{R}), \\ a(x_0)b(x_0) + \\int_{x_0}^{x_1} a'(x)b'(x) \\ dx \\big)$\n", "6. $\\big(\\mathbf{V}, \\langle a, b\\rangle\\big) = \\big(\\mathcal{C}^2([x_0, x_1]; \\mathbb{R}), \\ a(x_0)b(x_0) + \\int_{x_0}^{x_1} a''(x)b''(x) \\ dx \\big)$\n", "\n", "### Remarques\n", "- $\\mathbb{P}^n(\\mathbb{R})$ désigne l'ensemble des polynômes a valeurs réelles de degré au plus $n$, qui peuvent alors être exprimés de manière unique en termes de $n+1$ coefficients scalaires\n", "- $\\mathcal{C}^n([x_0, x_1]; \\mathbb{R})$ désigne l'ensemble des fonctions à valeurs réelles qui sont $n$-fois différenciables, avec toutes les dérivées continues, sur l'intervalle $[x_0; x_1]$" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 2, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Cliquer sur CTRL pour sélectionner plusieurs réponses\n" + ] + }, + { + "data": { + "application/vnd.jupyter.widget-view+json": { + "model_id": "55c9b835b43446d5a1618d8ca995f5b4", + "version_major": 2, + "version_minor": 0 + }, + "text/plain": [ + "interactive(children=(SelectMultiple(description=\"L'opérateur est un produit scalaire dans les cas:\", layout=L…" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "corrections.Ex1Chapitre9_2()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise 2\n", "Étant donné les couples de matrices suivants, déterminez celles qui sont orthogonales par rapport au produit scalaire défini via l'opérateur de trace\n", "\n", "1. $ \\qquad A = \\begin{pmatrix} 1 & 2 & 0 \\\\ 0 & -1 & -1 \\\\ 1 & 3 & 1 \\end{pmatrix} \\qquad \\quad \\ \\ \n", " B = \\begin{pmatrix} 1 & -1 & 4 \\\\ 3 & 1 & -2 \\\\ 1 & 0 & -1 \\end{pmatrix}$ \n", "2. $ \\qquad A = \\begin{pmatrix} 0 & 2 \\\\ -1 & 3 \\end{pmatrix} \\qquad \\qquad \\quad\n", " B = \\begin{pmatrix} 3 & 1 \\\\ 1 & -1 \\end{pmatrix}$\n", "3. $ \\qquad A = \\begin{pmatrix} 0 & 1 & 3 & 0 \\\\ 1 & 0 & 1 & 0 \\\\ -1 & -2 & 2 & 1 \\\\ 3 & 4 & 1 & 2 \\end{pmatrix} \\qquad\n", " B = \\begin{pmatrix} 3 & 1 & -1 & 2 \\\\ 2 & 2 & 0 & 1 \\\\ -1 & 1 & -1 & 3 \\\\ -1 & 1 & 1 & -1 \\end{pmatrix}$\n", "4. $ \\qquad A = \\begin{pmatrix} 1 & -3 \\\\ 2 & 1 \\end{pmatrix} \\qquad \\qquad \\quad\n", " B = \\begin{pmatrix} 1 & -2 \\\\ -2 & 1 \\end{pmatrix}$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "corrections.Ex2Chapitre9_2()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise 3\n", "Étant donné les couples de fonctions suivants, déterminez s'ils sont orthogonaux par rapport au produit scalaire donné\n", "\n", "1. $\\qquad f(x) = 1 - x^2; \\quad g(x) = -(x-1)^2 \\qquad \\quad with: \\quad \\langle f, g \\rangle = c^f_0c^g_0 + c^f_1c^g_1 + c^f_2c^g_2$\n", "2. $\\qquad f(x) = 1 - x^2; \\quad g(x) = -(x-1)^2 \\qquad \\quad with: \\quad \\langle f, g \\rangle = \\int_{-1}^{1} f(x)g(x) \\ dx$\n", "3. $\\qquad f(x) = x - \\dfrac{1}{2}; \\quad g(x) = x^2 - x - \\dfrac{1}{6} \\qquad \\ \\ with: \\quad \\langle f, g \\rangle = 2c^f_0c^g_0 - c^f_0c^g_0 + 2c^f_1c^g_1 - c^f_1c^g_2 - c^f_2c^g_0 - c^f_2c^g_1 + 2c^f_2c^g_2$\n", "4. $\\qquad f(x) = x - \\dfrac{1}{2}; \\quad g(x) = x^2 - x - \\dfrac{1}{6} \\qquad \\ \\ with: \\quad \\langle f, g \\rangle = \\int_0^1 f(x)g(x) \\ dx$\n", "5. $\\qquad f(x) = \\sin(x); \\quad \\ g(x) = \\cos(x) \\qquad \\qquad \\ \\ with: \\quad \\langle f,g \\rangle = \\int_{-\\pi}^{\\pi} f(x)g(x) \\ dx$\n", "6. $\\qquad f(x) = \\sin(x); \\quad \\ g(x) = \\cos(x) \\qquad \\qquad \\ \\ with: \\quad \\langle f,g \\rangle = \\int_{0}^{\\pi / 2} f(x)g(x) \\ dx$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "corrections.Ex3Chapitre9_2()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Aide**: Vous pouvez vous aider en exécutant les cellules suivantes, qui permettent de tracer les 3 couples de fonctions considérés dans l'exercice 3" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# CASES 1-2\n", "corrections.Ex3Chapitre9_2_plotter(case_nb=1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# CASES 3-4\n", "corrections.Ex3Chapitre9_2_plotter(case_nb=3)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# CASES 5-6\n", "corrections.Ex3Chapitre9_2_plotter(case_nb=5)" ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "[Passez au notebook du chapitre 9.3-9.4: Norme, inegalité de Cauchy-Schwarz, orthogonalité, inegalité du triangle, Pythagore](./9.3-9.4%20Norme%2C%20inégalité%20de%20Cauchy-Schwarz%2C%20orthogonalité%2C%20inegalité%20du%20triangle%2C%20Pythagore.ipynb)" + ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 4 }