{"id":260,"date":"2023-07-10T06:51:49","date_gmt":"2023-07-10T06:51:49","guid":{"rendered":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/"},"modified":"2023-07-10T06:51:49","modified_gmt":"2023-07-10T06:51:49","slug":"contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/","title":{"rendered":"Poin koplanar (atau koplanar)."},"content":{"rendered":"<p>Di halaman ini Anda akan mengetahui apa itu titik-titik sebidang (atau sebidang) dan bagaimana mengetahui apakah titik-titik tertentu sebidang atau tidak. Selain itu, Anda akan dapat melihat contoh dan latihan dengan latihan poin koplanar yang diselesaikan. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-puntos-coplanarios\"><\/span> Apa yang dimaksud dengan titik koplanar?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dalam geometri analitik, arti titik sebidang (atau coplanar) adalah sebagai berikut:<\/p>\n<p> <strong>Titik sebidang adalah titik-titik yang berada pada bidang yang sama.<\/strong><\/p>\n<p> Oleh karena itu, 2 atau 3 titik selalu sebidang karena sebuah bidang dapat dibentuk dengan sedikitnya 3 titik. Sebaliknya, jika terdapat 4, 5 titik atau lebih, ada kemungkinan beberapa titik 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\/points-coplanaires.webp\" alt=\"titik-titik sebidang\" class=\"wp-image-3165\" width=\"411\" height=\"215\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Misalnya, pada representasi grafis di atas, Anda dapat melihat bahwa titik A, C, D, dan F saling sebidang karena terletak pada bidang yang sama. Sebaliknya keempat titik tersebut tidak sebidang dengan titik B, E, dan G, karena tidak ada bidang yang dapat terbentuk pada ruang yang memuat semua titik tersebut.<\/p>\n<p> Dari sifat ini kita dapat menyimpulkan bahwa vektor-vektor yang didefinisikan oleh titik-titik koplanar juga merupakan vektor-vektor koplanar, yaitu terletak pada bidang yang sama. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuando-los-puntos-son-coplanarios\"><\/span> Kapan titik-titiknya sebidang?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Seperti yang kita lihat dalam definisi titik-titik sebidang (atau sebidang), dua atau tiga titik selalu sebidang, tetapi lebih dari tiga titik tidak harus menghormati hubungan koplanaritas.<\/p>\n<p> Jadi, ada 2 metode utama untuk menentukan apakah empat titik atau lebih adalah koplanar:<\/p>\n<ul>\n<li> Salah satu cara untuk mengetahui apakah titik-titik tersebut sebidang adalah dengan vektor-vektor yang ditentukan oleh titik-titik tersebut: <strong>jika vektor-vektor tersebut sebidang<\/strong> , maka titik-titik tersebut juga sebidang.<\/li>\n<\/ul>\n<p> Tentunya, untuk menerapkan metode ini Anda perlu mengetahui kapan vektor-vektor tersebut koplanar. Namun karena ada juga beberapa cara untuk menentukan apakah suatu himpunan vektor bersifat koplanar, kami sarankan untuk mempelajari <a href=\"https:\/\/mathority.org\/id\/vektor-koplanar-atau-koplanar\/\">cara mengetahui apakah suatu vektor bersifat koplanar<\/a> . Di sini Anda akan menemukan semua prosedur yang ada untuk mencari 2, 3, 4 atau lebih vektor koplanar, serta contoh dan latihan penyelesaiannya.<\/p>\n<ul>\n<li> Cara lain untuk mengetahui apakah suatu himpunan titik-titik adalah koplanar adalah <strong><a href=\"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/\">dengan mencari persamaan bidang<\/a><\/strong> yang dibentuk oleh 3 titik dalam himpunan tersebut, dan jika titik-titik lainnya memenuhi persamaan tersebut, berarti semua titik dalam himpunan tersebut adalah koplanar.<\/li>\n<\/ul>\n<p> Meskipun bergantung pada masalahnya, sebaiknya gunakan metode pertama dari kedua metode tersebut, karena lebih mudah dan cepat untuk memeriksa apakah vektor-vektor tersebut koplanar daripada menghitung persamaan bidang. Tapi, yang jelas, gunakan mana saja yang Anda suka. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-puntos-coplanarios\"><\/span> Memecahkan masalah titik koplanar<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Tentukan apakah ketiga titik berikut ini sebidang: <\/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-9a6d39406adb983eaad3c40c2e5d352c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(3,5,1)\" 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-a2db4553e8942cf714a98783699f44f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(0,-2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" 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-93a412e1bb305cd785882fd481852c9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(4,-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" 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\"> Dalam hal ini tidak perlu melakukan perhitungan apapun karena <strong>3 titik selalu sebidang<\/strong> , apapun itu.<\/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 keempat titik berikut ini sebidang: <\/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-adbc97b4e3f0c30a1cd47fb6eea0b566_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(1,2,0)\" 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-d5ca7779c360bcb8ad9cf4d6b6a455a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(-1,3,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" 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-8910b93724e6f59b070ee26673c9426a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(1,0,-1)\" 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-6d40d5bb66afe9a8e8445cdb8b7cac0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D(2,5,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-e7922b55a4e968ce9ef945ddf4b0973e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B- A = (-1,3,4)-(1,2,0) = (-2,1,4)\" 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-8dfa4c2463f3a69890706b0387f1fbc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C- A = (1,0,-1)-(1,2,0) = (0,-2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"378\" 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-e5617a7317e2c18fcb39b3be782e26c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AD} = D- A = (2,5,-3)-(1,2,0) = (1,3,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"367\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita buat matriks yang dibentuk oleh vektor-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-873ec750665fdf215a648778d6fcf042_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} -2&amp;1&amp;4 \\\\[1.1ex]0&amp;-2&amp;-1 \\\\[1.1ex] 1&amp;3&amp;-3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"164\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agar vektor-vektor yang dihasilkan bersifat koplanar, pangkat matriks sebelumnya harus sama dengan 2. Dan untuk ini, determinan seluruh matriks 3\u00d73 harus sama dengan nol: <\/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-18eb85dd3870c1e7c865705200cf6414_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-3eadb62f19c07b8a646fe9b2a00f9003_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -2&amp;1&amp;4 \\\\[1.1ex]0&amp;-2&amp;-1 \\\\[1.1ex] 1&amp;3&amp;-3\\end{vmatrix} =-11\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"198\" 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-466d66279a42e8aeb422d568e1547ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"rg(A)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Akan tetapi determinan seluruh matriks bukan nol, sehingga pangkat matriks tersebut adalah 3, sehingga <strong>keempat titik tersebut tidak sebidang<\/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> Cari tahu apakah lima 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-083776221bb4167cd960c67da2435d9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(3,1,1)\" 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-1a43aebd7e3c897898050e7440b0fe9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(0,0,2)\" 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-59e69260b8e3baacdfd4f673aeb87fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(4,-2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" 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-e862d9c0d7c715b2268b594c199f506a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D(-1,1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" 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-f361ff1c29cf0a1dba88da8bdb1fb448_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"E(2,4,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" 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 kelima titik tersebut koplanar, vektor-vektor yang didefinisikan 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-9b6a321df8b7b3e37dc31bf0cb9542dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B- A = (0,0,2)-(3,1,1) = (-3,-1,1)\" 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-611bc1c76bcadfe27e2a793128a5a3c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C- A = (4,-2,-1)-(3,1,1) = (1,-3,-2)\" 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-0ff805267450b865998e62bbc20111b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AD} = D- A = (-1,1,3)-(3,1,1) = (-4,0,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"366\" 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-494b794700c9c99091abf0fd59523dd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AE} = E- A = (2,4,3)-(3,1,1) = (-1,3,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"350\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita buat matriks yang terdiri dari vektor-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-5d571ea172dbf5e6b7d85da6db466eb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} -3&amp;-1&amp;1 \\\\[1.1ex]1&amp;-3&amp;-2 \\\\[1.1ex] -4&amp;0&amp;2\\\\[1.1ex] -1&amp;3&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\"> Agar vektor-vektor yang dihasilkan bersifat koplanar, pangkat matriks sebelumnya harus sama dengan 2. Oleh karena itu, kita menghitung pangkat matriks vektor-vektor dengan determinan untuk memeriksa apakah vektor-vektor tersebut 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-18eb85dd3870c1e7c865705200cf6414_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-bec15883d9e3aa5a401afa41cd3992ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -3&amp;-1&amp;1 \\\\[1.1ex]1&amp;-3&amp;-2 \\\\[1.1ex] -4&amp;0&amp;2\\end{vmatrix} = 0 \\qquad \\begin{vmatrix} -3&amp;-1&amp;1 \\\\[1.1ex]1&amp;-3&amp;-2 \\\\[1.1ex] -1&amp;3&amp;2\\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"326\" 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-cd2f8cfa577e91b74c69e50cc36dfd9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -3&amp;-1&amp;1 \\\\[1.1ex] -4&amp;0&amp;2\\\\[1.1ex] -1&amp;3&amp;2\\end{vmatrix} = 0 \\qquad \\begin{vmatrix} 1&amp;-3&amp;-2 \\\\[1.1ex] -4&amp;0&amp;2\\\\[1.1ex] -1&amp;3&amp;2\\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"312\" 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-d551dddceb916cc6cf8bf4137903da1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -3&amp;-1 \\\\[1.1ex] 1&amp;-3\\end{vmatrix} =10\\neq  0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" 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-6865ac4ae80c8b5a6eff791a9da5a937_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 matriksnya setara dengan 2, sehingga vektor-vektornya koplanar dan oleh karena itu <strong>5 titiknya juga 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<p> Terakhir, jika artikel ini bermanfaat bagi Anda, Anda mungkin juga tertarik dengan cara <a href=\"https:\/\/mathority.org\/id\/rumus-jarak-antara-dua-titik-contoh-geometri-dan-latihan-soal-yang-diselesaikan\/\">menghitung jarak antara dua titik (rumus)<\/a> , karena terkadang dalam soal geometri analitik kita ditanyai berapa jarak antara dua titik. Di halaman tertaut Anda akan menemukan penjelasan yang sangat rinci, serta contoh dan latihan yang diselesaikan langkah demi langkah.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Di halaman ini Anda akan mengetahui apa itu titik-titik sebidang (atau sebidang) dan bagaimana mengetahui apakah titik-titik tertentu sebidang atau tidak. Selain itu, Anda akan dapat melihat contoh dan latihan dengan latihan poin koplanar yang diselesaikan. Apa yang dimaksud dengan titik koplanar? Dalam geometri analitik, arti titik sebidang (atau coplanar) adalah sebagai berikut: Titik sebidang &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/\"> <span class=\"screen-reader-text\">Poin 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":[47],"tags":[],"class_list":["post-260","post","type-post","status-publish","format-standard","hentry","category-titik-garis-dan-bidang"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Poin coplanar (atau coplanar) - 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\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Poin coplanar (atau coplanar) - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan mengetahui apa itu titik-titik sebidang (atau sebidang) dan bagaimana mengetahui apakah titik-titik tertentu sebidang atau tidak. Selain itu, Anda akan dapat melihat contoh dan latihan dengan latihan poin koplanar yang diselesaikan. Apa yang dimaksud dengan titik koplanar? Dalam geometri analitik, arti titik sebidang (atau coplanar) adalah sebagai berikut: Titik sebidang &hellip; Poin koplanar (atau koplanar). Selengkapnya &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-10T06:51:49+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/points-coplanaires.webp\" \/>\n<meta name=\"author\" content=\"Tim Mathority\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Ditulis oleh\" \/>\n\t<meta name=\"twitter:data1\" content=\"Tim Mathority\" \/>\n\t<meta name=\"twitter:label2\" content=\"Estimasi waktu membaca\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 menit\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Poin koplanar (atau koplanar).\",\"datePublished\":\"2023-07-10T06:51:49+00:00\",\"dateModified\":\"2023-07-10T06:51:49+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/\"},\"wordCount\":672,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/id\/#organization\"},\"articleSection\":[\"Titik, garis, dan bidang\"],\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/\",\"url\":\"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/\",\"name\":\"Poin coplanar (atau coplanar) - 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Mathority","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/","og_locale":"id_ID","og_type":"article","og_title":"Poin coplanar (atau coplanar) - Mathority","og_description":"Di halaman ini Anda akan mengetahui apa itu titik-titik sebidang (atau sebidang) dan bagaimana mengetahui apakah titik-titik tertentu sebidang atau tidak. Selain itu, Anda akan dapat melihat contoh dan latihan dengan latihan poin koplanar yang diselesaikan. Apa yang dimaksud dengan titik koplanar? Dalam geometri analitik, arti titik sebidang (atau coplanar) adalah sebagai berikut: Titik sebidang &hellip; Poin koplanar (atau koplanar). Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/","article_published_time":"2023-07-10T06:51:49+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/points-coplanaires.webp"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"3 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Poin koplanar (atau koplanar).","datePublished":"2023-07-10T06:51:49+00:00","dateModified":"2023-07-10T06:51:49+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/"},"wordCount":672,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"articleSection":["Titik, garis, dan bidang"],"inLanguage":"id","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/","url":"https:\/\/mathority.org\/id\/contoh-poin-coplanar-atau-coplanar-yang-diselesaikan-latihan\/","name":"Poin coplanar (atau coplanar) - 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