{"id":2039,"date":"2026-10-14T06:41:43","date_gmt":"2026-10-14T05:41:43","guid":{"rendered":"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-projecao-de-vetores-3d\/"},"modified":"2026-10-14T06:41:43","modified_gmt":"2026-10-14T05:41:43","slug":"calculadora-de-projecao-de-vetores-3d","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-projecao-de-vetores-3d\/","title":{"rendered":"Calculadora de proje\u00e7\u00e3o de vetores 3D"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Projetar um vetor 3D sobre outro<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2039\" data-favorite-title=\"Proje\u00e7\u00e3o de vetores 3D\" data-favorite-url=\"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-projecao-de-vetores-3d\/\" data-favorite-excerpt=\"Digite as componentes de dois vetores 3D e esta calculadora projeta um sobre o outro, mostrando a proje\u00e7\u00e3o escalar, a proje\u00e7\u00e3o vetorial, a componente perpendicular, o produto escalar, os m\u00f3dulos e o \u00e2ngulo entre os vetores.\" aria-label=\"Adicionar aos favoritos\" data-favorite-icon=\"icon icon-geometry\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-projecao-de-vetores-3d\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2039\" data-post=\"2039\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Campos de entrada<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Vetor a \u2014 componentes<\/span><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Componente x (a\u2093)<\/label><div class=\"term\">a\u2093<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"componente_x_a\" id=\"field-2\" placeholder=\"2\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Componente y (a\u1d67)<\/label><div class=\"term\">a\u1d67<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"componente_y_a\" id=\"field-3\" placeholder=\"3\" step=\"any\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Componente z (a_z)<\/label><div class=\"term\">a_z<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"componente_z_az\" id=\"field-4\" placeholder=\"1\" step=\"any\" value=\"1\"\/><\/div><div class=\"is-single field\" id=\"input-5\"><span class=\"font-semibold\">Vetor b \u2014 componentes<\/span><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">Componente x (b\u2093)<\/label><div class=\"term\">b\u2093<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"componente_x_b\" id=\"field-6\" placeholder=\"1\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">Componente y (b\u1d67)<\/label><div class=\"term\">b\u1d67<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"componente_y_b\" id=\"field-7\" placeholder=\"0\" step=\"any\" value=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-8\"><label for=\"field-8\">Componente z (b_z)<\/label><div class=\"term\">b_z<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"componente_z_bz\" id=\"field-8\" placeholder=\"2\" step=\"any\" value=\"2\"\/><\/div><div class=\"is-single field\" id=\"input-9\"><span class=\"font-semibold\">Dire\u00e7\u00e3o da proje\u00e7\u00e3o<\/span><\/div><div class=\"field is-single\" id=\"input-10\"><label for=\"field-10\">Projetar<\/label><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"projetar\"><option value=\"ab\" selected>a sobre b<\/option><option value=\"ba\">b sobre a<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalcular\" id=\"calculate-button\" data-post=\"2039\">Calcular<\/button><button type=\"reset\">Redefinir<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calcular automaticamente<\/label><small>Se ativado, o resultado \u00e9 atualizado automaticamente quando voc\u00ea altera qualquer valor.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Sua proje\u00e7\u00e3o<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Proje\u00e7\u00e3o escalar<\/span><span class=\"result-term\">comp<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Proje\u00e7\u00e3o vetorial<\/span><span class=\"result-term\">proj<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>Componente perpendicular<\/span><span class=\"result-term\">rej<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Produto escalar<\/span><span class=\"result-term\">a\u00b7b<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>M\u00f3dulo de a<\/span><span class=\"result-term\">|a|<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-6\"><span class=\"field-title\"><span>M\u00f3dulo de b<\/span><span class=\"result-term\">|b|<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-7\"><span class=\"field-title\"><span>\u00c2ngulo entre os vetores<\/span><span class=\"result-term\">\u03b8<\/span><\/span><span class=\"field-value\" id=\"result-7\" data-unit=\"\u00b0\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-8\"><span class=\"field-title\"><span>Etapas do c\u00e1lculo<\/span><\/span><span class=\"field-value\" id=\"result-8\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copiar link para este c\u00e1lculo<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Deixar um coment\u00e1rio<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"2039\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">O que voc\u00ea achou deste resultado?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Curti <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">\u00datil <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confuso <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Decepcionado <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Impreciso <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>F\u00f3rmula da proje\u00e7\u00e3o de vetores 3D<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">F\u00f3rmula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ \\operatorname{proj}_{\\mathbf{b}} \\mathbf{a} = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert \\mathbf{b} \\rvert^2}\\, \\mathbf{b} \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Onde:<\/strong><\/p>\n<ul>\n<li>\\(\\mathbf{a} = (a_x, a_y, a_z)\\) \u2014 o vetor que ser\u00e1 projetado<\/li>\n<li>\\(\\mathbf{b} = (b_x, b_y, b_z)\\) \u2014 o vetor que fornece a dire\u00e7\u00e3o<\/li>\n<li>\\(\\mathbf{a} \\cdot \\mathbf{b} = a_x b_x + a_y b_y + a_z b_z\\) \u2014 o produto escalar<\/li>\n<li>\\(\\lvert \\mathbf{b} \\rvert = \\sqrt{b_x^2 + b_y^2 + b_z^2}\\) \u2014 o m\u00f3dulo (comprimento) de \\(\\mathbf{b}\\)<\/li>\n<\/ul>\n<p>A f\u00f3rmula expressa quanto de \\(\\mathbf{a}\\) aponta na dire\u00e7\u00e3o de \\(\\mathbf{b}\\). A fra\u00e7\u00e3o \\(k = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert \\mathbf{b} \\rvert^2}\\) escala \\(\\mathbf{b}\\) de modo que o resultado \u00e9 a sombra que \\(\\mathbf{a}\\) projeta sobre a reta que passa por \\(\\mathbf{b}\\).<\/p>\n<p>Da mesma proje\u00e7\u00e3o saem duas grandezas relacionadas:<\/p>\n<p>\\[ \\operatorname{comp}_{\\mathbf{b}} \\mathbf{a} = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert \\mathbf{b} \\rvert} \\qquad \\operatorname{perp}_{\\mathbf{b}} \\mathbf{a} = \\mathbf{a} - \\operatorname{proj}_{\\mathbf{b}} \\mathbf{a} \\]<\/p>\n<p>A proje\u00e7\u00e3o escalar (componente) \u00e9 o comprimento com sinal da sombra, e a componente perpendicular (rejei\u00e7\u00e3o) \u00e9 a parte de \\(\\mathbf{a}\\) que sobra \u2014 sempre ortogonal a \\(\\mathbf{b}\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Proje\u00e7\u00e3o de vetores 3D \u2013 exemplo de c\u00e1lculo<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemplo:<\/strong> projetar \\(\\mathbf{a} = (2, 3, 1)\\) sobre \\(\\mathbf{b} = (1, 0, 2)\\) (os valores padr\u00e3o da calculadora).<\/p>\n<p>\\[ \\mathbf{a} \\cdot \\mathbf{b} = 2 \\cdot 1 + 3 \\cdot 0 + 1 \\cdot 2 = 4 \\]<\/p>\n<p>\\[ \\lvert \\mathbf{a} \\rvert = \\sqrt{2^2 + 3^2 + 1^2} = \\sqrt{14} \\approx 3.742, \\qquad \\lvert \\mathbf{b} \\rvert = \\sqrt{1^2 + 0^2 + 2^2} = \\sqrt{5} \\approx 2.236 \\]<\/p>\n<p>\\[ k = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert \\mathbf{b} \\rvert^2} = \\frac{4}{5} = 0.8 \\]<\/p>\n<p>\\[ \\operatorname{proj}_{\\mathbf{b}} \\mathbf{a} = 0.8 \\cdot (1, 0, 2) = (0.8,\\ 0,\\ 1.6) \\]<\/p>\n<p>\\[ \\operatorname{perp}_{\\mathbf{b}} \\mathbf{a} = (2, 3, 1) - (0.8, 0, 1.6) = (1.2,\\ 3,\\ -0.6) \\]<\/p>\n<p><strong>Resultado:<\/strong> a proje\u00e7\u00e3o escalar \u00e9 \\(4 \/ \\sqrt{5} \\approx 1.789\\), a proje\u00e7\u00e3o vetorial \u00e9 \\((0.8, 0, 1.6)\\) e a componente perpendicular \u00e9 \\((1.2, 3, -0.6)\\), com m\u00f3dulo \\(\\approx 3.286\\). O \u00e2ngulo entre os vetores \u00e9 \\(\\arccos\\!\\left(\\tfrac{4}{\\sqrt{70}}\\right) \\approx 61.4\u00b0\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>Como usar a calculadora<\/h2>\n<ol>\n<li>Digite a <strong>componente x (a\u2093)<\/strong>, a <strong>componente y (a\u1d67)<\/strong> e a <strong>componente z (a_z)<\/strong> do primeiro vetor em \"Vetor a \u2014 componentes\".<\/li>\n<li>Digite as tr\u00eas componentes do segundo vetor em \"Vetor b \u2014 componentes\". Use 0 para qualquer eixo ausente.<\/li>\n<li>Escolha a op\u00e7\u00e3o de <strong>Projetar<\/strong>: \"a sobre b\" determina quanto de \\(\\mathbf{a}\\) est\u00e1 na dire\u00e7\u00e3o de \\(\\mathbf{b}\\); \"b sobre a\" inverte os pap\u00e9is.<\/li>\n<li>Leia os resultados: a proje\u00e7\u00e3o escalar, a proje\u00e7\u00e3o vetorial, a componente perpendicular, al\u00e9m do produto escalar, dos dois m\u00f3dulos, do \u00e2ngulo entre os vetores e do c\u00e1lculo passo a passo.<\/li>\n<\/ol>\n<h2>O que \u00e9 a proje\u00e7\u00e3o de um vetor 3D?<\/h2>\n<p>Geometricamente, a proje\u00e7\u00e3o do vetor \\(\\mathbf{a}\\) sobre o vetor \\(\\mathbf{b}\\) \u00e9 a sombra que \\(\\mathbf{a}\\) projeta na reta que passa por \\(\\mathbf{b}\\), como se a luz incidisse perpendicularmente a essa reta. O resultado \u00e9 um novo vetor que aponta exatamente na mesma dire\u00e7\u00e3o (ou exatamente na dire\u00e7\u00e3o oposta) de \\(\\mathbf{b}\\), com comprimento igual ao quanto \\(\\mathbf{a}\\) se estende nessa dire\u00e7\u00e3o. Isso tamb\u00e9m \u00e9 chamado de componente vetorial de \\(\\mathbf{a}\\) ao longo de \\(\\mathbf{b}\\).<\/p>\n<p>Toda proje\u00e7\u00e3o decomp\u00f5e um vetor em duas partes ortogonais: a pr\u00f3pria proje\u00e7\u00e3o (paralela a \\(\\mathbf{b}\\)) e a componente perpendicular, frequentemente chamada de <em>rejei\u00e7\u00e3o<\/em>. As duas partes sempre somam de volta o vetor original: \\(\\mathbf{a} = \\operatorname{proj}_{\\mathbf{b}} \\mathbf{a} + \\operatorname{perp}_{\\mathbf{b}} \\mathbf{a}\\), e o produto escalar entre elas \u00e9 zero. Essa decomposi\u00e7\u00e3o \u00e9 a base do processo de Gram\u2013Schmidt, da resolu\u00e7\u00e3o de for\u00e7as em componentes na f\u00edsica e do ajuste por m\u00ednimos quadrados na estat\u00edstica.<\/p>\n<p>A <strong>proje\u00e7\u00e3o escalar<\/strong> responde \u00e0 pergunta \"at\u00e9 onde \\(\\mathbf{a}\\) alcan\u00e7a ao longo de \\(\\mathbf{b}\\)?\" com um \u00fanico n\u00famero com sinal. Ela \u00e9 positiva quando o \u00e2ngulo entre os vetores \u00e9 menor que 90\u00b0, zero quando os vetores s\u00e3o perpendiculares e negativa quando o \u00e2ngulo passa de 90\u00b0 \u2014 ou seja, \\(\\mathbf{a}\\) aponta parcialmente <em>contra<\/em> \\(\\mathbf{b}\\). O pr\u00f3prio \u00e2ngulo decorre do produto escalar por meio de \\(\\cos\\theta = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert\\mathbf{a}\\rvert\\,\\lvert\\mathbf{b}\\rvert}\\), e \u00e9 por isso que a calculadora o exibe junto com a proje\u00e7\u00e3o.<\/p>\n<p>Observe que a proje\u00e7\u00e3o escala com o comprimento de \\(\\mathbf{b}\\), mas n\u00e3o com sua dire\u00e7\u00e3o: dobrar \\(\\mathbf{b}\\) reduz \u00e0 metade o fator de escala \\(k\\), de modo que a proje\u00e7\u00e3o vetorial em si n\u00e3o muda. Em outras palavras, a proje\u00e7\u00e3o depende da <em>reta<\/em> que passa por \\(\\mathbf{b}\\), e n\u00e3o do vetor espec\u00edfico \u2014 uma rela\u00e7\u00e3o de proporcionalidade direta entre \\(k\\) e \\(\\lvert\\mathbf{b}\\rvert^{-2}\\). Voc\u00ea pode explorar rela\u00e7\u00f5es proporcionais desse tipo com a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/variacao-direta\/\">calculadora de Proporcionalidade Direta<\/a>.<\/p>\n<h2>Premissas, limita\u00e7\u00f5es e dicas<\/h2>\n<ul>\n<li><strong>Vetores n\u00e3o nulos s\u00e3o obrigat\u00f3rios.<\/strong> O vetor de dire\u00e7\u00e3o precisa ter m\u00f3dulo positivo, porque a f\u00f3rmula divide por \\(\\lvert\\mathbf{b}\\rvert^2\\). Um vetor nulo n\u00e3o tem dire\u00e7\u00e3o sobre a qual projetar; a calculadora indicar\u00e1 qual vetor precisa ser corrigido.<\/li>\n<li><strong>O sinal importa.<\/strong> Uma proje\u00e7\u00e3o escalar negativa \u00e9 um resultado v\u00e1lido: significa que a proje\u00e7\u00e3o aponta no sentido oposto a \\(\\mathbf{b}\\), e a proje\u00e7\u00e3o vetorial conter\u00e1 m\u00faltiplos negativos das componentes de \\(\\mathbf{b}\\).<\/li>\n<li><strong>A ordem importa.<\/strong> Projetar \\(\\mathbf{a}\\) sobre \\(\\mathbf{b}\\) e \\(\\mathbf{b}\\) sobre \\(\\mathbf{a}\\) geralmente d\u00e1 vetores diferentes (apenas as proje\u00e7\u00f5es escalares se relacionam por \\(\\lvert\\mathbf{a}\\rvert\\) e \\(\\lvert\\mathbf{b}\\rvert\\)). Use o menu suspenso \"Projetar\" para trocar de dire\u00e7\u00e3o.<\/li>\n<li><strong>Teste de perpendicularidade.<\/strong> Se o produto escalar for 0, os vetores s\u00e3o ortogonais: a proje\u00e7\u00e3o \u00e9 o vetor nulo e a componente perpendicular \u00e9 igual ao vetor original.<\/li>\n<li><strong>M\u00f3dulos como RMS.<\/strong> O m\u00f3dulo \\(\\lvert\\mathbf{v}\\rvert = \\sqrt{v_x^2 + v_y^2 + v_z^2}\\) \u00e9 a m\u00e9dia quadr\u00e1tica das tr\u00eas componentes multiplicada por \\(\\sqrt{3}\\); a mesma ideia de m\u00e9dia aparece na <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-media-quadratica\/\">calculadora de M\u00e9dia Quadr\u00e1tica (RMS)<\/a>.<\/li>\n<li><strong>Vetores paralelos.<\/strong> Se os dois vetores s\u00e3o paralelos (\u00e2ngulo de 0\u00b0 ou 180\u00b0), a proje\u00e7\u00e3o \u00e9 igual ao pr\u00f3prio \\(\\mathbf{a}\\) (a menos do sinal), e a componente perpendicular \u00e9 o vetor nulo.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Digite as componentes de dois vetores 3D e esta calculadora projeta um sobre o outro, mostrando a proje\u00e7\u00e3o escalar, a proje\u00e7\u00e3o vetorial, a componente perpendicular, o produto escalar, os m\u00f3dulos e o \u00e2ngulo entre os vetores.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[42],"class_list":["post-2039","post","type-post","status-publish","format-standard","hentry","category-matematica","tag-geometria"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/2039","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/comments?post=2039"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/2039\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/media?parent=2039"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/categories?post=2039"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/tags?post=2039"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}