{"id":259,"date":"2023-07-10T07:15:20","date_gmt":"2023-07-10T07:15:20","guid":{"rendered":"https:\/\/mathority.org\/id\/vektor-koplanar-atau-koplanar\/"},"modified":"2023-07-10T07:15:20","modified_gmt":"2023-07-10T07:15:20","slug":"vektor-koplanar-atau-koplanar","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/vektor-koplanar-atau-koplanar\/","title":{"rendered":"Vektor koplanar (atau koplanar)."},"content":{"rendered":"<p>Di halaman ini Anda akan mempelajari apa itu vektor koplanar dan cara mengetahui apakah 2, 3, 4 atau lebih vektor adalah koplanar. Selain itu, Anda akan dapat melihat contoh dan latihan yang diselesaikan selangkah demi selangkah dari vektor koplanar. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-vectores-coplanarios\"><\/span> Apa yang dimaksud dengan vektor koplanar?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dalam geometri analitik, pengertian vektor koplanar (atau coplanar) adalah sebagai berikut:<\/p>\n<p> <strong>Vektor koplanar adalah vektor yang berada pada bidang yang sama.<\/strong><\/p>\n<p> Oleh karena itu, dua vektor selalu koplanar karena sebuah bidang dapat dibentuk dengan sedikitnya 2 vektor. Sebaliknya, jika terdapat 3, 4 atau lebih vektor, ada kemungkinan salah satu vektor tersebut tidak berada pada bidang yang sama sehingga tidak sebidang. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/vecteurs-coplanaires-ou-coplanaires.webp\" alt=\"contoh vektor sebidang atau coplanar\" class=\"wp-image-3133\" width=\"353\" height=\"171\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Misalnya, pada grafik di atas Anda dapat melihat vektor-vektornya<\/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-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> mereka sebidang satu sama lain, karena mereka berada dalam bidang yang sama. Sebaliknya, kedua vektor ini tidak sebidang dengan vektor tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4bbbc56786695092eac40831aee80d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> , karena tidak ada bidang yang dapat terbentuk pada ruang yang memuat ketiga vektor tersebut.<\/p>\n<p> Dari sifat ini kita dapat menyimpulkan bahwa jika 3 vektor atau lebih bersifat koplanar, maka titik-titik yang membatasi vektor-vektor tersebut (awal dan akhir vektor) juga merupakan titik-titik koplanar. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuando-los-vectores-son-coplanarios\"><\/span> Kapankah vektor bersifat koplanar?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Seperti yang kita lihat dalam definisi vektor koplanar (atau koplanar), dua vektor selalu koplanar, tetapi lebih dari dua vektor tidak harus memperhatikan hubungan koplanaritas.<\/p>\n<p> Jadi, ada beberapa metode untuk menentukan apakah tiga vektor atau lebih bersifat koplanar:<\/p>\n<ul>\n<li> Jika hasil kali campuran tiga vektor (atau hasil kali tiga titik) sama dengan nol, berarti ketiga vektor tersebut sebidang. Jika Anda kurang paham tentang cara menghitung operasi ini, saya sarankan Anda untuk melihat <a href=\"https:\/\/mathority.org\/id\/contoh-hasil-kali-campuran-tiga-vektor-atau-hasil-kali-skalar-rangkap-tiga\/\">hasil kali campuran tiga vektor<\/a> , di sini Anda akan menemukan penjelasan serta contoh dan latihan yang diselesaikan.<\/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-849b07c1e268c4903e7bd13ef56bcaf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr] =0\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"92\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Jika suatu himpunan vektor dapat dinyatakan sebagai <a href=\"https:\/\/mathority.org\/id\/kombinasi-linier-vektor-contoh-latihan-yang-diselesaikan\/\">kombinasi linier dari dua vektor,<\/a> maka vektor-vektor tersebut koplanar, artinya 3 atau lebih vektor adalah koplanar jika dan hanya jika vektor-vektor tersebut bergantung linier. Untuk menunjukkan bahwa tiga vektor atau lebih merupakan kombinasi linier dari dua vektor, pangkat matriks yang dibentuk oleh semua vektor tersebut cukup sama dengan 2.<\/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-ef18656c1a261aa20598fc8f6a587323_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"rg(A) = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Penting bagi Anda untuk memiliki pemahaman yang baik tentang konsep <a href=\"https:\/\/mathority.org\/id\/vektor-bebas-dan-bergantung-linier-kemandirian-ketergantungan-linier\/\">ketergantungan dan kemandirian linier<\/a> , yaitu kapan dua vektor bergantung linier atau bebas linier, dan apa artinya. Jika Anda belum sepenuhnya paham, di tautan tersebut Anda akan menemukan penjelasan yang sangat mendetail, di mana, selain itu, Anda dapat melihat contoh dan latihan yang diselesaikan langkah demi langkah.<\/p>\n<ul>\n<li> Jika vektor-vektor yang dimaksud adalah <a href=\"https:\/\/mathority.org\/id\/vektor-paralel\/\">vektor-vektor sejajar<\/a> , berarti vektor-vektor tersebut juga koplanar, yaitu semua vektor-vektor sejajar terdapat pada bidang yang sama. <\/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-4efa93d26f00c6abc1180201f84d126a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\parallel  \\vv{\\text{v}} \\parallel \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-vectores-coplanarios\"><\/span> Memecahkan masalah vektor koplanar<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Tentukan apakah ketiga vektor berikut koplanar: <\/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-16f2fe8ce9dccfd2f5f2b26461ca54e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (3,1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" 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-ce008f944cfd9efa2c48d0083a479c89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}} = (2,3,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" 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-d8759d1ec233d68fc5f81dfb3b67beb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} = (-1,-5,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"119\" style=\"vertical-align: -5px;\"><\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Untuk memeriksa apakah ini adalah 3 vektor koplanar, kita harus menghitung hasil kali campuran antara ketiga vektor tersebut:<\/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-2a1e4b0655c0a3f0165c880f5e64cce0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]&amp; =\\begin{vmatrix} 3 &amp; 1 &amp; 2 \\\\[1.1ex] 2 &amp; 3 &amp; -1 \\\\[1.1ex] -1 &amp; -5 &amp; 4 \\end{vmatrix} \\\\[2ex] &amp;= 36+1-20+6-15-8 \\\\[2ex] &amp; = \\bm{0} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"271\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Hasil kali campuran ketiga vektor tersebut adalah nol, sehingga <strong>ketiga vektor tersebut koplanar<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 2<\/h3>\n<p> Tentukan apakah ketiga vektor berikut koplanar: <\/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-a7c6550cedc0ccb79a9bfdebdd9987cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,-2,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" 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-730594e946d69d5c0bd66b4b6d0f443c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}} = (-2,1,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" 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-a85ef59d300497c53b26259f19df79f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} = (6,-3,9)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -5px;\"><\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Salah satu cara untuk memeriksa apakah kita berhadapan dengan 3 vektor koplanar adalah dengan menyelesaikan hasil kali campuran antara ketiga vektor tersebut. Namun jika kita perhatikan lebih dekat komponen-komponen vektornya, kita dapat melihat bahwa vektor-vektor tersebut proporsional. Oleh karena itu, ketiga vektor tersebut sejajar satu sama lain.<\/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-0c5ac41bb15ea29bdc9736f100d1cf74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\parallel \\vv{\\text{v}} \\parallel \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan karena semua vektornya sejajar, <strong>maka keduanya merupakan 3 vektor koplanar<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 3<\/h3>\n<p> Tentukan apakah keempat vektor berikut koplanar: <\/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-10d37864162c9c1d2eae8f5b7c7df066_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{a}} = (2,1,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" 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-0ab59c17a2058de83ef95ee9b7021751_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{b}} = (1,-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" 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-1822f69d3738974e93084ea4c454d63f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{c}} = (-1,0,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" 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-f76c271b5fc0fe45fe9b2591346f083f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{d}} = (3,1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -5px;\"><\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Untuk mengetahui apakah keempat vektor tersebut koplanar, kita harus menghitung pangkat matriks yang terdiri dari semua vektor:<\/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-8384924c86edafd568505d5f80e1705d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} 2&amp;1&amp;1 \\\\[1.1ex] 1&amp;-1&amp;2 \\\\[1.1ex] -1&amp;0&amp;-1 \\\\[1.1ex] 3&amp;1&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"164\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini, kami menghitung ruang lingkup matriks tersebut berdasarkan determinan: <\/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-5db59e1c8bbf94b95483870d47cea1b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"rg(A) = \\ ?\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" 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-2778435c7f53952adf072419af8b268c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2&amp;1&amp;1 \\\\[1.1ex] 1&amp;-1&amp;2 \\\\[1.1ex] -1&amp;0&amp;-1 \\end{vmatrix}=0 \\quad  \\begin{vmatrix} 2&amp;1&amp;1 \\\\[1.1ex] 1&amp;-1&amp;2 \\\\[1.1ex]3&amp;1&amp;2\\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"280\" 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-82f278494a221879cc86da92ab4378c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2&amp;1&amp;1 \\\\[1.1ex] -1&amp;0&amp;-1 \\\\[1.1ex] 3&amp;1&amp;2\\end{vmatrix}=0 \\quad \\begin{vmatrix} 1&amp;-1&amp;2 \\\\[1.1ex] -1&amp;0&amp;-1 \\\\[1.1ex] 3&amp;1&amp;2\\end{vmatrix}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"294\" 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-889142ac348173dd6c838633007f2d06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2&amp;1 \\\\[1.1ex] 1&amp;-1\\end{vmatrix}= -3\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"136\" 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-ef18656c1a261aa20598fc8f6a587323_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"rg(A) = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pangkat matriks yang dibentuk oleh semua vektor ekuivalen dengan 2, sehingga <strong>keempat vektor tersebut koplanar<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 4<\/h3>\n<p> Hitung nilai parameter<\/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> sehingga 4 poin berikut ini koplanar: <\/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-9483cca4fc2a94923b7c72ed89fc2d5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(3,1,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" 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-a5b113e265916c03a6de0547cfeb380b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(2,1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" 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-add9ad10fb8badd9de84f8ad1dcfe38d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(0,-1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" 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-8d5054a3ce659090916cda61b74f60bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D(3,2,k)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Agar keempat titik tersebut koplanar, vektor-vektor yang ditentukan oleh titik-titik tersebut harus koplanar. Oleh karena itu kami menghitung vektor-vektor ini: <\/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-996a90c58f67665e4a68e9dd4de6c718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B- A = (2,1,2)-(3,1,4) = (-1,0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"365\" 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-ae552981d5729c931f5bbb26c133ecc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C- A = (0,-1,3)-(3,1,4) = (-3,-2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"392\" 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-91aec8ed49541d8c2d8ca0b3b1f8a20d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AD} = D- A = (3,2,k)-(3,1,4) = (0,1,k-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"371\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Yang matriks vektornya adalah:<\/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-c3d801efcf5b56dd858890720797d6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} -1&amp;0&amp;-2 \\\\[1.1ex] -3&amp;-2&amp;-1 \\\\[1.1ex] 0&amp;1&amp;k-4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"181\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agar vektor-vektor yang dihasilkan bersifat koplanar, pangkat matriksnya harus 2. Oleh karena itu, determinan seluruh matriks 3&#215;3 harus 0: <\/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-bb7d3b31c10096d100843d781a85b621_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -1&amp;0&amp;-2 \\\\[1.1ex] -3&amp;-2&amp;-1 \\\\[1.1ex] 0&amp;1&amp;k-4\\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"160\" 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-b265559f1f5505b8c40a89f0d69f0c10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2k-3 =0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Akhirnya, kami menyelesaikan hal yang tidak diketahui <\/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-7cf6d2c84f82625cb8a795ee1394251f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" 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-f79b9d3960668149408038b9cb1d1e0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2k =3\" 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-e67be6e218d206fe735f54a6125b3d2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{k =}\\mathbf{\\cfrac{3}{2}}\" 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>Di halaman ini Anda akan mempelajari apa itu vektor koplanar dan cara mengetahui apakah 2, 3, 4 atau lebih vektor adalah koplanar. Selain itu, Anda akan dapat melihat contoh dan latihan yang diselesaikan selangkah demi selangkah dari vektor koplanar. Apa yang dimaksud dengan vektor koplanar? Dalam geometri analitik, pengertian vektor koplanar (atau coplanar) adalah sebagai &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/vektor-koplanar-atau-koplanar\/\"> <span class=\"screen-reader-text\">Vektor koplanar (atau koplanar).<\/span> Selengkapnya &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":[54],"tags":[],"class_list":["post-259","post","type-post","status-publish","format-standard","hentry","category-vektor"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Vektor koplanar (atau koplanar) - 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\/id\/vektor-koplanar-atau-koplanar\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Vektor koplanar (atau koplanar) - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan mempelajari apa itu vektor koplanar dan cara mengetahui apakah 2, 3, 4 atau lebih vektor adalah koplanar. 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