{"id":253,"date":"2023-07-10T11:53:09","date_gmt":"2023-07-10T11:53:09","guid":{"rendered":"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/"},"modified":"2023-07-10T11:53:09","modified_gmt":"2023-07-10T11:53:09","slug":"angolo-tra-due-piani-nella-formula-spaziale-r3","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/","title":{"rendered":"Angolo tra due piani nello spazio (formula)"},"content":{"rendered":"<p>In questa pagina troverai come calcolare l&#8217;angolo formato da due piani nello spazio (formula). Inoltre, potrai vedere esempi ed esercitarti con esercizi risolti. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-104\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-angulo-entre-dos-planos\"><\/span> Formula dell&#8217;angolo tra due piani<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>L&#8217;angolo tra due piani \u00e8 uguale all&#8217;angolo formato dai vettori normali di detti piani. Pertanto, per trovare l&#8217;angolo tra due piani, si calcola l&#8217;angolo formato dai loro vettori normali, poich\u00e9 sono equivalenti.<\/strong> <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<p> Quindi, una volta che sappiamo esattamente qual \u00e8 l&#8217;angolo tra due piani, diamo un&#8217;occhiata alla formula per calcolare l&#8217;angolo tra due piani nello spazio (in R3), che si deduce dalla <a href=\"https:\/\/mathority.org\/it\/come-calcolare-langolo-tra-due-vettori-esempi-esercizi-risolti\/\">formula per l&#8217;angolo tra due vettori<\/a> : <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Data l&#8217;equazione generale (o implicita) di due piani diversi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dfa3d7e6f1ece8353327be7c9227d75b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ A_1x+B_1y+C_1z+D_1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"249\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c3966346685421fe3e535cf57a5491d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ A_2x+B_2y+C_2z+D_2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"249\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Il vettore normale di ciascun piano \u00e8:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb0ca06882e0d61d6f8134368946ef29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1=(A_1,B_1,C_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-22fba6a063a544bdf257e64d8d139238_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2=(A_2,B_2,C_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> E l&#8217;angolo formato da questi due piani si determina calcolando l&#8217;angolo formato dai loro vettori normali utilizzando la seguente formula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<\/div>\n<p> Quindi, per determinare l&#8217;angolo tra due piani, devi padroneggiare il calcolo del <a href=\"https:\/\/mathority.org\/it\/calcolare-il-prodotto-scalare-tra-due-vettori-esempi-esercizi-risolti\/\">prodotto scalare di due vettori<\/a> . Se non ricordi come \u00e8 stato fatto, nel link troverai i passaggi per risolvere il prodotto scalare tra due vettori. Inoltre, potrai vedere esempi ed esercizi risolti passo dopo passo.<\/p>\n<p> Quando invece i due piani sono perpendicolari o paralleli, non \u00e8 necessario applicare la formula, perch\u00e9 l&#8217;angolo tra i 2 piani pu\u00f2 essere determinato direttamente:<\/p>\n<ul>\n<li> L&#8217;angolo tra due <strong>piani paralleli<\/strong> \u00e8 0\u00ba, poich\u00e9 i loro vettori normali hanno la stessa direzione.<\/li>\n<li> L&#8217;angolo tra due <strong>piani perpendicolari<\/strong> \u00e8 90\u00ba, perch\u00e9 anche i loro vettori normali sono perpendicolari (o ortogonali) tra loro e quindi formano un angolo retto. <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-calcular-el-angulo-entre-dos-planos\"><\/span> Esempio di calcolo dell&#8217;angolo tra due piani <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<p> Ecco un esempio concreto per vedere come determinare l&#8217;angolo tra due piani diversi:<\/p>\n<ul>\n<li> Calcola l&#8217;angolo tra i due piani seguenti:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-60b094e253552cbfa84175e60ef18801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ 3x-5y+z+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"191\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-364b525ebfd0b5ef85128562d1641cb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ 4x+2y+3z-1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"200\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> La prima cosa che dobbiamo fare \u00e8 trovare il vettore normale di ciascun piano. Pertanto, le coordinate X, Y, Z del vettore perpendicolare a un piano coincidono rispettivamente con i coefficienti A, B e C della sua equazione generale (o implicita):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-17c8988d3ee87dd9e708e46aadbb9086_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1 = (3,-5,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5783f55ed327128d1da574f38c8336e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2 = (4,2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E una volta conosciuto il vettore normale a ciascun piano, calcoliamo l&#8217;angolo che formano con la formula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Dobbiamo quindi trovare il modulo di ciascun vettore normale:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96718e6e72f0f3045ee39364f636b419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{3^2+(-5)^2+1^2}= \\sqrt{9+25+1} = \\sqrt{35}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e3558cfa79b7113b469df5b36e00f1ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4^2+2^2+3^2}= \\sqrt{16+4+9} = \\sqrt{29}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"280\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Ora sostituiamo il valore di ciascuna incognita nella formula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5cdc9e54b22394cf29f35c6c26ee1308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}=\\cfrac{\\lvert (3,-5,1) \\cdot (4,2,3)\\rvert}{\\sqrt{35} \\cdot \\sqrt{29} }\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"318\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Calcoliamo il coseno dell&#8217;angolo risolvendo il prodotto scalare dei due vettori:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1923cb6ac441ba0e980bc984bc4804b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert 3\\cdot 4 + (-5)\\cdot 2 +1 \\cdot 3 \\rvert}{\\sqrt{35}\\cdot \\sqrt{29} }=\\cfrac{\\lvert 12-10+3 \\rvert}{\\sqrt{1015}}= \\cfrac{5}{31,86}=0,16\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"497\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> E, infine, determiniamo l&#8217;angolo facendo l&#8217;inverso del coseno usando la calcolatrice: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-57297e96c6a14eed6ff87abfe7699df5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\alpha = \\cos^{-1}(0,16)=\\bm{80,97\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-del-angulo-entre-dos-planos\"><\/span> Risolti problemi dell&#8217;angolo tra due piani <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Trova l&#8217;angolo tra i due piani seguenti: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac5eb1f4c801eae9ea93342c56e7aa60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ x+2z-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"151\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f7b262c6e50b562858b1c5043548ba75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ 3x+y-4z+7=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"191\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La prima cosa che dobbiamo fare \u00e8 trovare il vettore normale di ciascun piano. Pertanto, le coordinate X, Y, Z del vettore perpendicolare a un piano sono rispettivamente equivalenti ai coefficienti A, B e C della sua equazione generale (o implicita): <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef6a043fab595442ed8b17da7798fc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1 = (1,0,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-158dadd8329a3954114f9d99160e9800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2 = (3,1,-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Una volta conosciuto il vettore normale di ciascun piano, calcoliamo l&#8217;angolo che formano con la formula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dobbiamo quindi trovare il modulo di ciascun vettore normale: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-303393672cb4ca6c891bfcf49d3e30b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{1^2+0^2+2^2}= \\sqrt{1+4} = \\sqrt{5}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"232\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede42afce2c7b11dd96617df978ee401_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{3^2+1^2+(-4)^2}= \\sqrt{9+1+16} = \\sqrt{26}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sostituiamo il valore di ciascuna incognita nella formula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-90594e9bdf100d52e0221a86d0878d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}=\\cfrac{\\lvert (1,0,2) \\cdot (3,1,-4)\\rvert}{\\sqrt{5} \\cdot \\sqrt{26} }\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"318\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo il coseno dell&#8217;angolo:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fb411278d0746c45c8ef0490e51862b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert 1\\cdot 3 + 0\\cdot 1 +2 \\cdot (-4) \\rvert}{\\sqrt{5}\\cdot \\sqrt{26} }=\\cfrac{\\lvert 3-8 \\rvert}{\\sqrt{130}}= \\cfrac{5}{11,4}=0,44\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"440\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E infine, troviamo l&#8217;angolo tra i due piani invertendo il coseno con la calcolatrice: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa9117c27a8c072f275a8cb5fac99528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\alpha = \\cos^{-1}(0,44)=\\bm{63,99\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 2<\/h3>\n<p> Qual \u00e8 l&#8217;angolo formato dai seguenti due piani? <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7566850c5f664f60ba826bb1ffbf7a3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ 3x-2y+5z=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ead661dfbeded4be15602ccda3f2864c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ 6x+3y-z-2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"191\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La prima cosa che dobbiamo fare \u00e8 trovare il vettore normale di ciascun piano. Pertanto, le coordinate X, Y, Z del vettore perpendicolare a un piano sono rispettivamente uguali ai parametri A, B e C della sua equazione generale (o implicita): <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-54788d8fdba6722dc87ae3c06c350400_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1 = (3,-2,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6dee74c50bac2e8859c3f4a3c9742f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2 = (6,3,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Una volta conosciuto il vettore normale di ciascun piano, calcoliamo l&#8217;angolo che formano con la formula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dobbiamo quindi trovare il modulo di ciascun vettore normale: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06daed2734e23937ffc63d52314656ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{3^2+(-2)^2+5^2}= \\sqrt{9+4+25} = \\sqrt{38}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0f0bafe4966148f6e199dbd2bdc7ede1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{6^2+3^2+(-1)^2}= \\sqrt{36+9+1} = \\sqrt{46}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sostituiamo il valore di ciascuna variabile nella formula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d635cbe5718a970df439157ca6346cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}=\\cfrac{\\lvert (3,-2,5) \\cdot (6,3,-1)\\rvert}{\\sqrt{38} \\cdot \\sqrt{46} }\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"332\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo il coseno dell&#8217;angolo:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a24e1ac8ad5f23b1d0734a564fa92a90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert 3\\cdot 6 + (-2)\\cdot 3 +5 \\cdot (-1) \\rvert}{\\sqrt{38}\\cdot \\sqrt{46} }=\\cfrac{\\lvert 18-6-5 \\rvert}{\\sqrt{1748}}= \\cfrac{7}{41,81}=0,17\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"516\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, infine, determiniamo l&#8217;angolo invertendo il coseno con la calcolatrice: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8fddc88af864b95ff37c56b126af7a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\alpha = \\cos^{-1}(0,17)=\\bm{80,36\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 3<\/h3>\n<p> Calcolare il valore del parametro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> in modo che i seguenti due piani siano perpendicolari: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-faed0ccf5807a84f20057b0128bbcee5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ x+2y-3z+1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"191\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9872d7160fb07e1a3ca43e18e24b3435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ -2x+5y+kz+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"215\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Innanzitutto per calcolare gli angoli tra i piani bisogna sempre trovare il vettore normale di ciascun piano: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-add7176bfadc88b0c27dd09e33b340cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1 = (1,2,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a3f04e685fb9ce2300340f149aa93059_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2 = (-2,5,k)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Due piani perpendicolari formano un angolo di 90\u00ba, quindi anche i loro vettori normali saranno di 90\u00ba. Possiamo quindi determinare il valore dell&#8217;incognita.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> con la formula per l&#8217;angolo tra due vettori: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a35dde3971efe5cdf36340847dcb02e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(90\u00ba) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"151\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb4652afcc23693b300da306909293ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"104\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il denominatore della frazione divide l&#8217;intero lato destro dell&#8217;equazione, quindi possiamo passarlo moltiplicando dall&#8217;altro lato: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f94a174ae36525a6e6ef055e53bb154a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 \\cdot \\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert =\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e4cffd0cb6eef3c14b8924706cca4a43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =\\vv{n}_1 \\cdot \\vv{n}_2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Risolviamo ora il prodotto scalare tra i due vettori normali: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3818384c903fe4f03fd785f0fbfb0197_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =(1,2,-3) \\cdot (-2,5,k)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a0317a3a3ce7f7a3b19d199026305b93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =1 \\cdot (-2) + 2\\cdot 5 +(-3)\\cdot k\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"223\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9a5497303548b34fd3bba83bdc588b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-2 +10-3k\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"134\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3f68c05212b2541bc6d1142287eebc18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =8-3k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"81\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, infine, chiariamo l&#8217;incognita: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c13e299367327b12f0f781a10c47f12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 3k=8\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b126ac6d6422d2ae3c536589c2461e2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{k =}\\mathbf{\\cfrac{8}{3}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"41\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In questa pagina troverai come calcolare l&#8217;angolo formato da due piani nello spazio (formula). Inoltre, potrai vedere esempi ed esercitarti con esercizi risolti. Formula dell&#8217;angolo tra due piani L&#8217;angolo tra due piani \u00e8 uguale all&#8217;angolo formato dai vettori normali di detti piani. Pertanto, per trovare l&#8217;angolo tra due piani, si calcola l&#8217;angolo formato dai loro &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/\"> <span class=\"screen-reader-text\">Angolo tra due piani nello spazio (formula)<\/span> Leggi altro &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[15],"tags":[],"class_list":["post-253","post","type-post","status-publish","format-standard","hentry","category-punti-rette-e-piani"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Angolo tra due piani nello spazio (formula) - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Angolo tra due piani nello spazio (formula) - Mathority\" \/>\n<meta property=\"og:description\" content=\"In questa pagina troverai come calcolare l&#8217;angolo formato da due piani nello spazio (formula). Inoltre, potrai vedere esempi ed esercitarti con esercizi risolti. Formula dell&#8217;angolo tra due piani L&#8217;angolo tra due piani \u00e8 uguale all&#8217;angolo formato dai vettori normali di detti piani. Pertanto, per trovare l&#8217;angolo tra due piani, si calcola l&#8217;angolo formato dai loro &hellip; Angolo tra due piani nello spazio (formula) Leggi altro &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-10T11:53:09+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dfa3d7e6f1ece8353327be7c9227d75b_l3.png\" \/>\n<meta name=\"author\" content=\"Squadra di Mathority\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Scritto da\" \/>\n\t<meta name=\"twitter:data1\" content=\"Squadra di Mathority\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tempo di lettura stimato\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minuti\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/\"},\"author\":{\"name\":\"Squadra di Mathority\",\"@id\":\"https:\/\/mathority.org\/it\/#\/schema\/person\/8d6f69ffbe48aea8b43675a9a3ddb9c8\"},\"headline\":\"Angolo tra due piani nello spazio (formula)\",\"datePublished\":\"2023-07-10T11:53:09+00:00\",\"dateModified\":\"2023-07-10T11:53:09+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/\"},\"wordCount\":701,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/it\/#organization\"},\"articleSection\":[\"Punti, rette e piani\"],\"inLanguage\":\"it-IT\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/\",\"url\":\"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/\",\"name\":\"Angolo tra due piani nello spazio (formula) - 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Mathority","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:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/","og_locale":"it_IT","og_type":"article","og_title":"Angolo tra due piani nello spazio (formula) - Mathority","og_description":"In questa pagina troverai come calcolare l&#8217;angolo formato da due piani nello spazio (formula). Inoltre, potrai vedere esempi ed esercitarti con esercizi risolti. Formula dell&#8217;angolo tra due piani L&#8217;angolo tra due piani \u00e8 uguale all&#8217;angolo formato dai vettori normali di detti piani. Pertanto, per trovare l&#8217;angolo tra due piani, si calcola l&#8217;angolo formato dai loro &hellip; Angolo tra due piani nello spazio (formula) Leggi altro &raquo;","og_url":"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/","article_published_time":"2023-07-10T11:53:09+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dfa3d7e6f1ece8353327be7c9227d75b_l3.png"}],"author":"Squadra di Mathority","twitter_card":"summary_large_image","twitter_misc":{"Scritto da":"Squadra di Mathority","Tempo di lettura stimato":"3 minuti"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/"},"author":{"name":"Squadra di Mathority","@id":"https:\/\/mathority.org\/it\/#\/schema\/person\/8d6f69ffbe48aea8b43675a9a3ddb9c8"},"headline":"Angolo tra due piani nello spazio (formula)","datePublished":"2023-07-10T11:53:09+00:00","dateModified":"2023-07-10T11:53:09+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/"},"wordCount":701,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/it\/#organization"},"articleSection":["Punti, rette e piani"],"inLanguage":"it-IT","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/","url":"https:\/\/mathority.org\/it\/angolo-tra-due-piani-nella-formula-spaziale-r3\/","name":"Angolo tra due piani nello spazio (formula) - 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