{"id":2518,"date":"2026-10-14T06:44:05","date_gmt":"2026-10-14T05:44:05","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-rzutu-wektora-3d\/"},"modified":"2026-10-14T06:44:05","modified_gmt":"2026-10-14T05:44:05","slug":"kalkulator-rzutu-wektora-3d","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-rzutu-wektora-3d\/","title":{"rendered":"Kalkulator rzutu wektora 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\">Rzut jednego wektora 3D na drugi<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"2518\" data-favorite-title=\"Rzut wektora 3D\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-rzutu-wektora-3d\/\" data-favorite-excerpt=\"Wpisz wsp\u00f3\u0142rz\u0119dne dw\u00f3ch wektor\u00f3w 3D, a kalkulator rzutuje jeden z nich na drugi, pokazuj\u0105c rzut skalarny, rzut wektorowy, sk\u0142adow\u0105 prostopad\u0142\u0105, iloczyn skalarny, d\u0142ugo\u015bci obu wektor\u00f3w oraz k\u0105t mi\u0119dzy nimi.\" aria-label=\"Dodaj do ulubionych\" 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\/pl\/matematyka\/kalkulator-rzutu-wektora-3d\/\" method=\"POST\" class=\"calculator\" id=\"calculator-2518\" data-post=\"2518\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Wektor a \u2014 wsp\u00f3\u0142rz\u0119dne<\/span><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Wsp\u00f3\u0142rz\u0119dna 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=\"wsprzdna_x_a\" id=\"field-2\" placeholder=\"2\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Wsp\u00f3\u0142rz\u0119dna 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=\"wsprzdna_y_a\" id=\"field-3\" placeholder=\"3\" step=\"any\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Wsp\u00f3\u0142rz\u0119dna 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=\"wsprzdna_z_az\" id=\"field-4\" placeholder=\"1\" step=\"any\" value=\"1\"\/><\/div><div class=\"is-single field\" id=\"input-5\"><span class=\"font-semibold\">Wektor b \u2014 wsp\u00f3\u0142rz\u0119dne<\/span><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">Wsp\u00f3\u0142rz\u0119dna 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=\"wsprzdna_x_b\" id=\"field-6\" placeholder=\"1\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">Wsp\u00f3\u0142rz\u0119dna 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=\"wsprzdna_y_b\" id=\"field-7\" placeholder=\"0\" step=\"any\" value=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-8\"><label for=\"field-8\">Wsp\u00f3\u0142rz\u0119dna 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=\"wsprzdna_z_bz\" id=\"field-8\" placeholder=\"2\" step=\"any\" value=\"2\"\/><\/div><div class=\"is-single field\" id=\"input-9\"><span class=\"font-semibold\">Kierunek rzutu<\/span><\/div><div class=\"field is-single\" id=\"input-10\"><label for=\"field-10\">Oblicz<\/label><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"oblicz\"><option value=\"ab\" selected>a na b<\/option><option value=\"ba\">b na 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=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"2518\">Oblicz<\/button><button type=\"reset\">Wyczy\u015b\u0107<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Obliczaj automatycznie<\/label><small>Po w\u0142\u0105czeniu wynik aktualizuje si\u0119 automatycznie po zmianie dowolnej warto\u015bci.<\/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> Tw\u00f3j rzut<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Rzut skalarny<\/span><span class=\"result-term\">comp<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Rzut wektorowy<\/span><span class=\"result-term\">proj<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Sk\u0142adowa prostopad\u0142a<\/span><span class=\"result-term\">rej<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Iloczyn skalarny<\/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=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>D\u0142ugo\u015b\u0107 wektora 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=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>D\u0142ugo\u015b\u0107 wektora 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=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>K\u0105t mi\u0119dzy wektorami<\/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=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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>Kroki oblicze\u0144<\/span><\/span><span class=\"field-value\" id=\"result-8\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Kopiuj wynik\" aria-label=\"Kopiuj wynik\"><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\">Kopiuj link do tego obliczenia<\/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>Zostaw komentarz<\/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=\"2518\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Jak oceniasz ten wynik?<\/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\">Lubi\u0119 to <\/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\">Pomocne <\/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\">Niejasne <\/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\">Rozczarowanie <\/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\">Niedok\u0142adne <\/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>Wz\u00f3r na rzut wektora 3D<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Wz\u00f3r<\/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>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(\\mathbf{a} = (a_x, a_y, a_z)\\) \u2014 wektor rzutowany<\/li>\n<li>\\(\\mathbf{b} = (b_x, b_y, b_z)\\) \u2014 wektor wyznaczaj\u0105cy kierunek<\/li>\n<li>\\(\\mathbf{a} \\cdot \\mathbf{b} = a_x b_x + a_y b_y + a_z b_z\\) \u2014 iloczyn skalarny<\/li>\n<li>\\(\\lvert \\mathbf{b} \\rvert = \\sqrt{b_x^2 + b_y^2 + b_z^2}\\) \u2014 d\u0142ugo\u015b\u0107 (modu\u0142) wektora \\(\\mathbf{b}\\)<\/li>\n<\/ul>\n<p>Wz\u00f3r pokazuje, jaka cz\u0119\u015b\u0107 wektora \\(\\mathbf{a}\\) skierowana jest wzd\u0142u\u017c \\(\\mathbf{b}\\). U\u0142amek \\(k = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert \\mathbf{b} \\rvert^2}\\) przeskalowuje wektor \\(\\mathbf{b}\\) tak, \u017ceby wynik by\u0142 cieniem, jaki \\(\\mathbf{a}\\) rzuca na prost\u0105 poprowadzon\u0105 przez \\(\\mathbf{b}\\).<\/p>\n<p>Z tego samego rzutu wynikaj\u0105 dwie pokrewne wielko\u015bci:<\/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>Rzut skalarny (sk\u0142adowa) to d\u0142ugo\u015b\u0107 cienia z zachowanym znakiem, a sk\u0142adowa prostopad\u0142a (tzw. odrzucenie) to pozosta\u0142a cz\u0119\u015b\u0107 wektora \\(\\mathbf{a}\\), kt\u00f3ra jest zawsze prostopad\u0142a do \\(\\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>Rzut wektora 3D \u2014 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> rzutujemy \\(\\mathbf{a} = (2, 3, 1)\\) na \\(\\mathbf{b} = (1, 0, 2)\\) (warto\u015bci domy\u015blne kalkulatora).<\/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>Wynik:<\/strong> rzut skalarny wynosi \\(4 \/ \\sqrt{5} \\approx 1.789\\), rzut wektorowy to \\((0.8, 0, 1.6)\\), a sk\u0142adowa prostopad\u0142a \u2014 \\((1.2, 3, -0.6)\\) o d\u0142ugo\u015bci \\(\\approx 3.286\\). K\u0105t mi\u0119dzy wektorami wynosi \\(\\arccos\\!\\left(\\tfrac{4}{\\sqrt{70}}\\right) \\approx 61.4\u00b0\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wpisz <strong>wsp\u00f3\u0142rz\u0119dn\u0105 x (a\u2093)<\/strong>, <strong>wsp\u00f3\u0142rz\u0119dn\u0105 y (a\u1d67)<\/strong> i <strong>wsp\u00f3\u0142rz\u0119dn\u0105 z (a_z)<\/strong> pierwszego wektora w sekcji \u201eWektor a \u2014 wsp\u00f3\u0142rz\u0119dne\u201d.<\/li>\n<li>Wpisz trzy wsp\u00f3\u0142rz\u0119dne drugiego wektora w sekcji \u201eWektor b \u2014 wsp\u00f3\u0142rz\u0119dne\u201d. Dla brakuj\u0105cej osi wpisz 0.<\/li>\n<li>Wybierz opcj\u0119 <strong>Rzutuj<\/strong>: \u201ea na b\u201d sprawdza, jaka cz\u0119\u015b\u0107 wektora \\(\\mathbf{a}\\) le\u017cy wzd\u0142u\u017c \\(\\mathbf{b}\\); \u201eb na a\u201d odwraca role.<\/li>\n<li>Odczytaj wyniki: rzut skalarny, rzut wektorowy, sk\u0142adow\u0105 prostopad\u0142\u0105, a tak\u017ce iloczyn skalarny, d\u0142ugo\u015bci obu wektor\u00f3w, k\u0105t mi\u0119dzy nimi oraz szczeg\u00f3\u0142owe obliczenia krok po kroku.<\/li>\n<\/ol>\n<h2>Czym jest rzut wektora w przestrzeni 3D?<\/h2>\n<p>Geometrycznie rzut wektora \\(\\mathbf{a}\\) na wektor \\(\\mathbf{b}\\) to cie\u0144, jaki \\(\\mathbf{a}\\) rzuca na prost\u0105 poprowadzon\u0105 przez \\(\\mathbf{b}\\) \u2014 tak, jakby \u015bwiat\u0142o pada\u0142o prostopadle do tej prostej. Wynikiem jest nowy wektor skierowany dok\u0142adnie tak samo (albo dok\u0142adnie odwrotnie) jak \\(\\mathbf{b}\\), o d\u0142ugo\u015bci r\u00f3wnej temu, jak daleko \\(\\mathbf{a}\\) si\u0119ga w tym kierunku. Nazywa si\u0119 go tak\u017ce sk\u0142adow\u0105 wektorow\u0105 \\(\\mathbf{a}\\) wzd\u0142u\u017c \\(\\mathbf{b}\\).<\/p>\n<p>Ka\u017cdy rzut rozk\u0142ada wektor na dwie prostopad\u0142e cz\u0119\u015bci: sam rzut (r\u00f3wnoleg\u0142y do \\(\\mathbf{b}\\)) oraz sk\u0142adow\u0105 prostopad\u0142\u0105, zwan\u0105 cz\u0119sto <em>odrzuceniem<\/em>. Obie cz\u0119\u015bci zawsze sk\u0142adaj\u0105 si\u0119 z powrotem w wektor wyj\u015bciowy: \\(\\mathbf{a} = \\operatorname{proj}_{\\mathbf{b}} \\mathbf{a} + \\operatorname{perp}_{\\mathbf{b}} \\mathbf{a}\\), a ich iloczyn skalarny jest zerowy. Ten rozk\u0142ad le\u017cy u podstaw procesu Grama\u2013Schmidta, rozk\u0142adu si\u0142 na sk\u0142adowe w fizyce oraz dopasowania metod\u0105 najmniejszych kwadrat\u00f3w w statystyce.<\/p>\n<p><strong>Rzut skalarny<\/strong> odpowiada na pytanie jedn\u0105 liczb\u0105 ze znakiem: \u201ejak daleko wzd\u0142u\u017c \\(\\mathbf{b}\\) si\u0119ga \\(\\mathbf{a}\\)?\u201d. Jest dodatni, gdy k\u0105t mi\u0119dzy wektorami jest mniejszy ni\u017c 90\u00b0, zerowy, gdy wektory s\u0105 prostopad\u0142e, a ujemny, gdy k\u0105t przekracza 90\u00b0 \u2014 co oznacza, \u017ce \\(\\mathbf{a}\\) wskazuje cz\u0119\u015bciowo <em>przeciwko<\/em> \\(\\mathbf{b}\\). Sam k\u0105t wyznacza si\u0119 z iloczynu skalarnego ze wzoru \\(\\cos\\theta = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert\\mathbf{a}\\rvert\\,\\lvert\\mathbf{b}\\rvert}\\), dlatego kalkulator podaje go obok rzutu.<\/p>\n<p>Zwr\u00f3\u0107 uwag\u0119, \u017ce rzut zale\u017cy od d\u0142ugo\u015bci \\(\\mathbf{b}\\), ale nie od jego kierunku: podwojenie \\(\\mathbf{b}\\) zmniejsza o po\u0142ow\u0119 wsp\u00f3\u0142czynnik \\(k\\), wi\u0119c sam rzut wektorowy pozostaje niezmieniony. Inaczej m\u00f3wi\u0105c, rzut zale\u017cy od <em>prostej<\/em> wyznaczonej przez \\(\\mathbf{b}\\), a nie od konkretnego wektora \u2014 zachodzi tu zwi\u0105zek prostej proporcjonalno\u015bci mi\u0119dzy \\(k\\) a \\(\\lvert\\mathbf{b}\\rvert^{-2}\\). Z takimi zale\u017cno\u015bciami proporcjonalnymi mo\u017cesz poeksperymentowa\u0107 w <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/proporcjonalnosc-prosta\/\">kalkulatorze proporcjonalno\u015bci prostej<\/a>.<\/p>\n<h2>Za\u0142o\u017cenia, ograniczenia i wskaz\u00f3wki<\/h2>\n<ul>\n<li><strong>Wektory nie mog\u0105 by\u0107 zerowe.<\/strong> Wektor kierunkowy musi mie\u0107 dodatni\u0105 d\u0142ugo\u015b\u0107, poniewa\u017c we wzorze dzielimy przez \\(\\lvert\\mathbf{b}\\rvert^2\\). Wektor zerowy nie wyznacza kierunku rzutowania; kalkulator podpowie, kt\u00f3ry wektor trzeba poprawi\u0107.<\/li>\n<li><strong>Znak ma znaczenie.<\/strong> Ujemny rzut skalarny to poprawny wynik: oznacza, \u017ce rzut wskazuje kierunek przeciwny do \\(\\mathbf{b}\\), a rzut wektorowy b\u0119dzie zawiera\u0142 ujemne wielokrotno\u015bci wsp\u00f3\u0142rz\u0119dnych \\(\\mathbf{b}\\).<\/li>\n<li><strong>Kolejno\u015b\u0107 ma znaczenie.<\/strong> Rzut \\(\\mathbf{a}\\) na \\(\\mathbf{b}\\) i rzut \\(\\mathbf{b}\\) na \\(\\mathbf{a}\\) to na og\u00f3\u0142 r\u00f3\u017cne wektory (powi\u0105zane s\u0105 jedynie rzuty skalarne \u2014 przez \\(\\lvert\\mathbf{a}\\rvert\\) i \\(\\lvert\\mathbf{b}\\rvert\\)). Aby zmieni\u0107 kierunek, u\u017cyj listy rozwijanej \u201eRzutuj\u201d.<\/li>\n<li><strong>Sprawdzenie prostopad\u0142o\u015bci.<\/strong> Je\u015bli iloczyn skalarny wynosi 0, wektory s\u0105 prostopad\u0142e: rzut jest wektorem zerowym, a sk\u0142adowa prostopad\u0142a r\u00f3wna si\u0119 wektorowi wyj\u015bciowemu.<\/li>\n<li><strong>D\u0142ugo\u015b\u0107 jako \u015brednia kwadratowa.<\/strong> D\u0142ugo\u015b\u0107 \\(\\lvert\\mathbf{v}\\rvert = \\sqrt{v_x^2 + v_y^2 + v_z^2}\\) to \u015brednia kwadratowa (RMS) trzech wsp\u00f3\u0142rz\u0119dnych pomno\u017cona przez \\(\\sqrt{3}\\); t\u0119 sam\u0105 ide\u0119 u\u015bredniania znajdziesz w <a href=\"https:\/\/wpcalc.com\/pl\/matematyka\/kalkulator-sredniej-kwadratowej-rms\/\">kalkulatorze \u015bredniej kwadratowej (RMS)<\/a>.<\/li>\n<li><strong>Wektory r\u00f3wnoleg\u0142e.<\/strong> Je\u015bli oba wektory s\u0105 r\u00f3wnoleg\u0142e (k\u0105t 0\u00b0 lub 180\u00b0), rzut r\u00f3wna si\u0119 \\(\\mathbf{a}\\) (co najwy\u017cej ze zmienionym znakiem), a sk\u0142adowa prostopad\u0142a jest wektorem zerowym.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Wpisz wsp\u00f3\u0142rz\u0119dne dw\u00f3ch wektor\u00f3w 3D, a kalkulator rzutuje jeden z nich na drugi, pokazuj\u0105c rzut skalarny, rzut wektorowy, sk\u0142adow\u0105 prostopad\u0142\u0105, iloczyn skalarny, d\u0142ugo\u015bci obu wektor\u00f3w oraz k\u0105t mi\u0119dzy nimi.<\/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":[6],"tags":[41],"class_list":["post-2518","post","type-post","status-publish","format-standard","hentry","category-matematyka","tag-geometria"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2518","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/comments?post=2518"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2518\/revisions"}],"predecessor-version":[{"id":2806,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/2518\/revisions\/2806"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=2518"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=2518"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=2518"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}