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 @@
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"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 $\\mathbb{R}^3$."
]
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"cell_type": "markdown",
"metadata": {},
"source": [
"# Exercises et Examples"
]
},
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- " \n",
- " "
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"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."
]
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"source": [
"corrections.Ex1Chapitre9_1()"
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{
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"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",
"$$\n",
"u_1 = \n",
"\\begin{pmatrix}\n",
"1 \\\\ 1\n",
"\\end{pmatrix}; \\quad\n",
"v_1 =\n",
"\\begin{pmatrix}\n",
"-1 \\\\ -1\n",
"\\end{pmatrix}; \\qquad \\qquad \n",
"u_2 = \n",
"\\begin{pmatrix}\n",
"\\dfrac{1}{2}\\\\ -\\dfrac{\\sqrt{3}}{2}\n",
"\\end{pmatrix}; \\quad\n",
"v_2 =\n",
"\\begin{pmatrix}\n",
"-\\dfrac{\\sqrt{3}}{2} \\\\ \\dfrac{1}{2}\n",
"\\end{pmatrix}; \\qquad \\qquad \n",
"u_3 = \n",
"\\begin{pmatrix}\n",
"2 \\\\ 1\n",
"\\end{pmatrix}; \\quad\n",
"v_3 =\n",
"\\begin{pmatrix}\n",
"-2 \\\\ \\sqrt{2}\n",
"\\end{pmatrix}; \\\\ \\quad\n",
"u_4 = \n",
"\\begin{pmatrix}\n",
"\\dfrac{1}{2} \\\\ \\dfrac{2}{3}\n",
"\\end{pmatrix}; \\quad \n",
"v_4 =\n",
"\\begin{pmatrix}\n",
"-1 \\\\ 2\n",
"\\end{pmatrix}; \\qquad \\qquad \n",
"u_5 = \n",
"\\begin{pmatrix}\n",
"\\dfrac{\\sqrt{3}}{2} \\\\ -\\dfrac{1}{2} \n",
"\\end{pmatrix}; \\quad \n",
"v_5 =\n",
"\\begin{pmatrix}\n",
"-\\dfrac{\\sqrt{2}}{2}\\\\ \\dfrac{\\sqrt{2}}{2}\n",
"\\end{pmatrix}; \\qquad \\qquad \n",
"u_6 = \n",
"\\begin{pmatrix}\n",
"0 \\\\ 0\n",
"\\end{pmatrix}; \\quad\n",
"v_6 =\n",
"\\begin{pmatrix}\n",
"0 \\\\ 1\n",
"\\end{pmatrix}; \\qquad \\quad\n",
"$$"
]
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"# Case 1\n",
"u_1 = [1, 1]\n",
"v_1 = [-1, -1]\n",
"corrections.Ex2Chapitre9_1(u_1, v_1)"
]
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"# 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)"
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"# Case 3\n",
"u_3 = [2, 1]\n",
"v_3 = [-2, np.sqrt(2)]\n",
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"# Case 4\n",
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- "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)"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
- {
- "data": {
- "application/vnd.jupyter.widget-view+json": {
- "model_id": "a15108ad0dd545c4a48a1fb328cfef5a",
- "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": "821f52f7062048ad9c7fcdc0df377f8b",
- "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": "6aa92966b3314aab8e09f2abe5859283",
- "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": "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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