{"id":34269,"date":"2022-03-29T13:00:52","date_gmt":"2022-03-29T13:00:52","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34269"},"modified":"2025-08-27T22:00:56","modified_gmt":"2025-08-27T22:00:56","slug":"algebra-et-projectiones-in-rn-productum-vectoriale-in-r3","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/","title":{"rendered":"Algebra et Projectiones in Rn, Productum Vectoriale in R3"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Algebra et Projectiones in Rn, Productum Vectoriale in <span class=\"katex-eq\" data-katex-display=\"false\">{\\mathbb{R}^3}<\/span><\/h1>\n<p style=\"text-align:center;\"><em><strong>Summarium:<\/strong><\/br>Haec series est continuatio directa seriei de Spatio Euclidiano dimensionis n. Hic recognoscemus quaedam notiones algebrae linearis quae adiuvant melius comprehendere spatium euclidianum n-dimensionale, recognoscemus notiones projectionum vectoris in alium, demonstraemus theorema Pythagorae et concludetur cum recognitione producti vectorialis in <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> eiusque relatione cum aliis productis spatii Euclidiani 3-dimensionalis. <\/p>\n<p style=\"text-align:center;\"><strong>INDEX<\/strong><br \/>\n<a href=\"#Independencia-Lineal-Ortogonalidad-y-Proyecciones\">Independentia Linearis, Orthogonalitas et Projectiones<\/a><br \/>\n<a href=\"#El-Teorema-de-Pitagoras-y-la-Proyecci\u00f3n-sobre-un-Subespacio\">Theorema Pythagorae et Projectio in Subspatium<\/a><br \/>\n<a href=\"#El-Producto-Escalar-y-Vectorial-en-R3\">Productum Scalaris et Vectorialis in <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/a>\n<\/p>\n<p><a name=\"Independencia-Lineal-Ortogonalidad-y-Proyecciones\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/vtNHkaHD3aA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>Independentia Linearis, Orthogonalis et Projectiones<\/h2>\n<h3>Combinatio linearis et independentia linearis<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=138s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Vector non nullus<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> potest constitui ut <strong>combinatio linearis<\/strong> respectu aliorum vectorum non nullorum <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> si exstat par numerorum realium <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span>, non ambo simul nulli, tales ut:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z} = \\alpha \\vec{x} + \\beta\\vec{y}<\/span>\n<p>Id est, vector <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> constitui potest ut summa ponderata vectorum <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}.<\/span>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=609s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Similiter dicitur<\/span><\/strong><\/a> vectores <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> esse <strong>lineariter independentes<\/strong> si <\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\alpha \\vec{x} + \\beta\\vec{y} = \\vec{0} ) \\longleftrightarrow (\\alpha=0 \\wedge \\beta=0 )<\/span>\n<p>Independentia linearis inter vectores <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> significat <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> non posse obtineri ut multiplicatio scalaris (non nulla) <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> nec e converso.<\/p>\n<p>Conceptus independentiae linearis quem modo recognovimus extendi potest ad maiores collectiones vectorum. Collectio vectorum non nullorum <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x}_1, \\cdots, \\vec{x}_n\\}<\/span> dicitur lineariter independens cum<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\displaystyle \\left[\\left(\\sum_{i=1}^n \\alpha_i \\vec{x}_i \\right) = \\vec{0} \\right] \\longleftrightarrow \\left[\\bigwedge_{i=1}^n (\\alpha_i = 0) \\right]<\/span>\n<h3>Angulus a duobus vectoribus formatus et orthogonalitas<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=1289s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Si memoramus inaequalitatem Cauchy-Schwarz,<\/span><\/strong><\/a> haec nobis dicit <span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\vec{x},\\vec{y}\\in\\mathbb{R}^n)(|\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\| \\|\\vec{y}\\|).<\/span> Hoc considerato facile est videre quod pro quolibet par vectorum <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> valet relatio:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle -1 \\leq \\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{x}\\|\\|\\vec{y}\\|}\\leq 1<\/span>\n<p>Nunc possumus intueri relationem inter productum puncti et angulum a vectoribus <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> formatum, quia hi efficiunt planum isometricum ad <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span>. Propter hoc, sine iactura generalitatis, eos imaginari possumus ut elementa <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2<\/span> cum angulis respectu axis <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_y,<\/span> respective, ita ut vectores scribantur in forma polari sic:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\vec{x} &amp;= \\|\\vec{x}\\|(\\cos(\\theta_x) , \\sin(\\theta_x)) \\\\ \\\\ \\vec{y} &amp;= \\|\\vec{y}\\|(\\cos(\\theta_y) , \\sin(\\theta_y))\n\n\\end{array}<\/span>\n<p>Ita possumus supponere (sine iactura generalitatis, iterum) quod <span class=\"katex-eq\" data-katex-display=\"false\">\\theta_x \\lt \\theta_y,<\/span> deinde computare productum puncti <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}.<\/span> Hoc facientes consequimur sequentem resultatum:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\\vec{x}\\cdot \\vec{y} &amp;=  \\|\\vec{x}\\|  \\|\\vec{y}\\| (\\cos(\\theta_x)\\cos(\\theta_y) + \\sin(\\theta_x)\\sin(\\theta_y)) \\\\ \\\\ &amp;=  \\|\\vec{x}\\|  \\|\\vec{y}\\| \\cos(\\theta_y-\\theta_x)\n\n\\end{array}<\/span>\n<p>Nunc, sumendo differentiam inter positionem angularem maiorem et minorem obtinemus angulum inter vectores comprehensum, <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})=\\theta_y - \\theta_x.<\/span> Et hoc nunc scribere possumus:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\cos\\left(\\angle(\\vec{x},\\vec{y}) \\right) = \\frac{\\vec{x} \\cdot \\vec{y}}{\\|\\vec{x}\\|\\|\\vec{y}\\|}\n\n<\/span>\n<p>Hic commemorandum est <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in [0, \\pi]<\/span>\n<p>Ex hoc coniungere possumus inaequalitatem Cauchy-Schwarz cum geometria angulorum, et insuper nobis concedit obtinere notionem rigorosam orthogonalitatis. Duo vectores dicuntur <strong>Orthogonales<\/strong> cum inter se teneant angulum <span class=\"katex-eq\" data-katex-display=\"false\">\\pi\/2<\/span> radianorum, eo sensu qui in paragrapho superiore explicatus est. Hoc idem est ac dicere <span class=\"katex-eq\" data-katex-display=\"false\">\\cos\\left(\\angle(\\vec{x},\\vec{y})\\right) = 0,<\/span> quod rursus idem est ac dicere <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = 0.<\/span> Hac de causa dicitur affirmare orthogonalitatem vectorum <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> idem esse ac dicere <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0.<\/span>\n<h4>Si duo vectores non nulli sunt orthogonales, tunc sunt lineariter independentes<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=2365s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Haec est proprietas quodammodo intuitiva vectorum<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> cuius demonstratio formalis non est tam directa, atque etiam proprietas quae interdum confusionem parere potest: Orthogonalitas duorum vectorum implicat independentiam linearim inter eos, sed independentia linearis inter duos vectores non necessario implicat eorum orthogonalitatem. Ad hoc ultimum videndum sufficit simplex contraexemplum:<\/p>\n<p>Si sumamus vectores <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}=(1,0)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}=(1,1),<\/span> qui plane non sunt orthogonales quia <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}\\cdot\\vec{B}=1,<\/span> videbimus quod si facimus<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\alpha\\vec{A} + \\beta\\vec{B} = \\vec{0}\n\n<\/span>\n<p>Tum habetur<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\alpha + \\beta &amp;= 0 \\\\ \\beta &amp;= 0\n\n\\end{array}<\/span>\n<p>et proinde: <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha = 0  \\wedge \\beta=0.<\/span> Et hoc concluditur quod:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\alpha\\vec{A} + \\beta\\vec{B} = \\vec{0} \\longleftrightarrow  \\alpha = 0  \\wedge \\beta=0\n\n<\/span>\n<p>Quod idem est ac dicere <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{A}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{B}<\/span> esse lineariter independentes. Hoc clare ostenditur modo valde explicito non verum esse quod independentia linearis implicet orthogonalitatem. Attamen, orthogonalitas quidem implicat independentiam linearim et hoc formaliter demonstrabo infra, et ad hoc consideremus sequentem collectionem praemissarum:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\mathcal{H}= \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\cdot\\vec{y}=0, \\alpha\\vec{x}+\\beta\\vec{y} = \\vec{0}\\}<\/span>\n<p>Ex hoc possumus producere sequentem ratiocinationem:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp;\\mathcal{H}\\vdash \\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\} &amp;{;\\;Praesumptio}\\\\ \\\\\n\n(2) &amp;\\mathcal{H}\\vdash \\vec{x}\\cdot\\vec{y}=0 &amp;{\\;Praesumptio} \\\\ \\\\\n\n(3) &amp;\\mathcal{H}\\vdash \\alpha\\vec{x} + \\beta\\vec{y} = \\vec{0} &amp;{\\;Praesumptio} \\\\ \\\\\n\n(4) &amp;\\mathcal{H}\\vdash (\\alpha\\vec{x} + \\beta\\vec{y})\\cdot\\vec{x} = \\alpha\\|\\vec{x}\\|^2 + \\beta(\\vec{x}\\cdot\\vec{y}) &amp;{;\\; Bilinearitas} \\\\ \\\\\n\n(5) &amp;\\mathcal{H}\\vdash  \\alpha\\|\\vec{x}\\|^2 = 0 &amp; {;\\; Ex(2,3,4)} \\\\ \\\\\n\n(6) &amp;\\mathcal{H}\\vdash  \\alpha  = 0 &amp; {;\\; Ex(1,5)} \\\\ \\\\\n\n(7) &amp;\\mathcal{H}\\vdash (\\alpha\\vec{x} + \\beta\\vec{y})\\cdot\\vec{y} = \\alpha(\\vec{x}\\cdot\\vec{y}) + \\beta\\|\\vec{y}\\|^2 &amp; {;\\;Bilinearitas} \\\\ \\\\\n\n(8) &amp;\\mathcal{H}\\vdash \\beta\\|\\vec{y}\\|^2 = 0 &amp;{;\\;Ex(2,3,7)} \\\\ \\\\\n\n(9) &amp;\\mathcal{H}\\vdash \\beta = 0 &amp;{;\\;Ex(1,8)} \\\\ \\\\\n\n(10) &amp;\\mathcal{H}\\vdash \\alpha= 0 \\wedge \\beta = 0 &amp;{;\\;\\wedge-int(6,9)}\n\n\\end{array}<\/span>\n<p>Hoc concludimus quod<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\cdot\\vec{y}=0, \\alpha\\vec{x}+\\beta\\vec{y} = \\vec{0}\\} \\vdash \\alpha= 0 \\wedge \\beta = 0  <\/span>\n<p>Denique, applicando theorema deductionis super hanc ultimam expressionem habetur:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\cdot\\vec{y}=0\\} \\vdash (\\alpha\\vec{x}+\\beta\\vec{y} = \\vec{0}) \\rightarrow (\\alpha= 0 \\wedge \\beta = 0)<\/span>\n<p>Probatio qua obtinetur sagitta in directione contraria est trivialis.<\/p>\n<p>Id est: si <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> sunt vectores non nulli et orthogonales, tunc sunt lineariter independentes.<\/p>\n<h3>Projectio vectoris in alium<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vtNHkaHD3aA&#038;t=3055s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Supponamus nos habere duos vectores non nullos<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> qui tenent inter se angulum <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})<\/span> et quaerimus \u00abQuanta parte vector <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> super vectorem <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> iacet?\u00bb vel \u00abQuam magnitudo est umbra vectoris <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> cum projicitur in directionem vectoris <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span>?\u00bb. Hanc quaestionem resolvere possumus per trigonometriam, atque ita definire projectionem vectoris <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> in alium <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y},<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Proy_{\\vec{y}}(\\vec{x}),<\/span> per expressionem:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Proy_{\\vec{y}}(\\vec{x}) = \\| \\vec{x}\\| \\cos(\\angle(\\vec{x},\\vec{y})) \\hat{y}<\/span>\n<p>Si hoc coniungimus cum iis quae in paragraphis superioribus visa sunt scribere possumus:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle Proy_{\\vec{y}}(\\vec{x}) = {\\| \\vec{x}\\|} \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{{\\|\\vec{x}\\|} \\|\\vec{y}\\|}\\right)\\color{red}{\\hat{y}} =  \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{y}\\|} \\right)\\color{red}{\\frac{\\vec{y}}{\\|\\vec{y}\\|}} = \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{y}\\|^2}\\right)\\vec{y} = \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\vec{y}\\cdot\\vec{y}}\\right)\\vec{y}<\/span>\n<p>cum, meminerimus<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\cos(\\angle(\\vec{x},\\vec{y}))  = \\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{x}\\| \\|\\vec{y}\\|}<\/span>\n<p>Projectiones magni momenti sunt quia nobis permittunt exprimere vectores secundum quamlibet basim ut summam suarum projectionum:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} = \\displaystyle \\sum_{i=1}^n \\alpha_i \\hat{u}_i<\/span>\n<p>Ubi <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{u}_i\\}_{i=1,\\cdots, n}<\/span> est basis vectorum lineariter independentium <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> et coefficientes <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha_i = (\\vec{x}\\cdot\\vec{u}_i)\/\\|\\vec{u}_i\\|<\/span> sunt ipsae projectiones in unumquemque elementum basis et constituunt coordinatas <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> respectu basis <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{u}_i\\}_{i=1,\\cdots, n}<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span>\n<p><a name=\"El-Teorema-de-Pitagoras-y-la-Proyecci\u00f3n-sobre-un-Subespacio\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/CGrr6IDnvjs\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>Theorema Pythagorae et Projectio in Subspatium<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=254s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Theorema Pythagorae est proventus<\/span><\/strong><\/a> omnibus notus et qui innumeras demonstrationes habet. Una demonstratio possibilis huius theorematos emergit ex materiis quas de spatio euclidiano elaboravimus cum addito quod valet pro qualibet numerorum dimensionum.<\/p>\n<h3>Demonstratio Theorematis Pythagorae<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=533s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Si habemus triangulum rectangulum cum cathetis<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span> et hypotenusa <span class=\"katex-eq\" data-katex-display=\"false\">c,<\/span> theorema Pythagorae dicit nobis <span class=\"katex-eq\" data-katex-display=\"false\">a^2+b^2=c^2.<\/span> Hoc intellecto possumus repraesentare quemque cathetum per par vectorum orthogonalium <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}<\/span> et scribere theorema Pythagorae hoc modo:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}\\} \\vdash\n\n \\vec{x}\\bot\\vec{y} \\leftrightarrow (\\|\\vec{x} + \\vec{y}\\|^2 = \\|\\vec{x}\\|^2 + \\|\\vec{y}\\|^2)<\/span>\n<p>Ubi expressio <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\bot\\vec{y}<\/span> significat ambos vectores esse orthogonales, id est: non nullos et tales ut <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}=0.<\/span> Hoc modo, constituitur relatio biconditionalitatis inter orthogonalitatem et summam magnitudinum in quadrato duorum vectorum.<\/p>\n<p>Haec forma vectorialis ad repraesentandum theorema Pythagorae demonstrari potest per sequentia duo ratiocinia:<\/p>\n<p><strong>Primum in directum:<\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\bot\\vec{y}\\} \\vdash \\vec{x}\\bot\\vec{y} &amp; {;\\;Praesumptio} \\\\ \\\\\n\n(2) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\bot\\vec{y}\\} \\vdash \\vec{x}\\cdot\\vec{y}= 0 &amp; {;\\;Ex(1)} \\\\ \\\\\n\n(3) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\bot\\vec{y}\\} \\vdash \\|\\vec{x} + \\vec{y}\\|^2 = (\\vec{x} + \\vec{y})\\cdot(\\vec{x} + \\vec{y}) = \\|\\vec{x}\\|^2 + 2(\\vec{x}\\cdot\\vec{y}) + \\|\\vec{y}\\|^2 &amp; \\\\\n\n&amp;;\\;Proprietas\\;normae\\;euclidianae\\;et\\;producti\\;scalari &amp; \\\\ \\\\\n\n(4) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\vec{x}\\bot\\vec{y}\\} \\vdash \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2 &amp; {;\\;Ex(2,3)} \\\\ \\\\\n\n(5) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}\\} \\vdash \\vec{x}\\bot\\vec{y} \\rightarrow ( \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2) &amp; {;\\;TD(4)} \\end{array}<\/span>\n<p><strong>Et nunc in regressum:<\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2\\} \\vdash \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2 &amp; {;\\;Praesumptio} \\\\ \\\\\n\n(2) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2\\} \\vdash \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2 +2(\\vec{x}\\cdot\\vec{y}) + \\|\\vec{y}\\|^2 &amp;  \\\\\n\n&amp;;\\;Proprietas\\;normae\\;euclidianae\\;et\\;producti\\;scalari &amp;\\\\ \\\\\n\n(3) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2\\} \\vdash  \\vec{x}\\cdot\\vec{y}=0 &amp; {;\\;Ex(1,2)} \\\\ \\\\\n\n(4) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}, \\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2\\} \\vdash  \\vec{x}\\bot\\vec{y} &amp; {;\\;Ex(3)} \\\\ \\\\\n\n(5) &amp; \\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}\\} \\vdash (\\|\\vec{x} + \\vec{y}\\|^2 =  \\|\\vec{x}\\|^2  + \\|\\vec{y}\\|^2) \\rightarrow  \\vec{x}\\bot\\vec{y} &amp; {;\\;TD(4)} \\end{array}<\/span>\n<p><strong>Et denique, coniungendo ambos ratiocinia habetur quod demonstrandum erat:<\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{\\vec{x},\\vec{y}\\in \\mathbb{R}^n\\setminus\\{\\vec{0}\\}\\} \\vdash   \\vec{x}\\bot\\vec{y} \\leftrightarrow (\\|\\vec{x} + \\vec{y}\\|^2 = \\|\\vec{x}\\|^2 + \\|\\vec{y}\\|^2)<\/span>\n<h3>Projectio vectoris in Subspatium <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1545s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Consideremus subspatium<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> formatum per basim vectorum unitariorum <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}.<\/span> Si sumimus vectorem <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\},<\/span> definitur projectio vectoris <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> in spatium <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> per expressionem:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Proy_{H}(\\vec{x}) = \\displaystyle \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j<\/span>\n<p>Quod collectio sit orthonormalis significat omnes eius elementos inter se esse orthogonales et unumquemque habere normam aequalem unitati.<\/p>\n<p>Hoc est, ut ita dicamus, umbra quam projicit vector in unamquamque partem subspatii <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> <\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEga986LBrInk-B_9gUKPe01TF10dNECXU54KK1bSf3mAPakWE-FqdqyPbb0TVy88OfGxQmJRd-yW4dwAfcC21i2dM0KZqQjPe_Qx0M5OUz4f_P6IipJQ6PcxtkOmcO7-GqRiGZ-3StQpzy8FMIfPYE89Wae6JZIC2Jk9dSTPFTK1L4TsnpkcdpV1Dbr\" width=\"578\" height=\"591\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEga986LBrInk-B_9gUKPe01TF10dNECXU54KK1bSf3mAPakWE-FqdqyPbb0TVy88OfGxQmJRd-yW4dwAfcC21i2dM0KZqQjPe_Qx0M5OUz4f_P6IipJQ6PcxtkOmcO7-GqRiGZ-3StQpzy8FMIfPYE89Wae6JZIC2Jk9dSTPFTK1L4TsnpkcdpV1Dbr\" width=\"578\" height=\"591\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<h3>Distantia inter Punctum vel Vectorem <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> et Subspatium <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=1974s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Ex projectione vectoris<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n\\setminus\\{\\vec{0}\\}<\/span> in subspatium <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> construere potest vector huius formae<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} - Proy_{H}(\\vec{x})<\/span>\n<p>Vector hoc modo formatus erit vector qui coniungit punctum subspatii <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> cum puncto coordinatarum <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},<\/span> qui egreditur orthogonaliter ad subspatium <span class=\"katex-eq\" data-katex-display=\"false\">H.<\/span> Hoc non difficile est probare, si sumamus vectorem quendam <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> et computemus productum puncti <span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z},<\/span> satis est videre quod huius operationis exitus est zero. Faciamus computationes ut videamus si hoc revera ita est:<\/p>\n<p>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H,<\/span> tunc erit formae<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}=\\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j<\/span>\n<p>Ubi <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_j\\}_{j=1}^k<\/span> est basis orthonormalis <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\beta_j \\in\\mathbb{R}<\/span> sunt coefficientes <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span> in <span class=\"katex-eq\" data-katex-display=\"false\">H.<\/span> Hoc intellecto, calculus producti puncti <span class=\"katex-eq\" data-katex-display=\"false\">(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z},<\/span> dabit:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} (\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z} &amp;= \\left(\\vec{x} - \\displaystyle \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\right) \\cdot \\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j \\\\ \\\\ &amp;= \\vec{x} \\cdot \\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j - \\displaystyle \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\cdot \\displaystyle \\sum_{j=1}^k \\beta_j\\hat{v}_j \\end{array}<\/span>\n<p>Sed quia <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> est vector <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> cuius <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> est subspatium, possibile est invenire collectionem <span class=\"katex-eq\" data-katex-display=\"false\">n-k<\/span> vectorum orthonormalium inter se et simul orthonormalium omnibus vectoribus <span class=\"katex-eq\" data-katex-display=\"false\">H,<\/span> dicamus <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_{k+1}, \\cdots, \\hat{v}_n\\},<\/span> ita ut una cum basi <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> formet basim <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> et scribi possit<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} = \\displaystyle  \\sum_{j=1}^k (\\vec{x}\\cdot\\hat{v}_j )\\hat{v}_j + \\sum_{j=k+1}^n \\alpha_j \\hat{v}_j <\/span>\n<p>Ita ut expositio superior sequatur hoc modo:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(\\vec{x}-Proy_{H}(\\vec{x}))\\cdot \\vec{z} &amp;= \\displaystyle \\left( \\sum_{j=1}^k (\\vec{x}\\cdot\\hat{v}_j )\\hat{v}_j + \\sum_{j=k+1}^n \\alpha_j \\hat{v}_j\\right) \\cdot  \\sum_{j=1}^k \\beta_j\\hat{v}_j -  \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\cdot  \\sum_{j=1}^k \\beta_j\\hat{v}_j \\\\ \\\\\n\n&amp;=  \\displaystyle \\sum_{j=1}^k (\\vec{x}\\cdot\\hat{v}_j )\\hat{v}_j \\cdot \\sum_{j=1}^k \\beta_j\\hat{v}_j + \\underbrace{\\color{red}{\\sum_{j=k+1}^n \\alpha_j \\hat{v}_j \\cdot \\sum_{j=1}^k \\beta_j\\hat{v}_j}}_{(*)} - \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\cdot  \\sum_{j=1}^k \\beta_j\\hat{v}_j \\\\ \\\\\n\n&amp;=  \\displaystyle \\sum_{j=1}^k (\\vec{x}\\cdot\\hat{v}_j )\\hat{v}_j \\cdot \\sum_{j=1}^k \\beta_j\\hat{v}_j  - \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j \\cdot  \\sum_{j=1}^k \\beta_j\\hat{v}_j \\\\ \\\\\n\n&amp;= 0  \\end{array}<\/span>\n<p>(*) Summa nulla quia <span class=\"katex-eq\" data-katex-display=\"false\">\\{v_j\\}_{j=1}^n<\/span> est basis orthonormalis <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span>\n<p>Ex hoc demonstrare possumus quod distantia inter subspatium <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> et vectorem <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> datur per:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\|<\/span>\n<h4>Demonstratio<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=CGrr6IDnvjs&#038;t=2995s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Ad hunc exitum demonstrandum ostendetur<\/span><\/strong><\/a> quod pro omni <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}\\in H<\/span> semper valebit <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\| \\leq \\|\\vec{x} - \\vec{z}\\|,<\/span> ad hoc utemur theorema Pythagorae hoc modo:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\|\\vec{x} - \\vec{z}\\|^2 &amp;= \\| \\left(\\vec{x} -Proy_{H}(\\vec{x}) \\right) + \\left(Proy_{H}(\\vec{x}) - \\vec{z}\\right)\\|^2 \\\\ \\\\ &amp;= \\| \\vec{x} -Proy_{H}(\\vec{x}) \\|^2 + \\|Proy_{H}(\\vec{x}) - \\vec{z}\\|^2 \\\\ \\\\ \\end{array}<\/span>\n<p>Haec ultima aequalitas obtinetur quia vectores <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x} -Proy_{H}(\\vec{x})<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Proy_{H}(\\vec{x}) - \\vec{z}<\/span> sunt orthogonales. Itaque:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} - Proy_{H}(\\vec{x})\\|^2 \\leq \\|\\vec{x} - \\vec{z}\\|^2<\/span>\n<p>quod erat demonstrandum.<\/p>\n<p>Iam hoc effectu obtento, dicere possumus distantiam inter punctum <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span> et subspatium <span class=\"katex-eq\" data-katex-display=\"false\">H<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> generatum per vectores orthonormales <span class=\"katex-eq\" data-katex-display=\"false\">\\{\\hat{v}_1, \\cdots, \\hat{v}_k\\}<\/span> dari per:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">dist(\\vec{x},H) =\\left\\|\\vec{x} - Proy_{H}(\\vec{x})\\right\\|= \\left\\|\\vec{x} - \\displaystyle \\sum_{j=1}^k (\\vec{x} \\cdot \\hat{v}_j)\\hat{v}_j\\right\\|<\/span>\n<p><a name=\"El-Producto-Escalar-y-Vectorial-en-R3\"><\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/uei6y2tniOc\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h2>Productum Scalaris et Vectorialis in <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/h2>\n<p><strong><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=242s\" rel=\"noopener\" target=\"_blank\"><span style=\"color: #ff0000;\">Nunc parum mutabimus nostrum prospectum<\/span><\/a><\/strong> ut intendamus in vectores <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span> Hic, praeter operationes quas iam in genere de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span> recognovimus, possibile est etiam productum vectoriale quod ex duobus vectoribus resultat in alium vectorem. Hoc est productum proprium <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> (et fortasse <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^7<\/span>, cuius casum hic non examinabimus). Generaliter repraesentantur vectores basis canonicae <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span> per litteras <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}, \\hat{y}, \\hat{z}<\/span> vel ut <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{\\imath}, \\hat{\\jmath}, \\hat{k}<\/span>. Praeferentia unius vel alterius est personalis.<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\hat{\\imath} = \\hat{x}&amp;=(1,0,0)\\\\ \\hat{\\jmath} =\\hat{y}&amp;=(0,1,0)\\\\ \\hat{k} =\\hat{z}&amp;=(0,0,1)\\\\ \\end{array}<\/span>\n<p>Itaque, si habemus vectorem formae <span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c),<\/span> scribi potest in forma algebraica hoc modo:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">(a,b,c) = a\\hat{x} + b\\hat{y} + c\\hat{z}<\/span>\n<h3>Productum vectoriale in <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=330s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Sint <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span> vectores <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span><\/span><\/strong><\/a> Definimus productum vectoriale <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y},<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y}<\/span> per:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl} \\vec{x}\\times\\vec{y} &amp;= \\left|\\begin{array}{ccc} \\hat{x} &amp; \\hat{y} &amp; \\hat{z} \\\\ x_1 &amp; x_2 &amp; x_3 \\\\ y_1 &amp; y_2 &amp; y_3 \\end{array}\\right| \\\\ \\\\ &amp;=\\hat{x}x_2y_3 + \\hat{y}x_3y_1 + \\hat{z} x_1y_2 - \\left( \\hat{z} x_2 y_1 + \\hat{y} x_1 y_3 + \\hat{x}x_3y_2\\right) \\\\ \\\\ &amp;=\\hat{x}(x_2y_3 - x_3y_2) + \\hat{y}(x_3y_1 - x_1y_3) + \\hat{z}(x_1y_2 - x_2y_1) \\end{array}<\/span>\n<h3>Identitas Lagrange<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=1399s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Pro casu vectorum <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3<\/span><\/span><\/strong><\/a> possumus agnoscere tria genera \u00abproductorum\u00bb: Scalaris <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y},<\/span> vectorialis <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\times\\vec{y},<\/span> et normarum <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|\\|\\vec{y}\\|.<\/span> Haec tria producta inter se connectuntur per identitatem Lagrange<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\|^2  = \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2- (\\vec{x}\\cdot\\vec{y})^2 <\/span>\n<h4>Demonstratio identitatis Lagrange<\/h4>\n<p>Sint <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,x_2,x_3)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,y_2,y_3)<\/span> vectores <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3,<\/span> tunc habemus:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl} \\vec{x}\\times\\vec{y} &amp;=(x_2y_3 - x_3y_2) \\hat{x} + (x_3y_1 - x_1y_3)\\hat{y} + (x_1y_2 - x_2y_1)\\hat{z} \\end{array}<\/span>\n<p>Quare:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl}\n\n\\|\\vec{x}\\times\\vec{y}\\|^2 &amp;=(x_2y_3 - x_3y_2)^2 + (x_3y_1 - x_1y_3)^2 + (x_1y_2 - x_2y_1)^2 \\\\ \\\\\n\n&amp;= \\color{green}{x_2^2y_3^2 - 2x_2x_3y_3y_2 + x_3^2y_2^2} + \\cdots\\\\ \\\\\n\n&amp;\\cdots + \\color{blue}{x_3^2y_1^2 - 2x_3x_1y_1y_3 + x_1^2y_3^2} + \\cdots \\\\ \\\\\n\n&amp;\\cdots + \\color{red}{x_1^2y_2^2 - 2x_1x_2y_2y_1 + x_2^2y_1^2} \\end{array}<\/span>\n<p>Altera ex parte:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rl}\n\n\\|\\vec{x}\\|^2 \\|\\vec{y}\\|^2 - (\\vec{x}\\cdot\\vec{y})^2 &amp;= (x_1^2 + x_2^2 + x_3^2)(y_1^2+y_2^2 + y_3^2) - (x_1y_1 + x_2y_2 + x_3 y_3)^2 \\\\ \\\\ \\\\\n\n&amp;=  {x_1^2y_1^2} + \\color{red}{x_1^2y_2^2} + \\color{blue}{x_1^2y_3^2} + \\cdots \\\\ \\\\\n\n&amp;\\cdots + \\color{red}{x_2^2y_1^2} +  {x_2^2y_2^2} + \\color{green}{x_2^2y_3^2} + \\cdots \\\\ \\\\\n\n&amp;\\cdots + \\color{blue}{x_3^2y_1^2} + \\color{green}{x_3^2y_2^2} +  {x_3^2y_3^2} + \\cdots \\\\ \\\\\n\n&amp;\\cdots - \\left[ {x_1^2y_1^2} +  {x_2^2y_2^2} +  {x_3^2y_3^2} + \\right. \\cdots \\\\ \\\\\n\n&amp;\\cdots + 2\\left(\\color{red}{x_1x_2y_1y_2} + \\color{blue}{x_1x_3y_1y_3} + \\color{green}{x_2x_3y_2y_3} \\right)\\left.\\right] \\\\ \\\\ \\\\\n\n&amp;= \\color{red}{x_1^2y_2^2 - 2x_1x_2y_2y_1 + x_2^2y_1^2} + \\cdots \\\\ \\\\\n\n&amp; \\cdots + \\color{blue}{x_1^2y_3^2 - 2x_1x_3y_3y_1 + x_3^2y_1^2} + \\cdots \\\\ \\\\\n\n&amp; \\cdots + \\color{green}{x_2^2y_3^2 - 2x_2x_3y_3y_2 + x_3^2y_2^2}\n\n\\end{array}<\/span>\n<p>Denique, comparando expressiones coloribus signatas habetur quod erat demonstrandum.<\/p>\n<h3>Productum Crucis et angulus inter vectores<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=uei6y2tniOc&#038;t=1954s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Superius vidimus exstare arctam relationem<\/span><\/strong><\/a> inter angulum a duobus vectoribus sustentum et effectum producti scalaris, quod datur relatione <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = \\|\\vec{x}\\|\\|\\vec{y}\\|\\cos(\\angle(\\vec{x},\\vec{y})).<\/span> Fit autem aliquid simile cum producto vectoriali, et datur relatione sequenti:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\| = \\|\\vec{x}\\|\\|\\vec{y}\\| \\sin(\\angle(\\vec{x},\\vec{y}))<\/span>\n<p>Haec expressio est effectus directus identitatis Lagrange supra demonstratae, demonstratio sic se habet:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\|\\vec{x}\\times\\vec{y}\\|^2 &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 - (\\vec{x}\\cdot\\vec{y})^2 \\\\ \\\\ &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 - (\\|\\vec{x}\\|\\|\\vec{y}\\|\\cos(\\angle(\\vec{x},\\vec{y})))^2 \\\\ \\\\ &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 - \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2\\cos^2(\\angle(\\vec{x},\\vec{y})) \\\\ \\\\ &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 (1 - \\cos^2(\\angle(\\vec{x},\\vec{y}))) \\\\ \\\\ &amp;= \\|\\vec{x}\\|^2\\|\\vec{y}\\|^2 \\sin^2(\\angle(\\vec{x},\\vec{y})) \\end{array}<\/span>\n<p>Denique, radices sumendo pervenimus ad:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\| = \\|\\vec{x}\\|\\|\\vec{y}\\|\\; |\\sin(\\angle(\\vec{x},\\vec{y}))|<\/span>\n<p>At meminerimus <span class=\"katex-eq\" data-katex-display=\"false\">\\angle(\\vec{x},\\vec{y})\\in[0,\\pi],<\/span> et in hoc intervallo valorum functio sinus semper est non-negativa, ita removere possumus valorem absolutum et pervenimus ad id quod erat demonstrandum.<\/p>\n<p>Ex hac expressione possumus intueri quod effectus operationis <span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\times\\vec{y}\\|<\/span> dat nobis aream generatam a vectoribus <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}.<\/span>\n","protected":false},"excerpt":{"rendered":"<p>Algebra et Projectiones in Rn, Productum Vectoriale in Summarium:Haec series est continuatio directa seriei de Spatio Euclidiano dimensionis n. Hic recognoscemus quaedam notiones algebrae linearis quae adiuvant melius comprehendere spatium euclidianum n-dimensionale, recognoscemus notiones projectionum vectoris in alium, demonstraemus theorema Pythagorae et concludetur cum recognitione producti vectorialis in eiusque relatione cum aliis productis spatii Euclidiani [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":34241,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[1340,1298],"tags":[],"class_list":["post-34269","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculus-multivariabilis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Algebra et Projectiones in Rn, Productum Vectoriale in R3 - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explora projectiones et vectores Rn: independentia linearis, orthogonalitas, theorema Pythagorae et productum vectoriale in spatiis euclidianis.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Algebra et Projectiones in Rn, Productum Vectoriale in R3\" \/>\n<meta property=\"og:description\" content=\"Explora projectiones et vectores Rn: independentia linearis, orthogonalitas, theorema Pythagorae et productum vectoriale in spatiis euclidianis.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2022-03-29T13:00:52+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-08-27T22:00:56+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn-1024x498.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Algebra et Projectiones in Rn, Productum Vectoriale in R3\" \/>\n<meta name=\"twitter:description\" content=\"Explora projectiones et vectores Rn: independentia linearis, orthogonalitas, theorema Pythagorae et productum vectoriale in spatiis euclidianis.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Algebra et Projectiones in Rn, Productum Vectoriale in R3\",\"datePublished\":\"2022-03-29T13:00:52+00:00\",\"dateModified\":\"2025-08-27T22:00:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/\"},\"wordCount\":3729,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg\",\"articleSection\":[\"Calculus Multivariabilis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/\",\"url\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/\",\"name\":\"Algebra et Projectiones in Rn, Productum Vectoriale in R3 - toposuranos.com\/material\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg\",\"datePublished\":\"2022-03-29T13:00:52+00:00\",\"dateModified\":\"2025-08-27T22:00:56+00:00\",\"description\":\"Explora projectiones et vectores Rn: independentia linearis, orthogonalitas, theorema Pythagorae et productum vectoriale in spatiis euclidianis.\",\"breadcrumb\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#primaryimage\",\"url\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg\",\"contentUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg\",\"width\":1536,\"height\":747},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Algebra et Projectiones in Rn, Productum Vectoriale in R3\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/toposuranos.com\/material\/#website\",\"url\":\"https:\/\/toposuranos.com\/material\/\",\"name\":\"toposuranos.com\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/toposuranos.com\/material\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\",\"name\":\"toposuranos.com\/material\",\"url\":\"https:\/\/toposuranos.com\/material\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\/material\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/groups\/toposuranos\",\"https:\/\/x.com\/topuranos\",\"https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\/\/www.linkedin.com\/company\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\/\/toposuranos.com\/material\"],\"url\":\"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Algebra et Projectiones in Rn, Productum Vectoriale in R3 - toposuranos.com\/material","description":"Explora projectiones et vectores Rn: independentia linearis, orthogonalitas, theorema Pythagorae et productum vectoriale in spatiis euclidianis.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/","og_locale":"es_ES","og_type":"article","og_title":"Algebra et Projectiones in Rn, Productum Vectoriale in R3","og_description":"Explora projectiones et vectores Rn: independentia linearis, orthogonalitas, theorema Pythagorae et productum vectoriale in spatiis euclidianis.","og_url":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2022-03-29T13:00:52+00:00","article_modified_time":"2025-08-27T22:00:56+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn-1024x498.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Algebra et Projectiones in Rn, Productum Vectoriale in R3","twitter_description":"Explora projectiones et vectores Rn: independentia linearis, orthogonalitas, theorema Pythagorae et productum vectoriale in spatiis euclidianis.","twitter_image":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#article","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/"},"author":{"name":"giorgio.reveco","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Algebra et Projectiones in Rn, Productum Vectoriale in R3","datePublished":"2022-03-29T13:00:52+00:00","dateModified":"2025-08-27T22:00:56+00:00","mainEntityOfPage":{"@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/"},"wordCount":3729,"commentCount":0,"publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg","articleSection":["Calculus Multivariabilis","Mathematica"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/","url":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/","name":"Algebra et Projectiones in Rn, Productum Vectoriale in R3 - toposuranos.com\/material","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#primaryimage"},"image":{"@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg","datePublished":"2022-03-29T13:00:52+00:00","dateModified":"2025-08-27T22:00:56+00:00","description":"Explora projectiones et vectores Rn: independentia linearis, orthogonalitas, theorema Pythagorae et productum vectoriale in spatiis euclidianis.","breadcrumb":{"@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#primaryimage","url":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg","contentUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectoresyproyeccionesrn.jpg","width":1536,"height":747},{"@type":"BreadcrumbList","@id":"https:\/\/toposuranos.com\/material\/la\/algebra-et-projectiones-in-rn-productum-vectoriale-in-r3\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Algebra et Projectiones in Rn, Productum Vectoriale in R3"}]},{"@type":"WebSite","@id":"https:\/\/toposuranos.com\/material\/#website","url":"https:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"https:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"https:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/34269","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=34269"}],"version-history":[{"count":0,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/34269\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/34241"}],"wp:attachment":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=34269"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=34269"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=34269"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}