{"id":52,"date":"2026-10-04T08:19:46","date_gmt":"2026-10-04T07:19:46","guid":{"rendered":"https:\/\/wpcalc.com\/pl\/matematyka\/kat-miedzy-wektorami-2d\/"},"modified":"2026-10-04T08:19:46","modified_gmt":"2026-10-04T07:19:46","slug":"kat-miedzy-wektorami-2d","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pl\/matematyka\/kat-miedzy-wektorami-2d\/","title":{"rendered":"Kalkulator k\u0105ta mi\u0119dzy wektorami 2D"},"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\">Obliczanie k\u0105ta mi\u0119dzy dwoma wektorami 2D<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"52\" data-favorite-title=\"K\u0105t mi\u0119dzy wektorami 2D\" data-favorite-url=\"https:\/\/wpcalc.com\/pl\/matematyka\/kat-miedzy-wektorami-2d\/\" data-favorite-excerpt=\"Wprowad\u017a sk\u0142adowe X i Y dw\u00f3ch wektor\u00f3w, aby znale\u017a\u0107 k\u0105t mi\u0119dzy nimi w stopniach. Kalkulator wykorzystuje wz\u00f3r na iloczyn skalarny i przydaje si\u0119 w fizyce, geometrii oraz grafice komputerowej.\" 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\/kat-miedzy-wektorami-2d\/\" method=\"POST\" class=\"calculator\" id=\"calculator-52\" data-post=\"52\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Pola danych<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Wektor V1 \u2013 sk\u0142adowa X<\/label><input type=\"number\" name=\"wektor_v1_skadowa_x\" id=\"field-1\" placeholder=\"1\" step=\"1\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Wektor V1 \u2013 sk\u0142adowa Y<\/label><input type=\"number\" name=\"wektor_v1_skadowa_y\" id=\"field-2\" placeholder=\"0\" step=\"1\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Wektor V2 \u2013 sk\u0142adowa X<\/label><input type=\"number\" name=\"wektor_v2_skadowa_x\" id=\"field-3\" placeholder=\"0\" step=\"1\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">Wektor V2 \u2013 sk\u0142adowa Y<\/label><input type=\"number\" name=\"wektor_v2_skadowa_y\" id=\"field-4\" placeholder=\"1\" step=\"1\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Przelicz ponownie\" id=\"calculate-button\" data-post=\"52\">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> Twoje wyniki<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>K\u0105t mi\u0119dzy wektorami (\u00b0)<\/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><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=\"52\" 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 k\u0105t mi\u0119dzy dwoma wektorami 2D<\/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\\[ \\theta = \\cos^{-1}\\left( \\frac{\\vec{A} \\cdot \\vec{B}}{|\\vec{A}| \\, |\\vec{B}|} \\right) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Gdzie:<\/strong><\/p>\n<ul>\n<li>\\(\\vec{A} \\cdot \\vec{B} = A_x B_x + A_y B_y\\) \u2014 iloczyn skalarny dw\u00f3ch wektor\u00f3w<\/li>\n<li>\\(|\\vec{A}| = \\sqrt{A_x^2 + A_y^2}\\) \u2014 d\u0142ugo\u015b\u0107 (modu\u0142) wektora \\(\\vec{A}\\)<\/li>\n<li>\\(|\\vec{B}| = \\sqrt{B_x^2 + B_y^2}\\) \u2014 d\u0142ugo\u015b\u0107 (modu\u0142) wektora \\(\\vec{B}\\)<\/li>\n<li>\\(\\theta\\) \u2014 k\u0105t mi\u0119dzy dwoma wektorami<\/li>\n<\/ul>\n<p>Wz\u00f3r pochodzi z geometrycznej definicji iloczynu skalarnego: \\(\\vec{A} \\cdot \\vec{B} = |\\vec{A}| |\\vec{B}| \\cos\\theta\\). Rozwi\u0105zuj\u0105c wzgl\u0119dem \\(\\theta\\), otrzymujemy powy\u017csze wyra\u017cenie. Wynik funkcji \\(\\cos^{-1}\\) jest w radianach, dlatego przelicza si\u0119 go na stopnie, mno\u017c\u0105c przez \\(\\frac{180}{\\pi}\\) \u2014 ten kalkulator robi to za Ciebie i podaje odpowied\u017a w stopniach.<\/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>K\u0105t mi\u0119dzy dwoma wektorami 2D \u2013 przyk\u0142ad obliczenia<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Przyk\u0142ad:<\/strong> znajd\u017a k\u0105t mi\u0119dzy \\(\\vec{V_1} = (1, 0)\\) i \\(\\vec{V_2} = (0, 1)\\).<\/p>\n<p>\\[ \\vec{V_1} \\cdot \\vec{V_2} = (1)(0) + (0)(1) = 0 \\]<\/p>\n<p>\\[ |\\vec{V_1}| = \\sqrt{1^2 + 0^2} = 1, \\qquad |\\vec{V_2}| = \\sqrt{0^2 + 1^2} = 1 \\]<\/p>\n<p>\\[ \\theta = \\cos^{-1}\\left( \\frac{0}{1 \\cdot 1} \\right) = \\cos^{-1}(0) = 90\u00b0 \\]<\/p>\n<p><strong>Wynik:<\/strong> k\u0105t mi\u0119dzy dwoma wektorami wynosi 90\u00b0. Wektory s\u0105 prostopad\u0142e, czego mo\u017cna si\u0119 spodziewa\u0107 po wektorach jednostkowych wzd\u0142u\u017c osi X i Y.<\/p>\n<p>\n\r\n<\/section>\n<h2>Jak korzysta\u0107 z kalkulatora<\/h2>\n<ol>\n<li>Wprowad\u017a sk\u0142adow\u0105 X pierwszego wektora w polu <strong>Wektor V1 - X<\/strong>.<\/li>\n<li>Wprowad\u017a sk\u0142adow\u0105 Y pierwszego wektora w polu <strong>Wektor V1 - Y<\/strong>.<\/li>\n<li>Wprowad\u017a sk\u0142adow\u0105 X drugiego wektora w polu <strong>Wektor V2 - X<\/strong>.<\/li>\n<li>Wprowad\u017a sk\u0142adow\u0105 Y drugiego wektora w polu <strong>Wektor V2 - Y<\/strong>.<\/li>\n<li>Odczytaj wynik <strong>K\u0105t mi\u0119dzy wektorami (\u00b0)<\/strong>. Odpowied\u017a podawana jest w stopniach, zaokr\u0105glona do trzech miejsc po przecinku.<\/li>\n<\/ol>\n<p>Akceptowane s\u0105 warto\u015bci dodatnie i ujemne, a tak\u017ce liczby dziesi\u0119tne. Nie trzeba najpierw normalizowa\u0107 wektor\u00f3w \u2014 we wzorze nast\u0119puje dzielenie przez d\u0142ugo\u015bci, wi\u0119c k\u0105t jest taki sam niezale\u017cnie od d\u0142ugo\u015bci wektor\u00f3w.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Wprowad\u017a sk\u0142adowe X i Y dw\u00f3ch wektor\u00f3w, aby znale\u017a\u0107 k\u0105t mi\u0119dzy nimi w stopniach. Kalkulator wykorzystuje wz\u00f3r na iloczyn skalarny i przydaje si\u0119 w fizyce, geometrii oraz grafice komputerowej.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[6],"tags":[41],"class_list":["post-52","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\/52","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=52"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/52\/revisions"}],"predecessor-version":[{"id":1693,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/posts\/52\/revisions\/1693"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/media?parent=52"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/categories?post=52"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pl\/wp-json\/wp\/v2\/tags?post=52"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}