{"id":3909,"date":"2026-10-14T06:43:29","date_gmt":"2026-10-14T05:43:29","guid":{"rendered":"https:\/\/wpcalc.com\/tr\/matematik\/3d-vektor-izdusumu\/"},"modified":"2026-10-14T06:43:29","modified_gmt":"2026-10-14T05:43:29","slug":"3d-vektor-izdusumu","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/tr\/matematik\/3d-vektor-izdusumu\/","title":{"rendered":"3B Vekt\u00f6r \u0130zd\u00fc\u015f\u00fcm\u00fc Hesaplay\u0131c\u0131s\u0131"},"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\">Bir 3B Vekt\u00f6r\u00fc Di\u011ferine \u0130zd\u00fc\u015f\u00fcr\u00fcn<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"3909\" data-favorite-title=\"3B Vekt\u00f6r \u0130zd\u00fc\u015f\u00fcm\u00fc\" data-favorite-url=\"https:\/\/wpcalc.com\/tr\/matematik\/3d-vektor-izdusumu\/\" data-favorite-excerpt=\"\u0130ki 3B vekt\u00f6r\u00fcn bile\u015fenlerini girin; hesaplay\u0131c\u0131 bunlardan birini di\u011ferinin \u00fczerine izd\u00fc\u015f\u00fcrerek skaler izd\u00fc\u015f\u00fcm\u00fc, vekt\u00f6rel izd\u00fc\u015f\u00fcm\u00fc, dik bile\u015feni, nokta \u00e7arp\u0131m\u0131, b\u00fcy\u00fckl\u00fckleri ve vekt\u00f6rler aras\u0131ndaki a\u00e7\u0131y\u0131 g\u00f6sterir.\" aria-label=\"Favorilere ekle\" 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\/tr\/matematik\/3d-vektor-izdusumu\/\" method=\"POST\" class=\"calculator\" id=\"calculator-3909\" data-post=\"3909\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Giri\u015f alanlar\u0131<\/legend><div class=\"is-single field\" id=\"input-1\"><span class=\"font-semibold\">Vekt\u00f6r a \u2014 bile\u015fenler<\/span><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">x bile\u015feni (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=\"x_bileeni_a\" id=\"field-2\" placeholder=\"2\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">y bile\u015feni (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=\"y_bileeni_a\" id=\"field-3\" placeholder=\"3\" step=\"any\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">z bile\u015feni (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=\"z_bileeni_az\" id=\"field-4\" placeholder=\"1\" step=\"any\" value=\"1\"\/><\/div><div class=\"is-single field\" id=\"input-5\"><span class=\"font-semibold\">Vekt\u00f6r b \u2014 bile\u015fenler<\/span><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">x bile\u015feni (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=\"x_bileeni_b\" id=\"field-6\" placeholder=\"1\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">y bile\u015feni (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=\"y_bileeni_b\" id=\"field-7\" placeholder=\"0\" step=\"any\" value=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-8\"><label for=\"field-8\">z bile\u015feni (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=\"z_bileeni_bz\" id=\"field-8\" placeholder=\"2\" step=\"any\" value=\"2\"\/><\/div><div class=\"is-single field\" id=\"input-9\"><span class=\"font-semibold\">\u0130zd\u00fc\u015f\u00fcm y\u00f6n\u00fc<\/span><\/div><div class=\"field is-single\" id=\"input-10\"><label for=\"field-10\">\u0130zd\u00fc\u015f\u00fcm<\/label><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"zdm\"><option value=\"ab\" selected>a&#039;n\u0131n b \u00fczerine izd\u00fc\u015f\u00fcm\u00fc<\/option><option value=\"ba\">b&#039;nin a \u00fczerine izd\u00fc\u015f\u00fcm\u00fc<\/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=\"Yeniden hesapla\" id=\"calculate-button\" data-post=\"3909\">Hesapla<\/button><button type=\"reset\">S\u0131f\u0131rla<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Otomatik hesapla<\/label><small>Etkinle\u015ftirilirse, herhangi bir de\u011feri de\u011fi\u015ftirdi\u011finizde sonu\u00e7 otomatik olarak g\u00fcncellenir.<\/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> \u0130zd\u00fc\u015f\u00fcm Sonucunuz<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Skaler izd\u00fc\u015f\u00fcm<\/span><span class=\"result-term\">comp<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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>Vekt\u00f6rel izd\u00fc\u015f\u00fcm<\/span><span class=\"result-term\">proj<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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>Dik bile\u015fen<\/span><span class=\"result-term\">rej<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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>\u0130\u00e7 \u00e7arp\u0131m<\/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=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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>a&#039;n\u0131n b\u00fcy\u00fckl\u00fc\u011f\u00fc<\/span><span class=\"result-term\">|a|<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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>b&#039;nin b\u00fcy\u00fckl\u00fc\u011f\u00fc<\/span><span class=\"result-term\">|b|<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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>Vekt\u00f6rler aras\u0131ndaki a\u00e7\u0131<\/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=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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>Hesaplama ad\u0131mlar\u0131<\/span><\/span><span class=\"field-value\" id=\"result-8\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><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\">Bu hesaplaman\u0131n ba\u011flant\u0131s\u0131n\u0131 kopyala<\/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>Yorum b\u0131rak<\/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=\"3909\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Bu sonu\u00e7 hakk\u0131nda ne d\u00fc\u015f\u00fcn\u00fcyorsunuz?<\/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\">Be\u011fendim <\/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\">Faydal\u0131 <\/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\">Kafam kar\u0131\u015ft\u0131 <\/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\">Hayal k\u0131r\u0131kl\u0131\u011f\u0131 <\/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\">Hatal\u0131 <\/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>3B vekt\u00f6r izd\u00fc\u015f\u00fcm\u00fc form\u00fcl\u00fc<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Form\u00fcl<\/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>Burada:<\/strong><\/p>\n<ul>\n<li>\\(\\mathbf{a} = (a_x, a_y, a_z)\\) \u2014 izd\u00fc\u015f\u00fcm\u00fc al\u0131nan vekt\u00f6r<\/li>\n<li>\\(\\mathbf{b} = (b_x, b_y, b_z)\\) \u2014 y\u00f6n\u00fc veren vekt\u00f6r<\/li>\n<li>\\(\\mathbf{a} \\cdot \\mathbf{b} = a_x b_x + a_y b_y + a_z b_z\\) \u2014 nokta \u00e7arp\u0131m<\/li>\n<li>\\(\\lvert \\mathbf{b} \\rvert = \\sqrt{b_x^2 + b_y^2 + b_z^2}\\) \u2014 \\(\\mathbf{b}\\) vekt\u00f6r\u00fcn\u00fcn b\u00fcy\u00fckl\u00fc\u011f\u00fc (uzunlu\u011fu)<\/li>\n<\/ul>\n<p>Form\u00fcl, \\(\\mathbf{a}\\) vekt\u00f6r\u00fcn\u00fcn ne kadar\u0131n\u0131n \\(\\mathbf{b}\\) y\u00f6n\u00fcnde uzand\u0131\u011f\u0131n\u0131 ifade eder. \\(k = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert \\mathbf{b} \\rvert^2}\\) \u00e7arpan\u0131 \\(\\mathbf{b}\\)'yi \u00f6yle \u00f6l\u00e7ekler ki sonu\u00e7, \\(\\mathbf{a}\\)'n\u0131n \\(\\mathbf{b}\\) do\u011frusu \u00fczerine d\u00fc\u015f\u00fcrd\u00fc\u011f\u00fc g\u00f6lge olur.<\/p>\n<p>Ayn\u0131 izd\u00fc\u015f\u00fcmden iki ili\u015fkili nicelik daha \u00e7\u0131kar:<\/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>Skaler izd\u00fc\u015f\u00fcm (bile\u015fen), g\u00f6lgenin i\u015faretli uzunlu\u011fudur; dik bile\u015fen (rejection) ise \\(\\mathbf{a}\\)'n\u0131n geri kalan k\u0131sm\u0131d\u0131r ve her zaman \\(\\mathbf{b}\\)'ye diktir.<\/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>3B vekt\u00f6r izd\u00fc\u015f\u00fcm\u00fc \u2013 \u00f6rnek hesaplama<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>\u00d6rnek:<\/strong> \\(\\mathbf{a} = (2, 3, 1)\\) vekt\u00f6r\u00fcn\u00fc \\(\\mathbf{b} = (1, 0, 2)\\) vekt\u00f6r\u00fcn\u00fcn \u00fczerine izd\u00fc\u015f\u00fcr\u00fcn (hesaplay\u0131c\u0131n\u0131n varsay\u0131lan de\u011ferleri).<\/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>Sonu\u00e7:<\/strong> skaler izd\u00fc\u015f\u00fcm \\(4 \/ \\sqrt{5} \\approx 1.789\\), vekt\u00f6rel izd\u00fc\u015f\u00fcm \\((0.8, 0, 1.6)\\), dik bile\u015fen ise b\u00fcy\u00fckl\u00fc\u011f\u00fc \\(\\approx 3.286\\) olan \\((1.2, 3, -0.6)\\) vekt\u00f6r\u00fcd\u00fcr. Vekt\u00f6rler aras\u0131ndaki a\u00e7\u0131 \\(\\arccos\\!\\left(\\tfrac{4}{\\sqrt{70}}\\right) \\approx 61.4\u00b0\\) olarak bulunur.<\/p>\n<p>\n\r\n<\/section>\n<h2>Hesaplay\u0131c\u0131 nas\u0131l kullan\u0131l\u0131r?<\/h2>\n<ol>\n<li>\u0130lk vekt\u00f6r\u00fcn <strong>x bile\u015fenini (a\u2093)<\/strong>, <strong>y bile\u015fenini (a\u1d67)<\/strong> ve <strong>z bile\u015fenini (a_z)<\/strong> \"Vekt\u00f6r a \u2014 bile\u015fenleri\" b\u00f6l\u00fcm\u00fcne girin.<\/li>\n<li>\u0130kinci vekt\u00f6r\u00fcn \u00fc\u00e7 bile\u015fenini \"Vekt\u00f6r b \u2014 bile\u015fenleri\" b\u00f6l\u00fcm\u00fcne girin. Eksik eksenler i\u00e7in 0 yaz\u0131n.<\/li>\n<li><strong>\u0130zd\u00fc\u015f\u00fcm<\/strong> se\u00e7ene\u011fini belirleyin: \"a, b \u00fczerine\" \\(\\mathbf{a}\\)'n\u0131n ne kadar\u0131n\u0131n \\(\\mathbf{b}\\) do\u011frultusunda yatt\u0131\u011f\u0131n\u0131 bulur; \"b, a \u00fczerine\" ise rolleri tersine \u00e7evirir.<\/li>\n<li>Sonu\u00e7lar\u0131 okuyun: skaler izd\u00fc\u015f\u00fcm, vekt\u00f6rel izd\u00fc\u015f\u00fcm, dik bile\u015fen; ayr\u0131ca nokta \u00e7arp\u0131m, her iki b\u00fcy\u00fckl\u00fck, vekt\u00f6rler aras\u0131ndaki a\u00e7\u0131 ve hesaplaman\u0131n ad\u0131m ad\u0131m d\u00f6k\u00fcm\u00fc.<\/li>\n<\/ol>\n<h2>3B vekt\u00f6r izd\u00fc\u015f\u00fcm\u00fc nedir?<\/h2>\n<p>Geometrik olarak, \\(\\mathbf{a}\\) vekt\u00f6r\u00fcn\u00fcn \\(\\mathbf{b}\\) vekt\u00f6r\u00fc \u00fczerine izd\u00fc\u015f\u00fcm\u00fc, \\(\\mathbf{a}\\)'n\u0131n \\(\\mathbf{b}\\) do\u011frusuna d\u00fc\u015f\u00fcrd\u00fc\u011f\u00fc g\u00f6lgedir; sanki \u0131\u015f\u0131k bu do\u011fruya dik olarak gelir. Sonu\u00e7, \\(\\mathbf{b}\\) ile tam ayn\u0131 (ya da tam ters) y\u00f6nde g\u00f6steren ve uzunlu\u011fu \\(\\mathbf{a}\\)'n\u0131n o y\u00f6ndeki uzan\u0131m\u0131na e\u015fit olan yeni bir vekt\u00f6rd\u00fcr. Bu nicelik, \\(\\mathbf{a}\\)'n\u0131n \\(\\mathbf{b}\\) do\u011frultusundaki vekt\u00f6r bile\u015feni olarak da adland\u0131r\u0131l\u0131r.<\/p>\n<p>Her izd\u00fc\u015f\u00fcm bir vekt\u00f6r\u00fc birbirine dik iki par\u00e7aya ay\u0131r\u0131r: izd\u00fc\u015f\u00fcm\u00fcn kendisi (\\(\\mathbf{b}\\)'ye paralel) ve <em>rejection<\/em> denen dik bile\u015fen. Bu iki par\u00e7an\u0131n toplam\u0131 her zaman orijinal vekt\u00f6r\u00fc verir: \\(\\mathbf{a} = \\operatorname{proj}_{\\mathbf{b}} \\mathbf{a} + \\operatorname{perp}_{\\mathbf{b}} \\mathbf{a}\\), ve ikisinin nokta \u00e7arp\u0131m\u0131 s\u0131f\u0131rd\u0131r. Bu ayr\u0131\u015ft\u0131rma; Gram\u2013Schmidt s\u00fcrecinin, fizikte kuvvetlerin bile\u015fenlere ayr\u0131lmas\u0131n\u0131n ve istatistikte en k\u00fc\u00e7\u00fck kareler uyumunun temelini olu\u015fturur.<\/p>\n<p><strong>Skaler izd\u00fc\u015f\u00fcm<\/strong>, \"\\(\\mathbf{a}\\), \\(\\mathbf{b}\\) boyunca ne kadar ilerler?\" sorusunu tek bir i\u015faretli say\u0131yla yan\u0131tlar. Vekt\u00f6rler aras\u0131ndaki a\u00e7\u0131 90\u00b0'den k\u00fc\u00e7\u00fckse pozitif, vekt\u00f6rler dikse s\u0131f\u0131r, a\u00e7\u0131 90\u00b0'yi a\u015farsa negatiftir \u2014 bu durumda \\(\\mathbf{a}\\) k\u0131smen \\(\\mathbf{b}\\)'ye <em>ters<\/em> y\u00f6ndedir. A\u00e7\u0131n\u0131n kendisi ise nokta \u00e7arp\u0131mdan \\(\\cos\\theta = \\frac{\\mathbf{a} \\cdot \\mathbf{b}}{\\lvert\\mathbf{a}\\rvert\\,\\lvert\\mathbf{b}\\rvert}\\) ba\u011f\u0131nt\u0131s\u0131yla elde edilir; hesaplay\u0131c\u0131n\u0131n izd\u00fc\u015f\u00fcm\u00fcn yan\u0131nda a\u00e7\u0131y\u0131 da g\u00f6stermesinin nedeni budur.<\/p>\n<p>\u0130zd\u00fc\u015f\u00fcm\u00fcn \\(\\mathbf{b}\\)'nin uzunlu\u011fuyla \u00f6l\u00e7eklendi\u011fine ama y\u00f6n\u00fcnden etkilenmedi\u011fine dikkat edin: \\(\\mathbf{b}\\) iki kat\u0131na \u00e7\u0131kar\u0131l\u0131rsa \u00f6l\u00e7ek \u00e7arpan\u0131 \\(k\\) yar\u0131ya iner, dolay\u0131s\u0131yla vekt\u00f6rel izd\u00fc\u015f\u00fcm de\u011fi\u015fmez. Ba\u015fka bir deyi\u015fle izd\u00fc\u015f\u00fcm, belirli bir vekt\u00f6re de\u011fil \\(\\mathbf{b}\\)'nin ge\u00e7ti\u011fi <em>do\u011fruya<\/em> ba\u011fl\u0131d\u0131r \u2014 \\(k\\) ile \\(\\lvert\\mathbf{b}\\rvert^{-2}\\) aras\u0131nda do\u011fru orant\u0131l\u0131 bir ili\u015fki vard\u0131r. Bu t\u00fcr orant\u0131 ili\u015fkilerini <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/dogru-oranti\/\">do\u011fru orant\u0131 hesaplay\u0131c\u0131s\u0131yla<\/a> inceleyebilirsiniz.<\/p>\n<h2>Varsay\u0131mlar, s\u0131n\u0131rlamalar ve ipu\u00e7lar\u0131<\/h2>\n<ul>\n<li><strong>S\u0131f\u0131rdan farkl\u0131 vekt\u00f6rler gerekir.<\/strong> Form\u00fcl \\(\\lvert\\mathbf{b}\\rvert^2\\)'ye b\u00f6ld\u00fc\u011f\u00fc i\u00e7in y\u00f6n vekt\u00f6r\u00fcn\u00fcn b\u00fcy\u00fckl\u00fc\u011f\u00fc pozitif olmal\u0131d\u0131r. S\u0131f\u0131r vekt\u00f6r\u00fcn\u00fcn \u00fczerine izd\u00fc\u015f\u00fcm al\u0131nacak bir y\u00f6n\u00fc yoktur; hesaplay\u0131c\u0131 hangi vekt\u00f6r\u00fcn d\u00fczeltilmesi gerekti\u011fini size bildirir.<\/li>\n<li><strong>\u0130\u015faret \u00f6nemlidir.<\/strong> Negatif bir skaler izd\u00fc\u015f\u00fcm ge\u00e7erli bir sonu\u00e7tur: izd\u00fc\u015f\u00fcm\u00fcn \\(\\mathbf{b}\\)'nin tersi y\u00f6n\u00fcnde oldu\u011funu g\u00f6sterir ve vekt\u00f6rel izd\u00fc\u015f\u00fcm, \\(\\mathbf{b}\\)'nin bile\u015fenlerinin negatif katlar\u0131n\u0131 i\u00e7erir.<\/li>\n<li><strong>S\u0131ra \u00f6nemlidir.<\/strong> \\(\\mathbf{a}\\)'y\u0131 \\(\\mathbf{b}\\)'ye izd\u00fc\u015f\u00fcrmek ile \\(\\mathbf{b}\\)'yi \\(\\mathbf{a}\\)'ya izd\u00fc\u015f\u00fcrmek genellikle farkl\u0131 vekt\u00f6rler verir (yaln\u0131zca skaler izd\u00fc\u015f\u00fcmler \\(\\lvert\\mathbf{a}\\rvert\\) ve \\(\\lvert\\mathbf{b}\\rvert\\) ile ili\u015fkilidir). Y\u00f6n de\u011fi\u015ftirmek i\u00e7in \"\u0130zd\u00fc\u015f\u00fcm\" a\u00e7\u0131l\u0131r men\u00fcs\u00fcn\u00fc kullan\u0131n.<\/li>\n<li><strong>Diklik denetimi.<\/strong> Nokta \u00e7arp\u0131m 0 ise vekt\u00f6rler diktir: izd\u00fc\u015f\u00fcm s\u0131f\u0131r vekt\u00f6r\u00fc, dik bile\u015fen ise orijinal vekt\u00f6r\u00fcn kendisidir.<\/li>\n<li><strong>B\u00fcy\u00fckl\u00fckler ve kareler ortalamas\u0131.<\/strong> \\(\\lvert\\mathbf{v}\\rvert = \\sqrt{v_x^2 + v_y^2 + v_z^2}\\) b\u00fcy\u00fckl\u00fc\u011f\u00fc, \u00fc\u00e7 bile\u015fenin kareler ortalamas\u0131n\u0131n \\(\\sqrt{3}\\) ile \u00e7arp\u0131m\u0131d\u0131r; ayn\u0131 ortalama fikri <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/karesel-ortalama-hesaplama\/\">kareler ortalamas\u0131 (RMS) hesaplay\u0131c\u0131s\u0131nda<\/a> da kar\u015f\u0131n\u0131za \u00e7\u0131kar.<\/li>\n<li><strong>Paralel vekt\u00f6rler.<\/strong> \u0130ki vekt\u00f6r paralelse (a\u00e7\u0131 0\u00b0 veya 180\u00b0), izd\u00fc\u015f\u00fcm (i\u015faret hari\u00e7) \\(\\mathbf{a}\\)'n\u0131n kendisine e\u015fit olur ve dik bile\u015fen s\u0131f\u0131r vekt\u00f6r\u00fcd\u00fcr.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u0130ki 3B vekt\u00f6r\u00fcn bile\u015fenlerini girin; hesaplay\u0131c\u0131 bunlardan birini di\u011ferinin \u00fczerine izd\u00fc\u015f\u00fcrerek skaler izd\u00fc\u015f\u00fcm\u00fc, vekt\u00f6rel izd\u00fc\u015f\u00fcm\u00fc, dik bile\u015feni, nokta \u00e7arp\u0131m\u0131, b\u00fcy\u00fckl\u00fckleri ve vekt\u00f6rler aras\u0131ndaki a\u00e7\u0131y\u0131 g\u00f6sterir.<\/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-3909","post","type-post","status-publish","format-standard","hentry","category-matematik","tag-geometri"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/3909","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/comments?post=3909"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/3909\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/media?parent=3909"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/categories?post=3909"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/tags?post=3909"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}