{"id":273,"date":"2023-07-10T00:23:44","date_gmt":"2023-07-10T00:23:44","guid":{"rendered":"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/"},"modified":"2023-07-10T00:23:44","modified_gmt":"2023-07-10T00:23:44","slug":"rumus-jarak-garis-ke-bidang-dalam-ruang","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/","title":{"rendered":"Jarak garis ke bidang di ruang angkasa"},"content":{"rendered":"<p>Di sini Anda akan menemukan cara menghitung jarak dari garis ke bidang dan, sebagai tambahan, Anda akan dapat melihat contoh dan latihan yang diselesaikan langkah demi langkah. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-distancia-de-una-recta-a-un-plano\"><\/span> Berapa jarak antara garis dan bidang?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dalam geometri analitik, jarak antara garis dan bidang dalam ruang bergantung pada posisi relatif antara dua elemen geometri berikut:<\/p>\n<ul>\n<li> Jika <strong>garis termasuk dalam bidang<\/strong> atau <strong>garis dan bidang sejajar<\/strong> , maka jarak yang memisahkan keduanya adalah nol.<\/li>\n<li> Jika <strong>garis sejajar dengan bidang<\/strong> , jarak garis ke bidang ditentukan dengan mengambil titik mana saja pada garis dan menghitung jarak titik tersebut ke bidang. <\/li>\n<\/ul>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/distance-entre-une-ligne-et-un-plan.webp\" alt=\"jarak garis ke bidang di ruang angkasa\" class=\"wp-image-4062\" width=\"459\" height=\"252\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Oleh karena itu, untuk menghitung jarak garis ke bidang, penting bagi Anda untuk mengetahui cara menentukan <a href=\"https:\/\/mathority.org\/id\/kedudukan-relatif-suatu-garis-dan-bidang\/\">posisi relatif antara garis dan bidang<\/a> dan cara menghitung <a href=\"https:\/\/mathority.org\/id\">jarak antara titik dan bidang<\/a> . Jadi jika Anda belum sepenuhnya paham atau belum mengetahui rumusnya, kami sarankan Anda melihat terlebih dahulu halaman tertaut, di mana Anda akan menemukan penjelasan, contoh, dan latihan yang diselesaikan langkah demi langkah. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-calcular-la-distancia-entre-una-recta-y-un-plano\"><\/span> Contoh penghitungan jarak antara garis dan bidang<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Agar Anda dapat melihat cara mencari jarak antara garis dan bidang di ruang angkasa (di R3), kita akan menyelesaikan soal sebagai contoh:<\/p>\n<ul>\n<li> Seberapa jauh garisnya\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> di pesawat<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26d6788550ffd50fe94542bb3e8ee615_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> ?<\/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-4785f2386b4ff9387b56cf8c5645c62d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ \\begin{cases}x=-2+t \\\\[1.7ex] y=1-3t \\\\[1.7ex] z=-1+2t\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"137\" 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-5d8872d79c59216f4a8489f75a751cdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi : \\ 4x+2y+z-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"184\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Untuk mencari jarak antara garis dan bidang, Anda harus mengetahui terlebih dahulu posisi relatif antara keduanya.<\/p>\n<p> Di satu sisi, garis didefinisikan dalam bentuk persamaan parametrik, sehingga vektor arahnya dan titik yang dilaluinya 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-0765b71f9eead3b644fd5de150cdf321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ \\begin{cases} \\vv{\\text{v}}_r =(1,-3,2) \\\\[2ex] P(-2,1,-1) \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"154\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dan sebaliknya, vektor normal bidang 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-f4fe9c16b8d08778c89ab62afcd8e34e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n} =(4,2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Jadi, untuk menentukan kedudukan relatif antara bidang dan garis, perlu dihitung hasil kali skalar antara vektor arah garis dan vektor tegak lurus bidang:<\/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-39d2d450c32a6fdf3182a3d16cc18116_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\vv{\\text{v}}_r \\cdot \\vv{n} &amp; = (1,-3,2) \\cdot (4,2,1) \\\\[2ex] &amp; = 1 \\cdot 4-3 \\cdot 2 +2\\cdot 1 \\\\[2ex] &amp;= 4 -6 +2 \\\\[2ex] &amp; = 0\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"140\" width=\"202\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Hasil perkalian titiknya adalah nol, sehingga garis hanya dapat dimuat pada bidang atau sejajar dengannya. Jadi, untuk mengetahui kasusnya, kita substitusikan koordinat kartesius titik pada garis tersebut ke dalam persamaan bidang: <\/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-e4916138781ab0b14c0b827b212fdd98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x+2y+z-6=0 \\ \\xrightarrow{P(-2,1,-1)} \\ 4\\cdot (-2) +2\\cdot 1 -1-6= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"460\" 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-15f3d9b73d0b2646f6d0173989e7b20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8+2-1-6 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"146\" style=\"vertical-align: -2px;\"><\/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-d93d0450119c6530b10ed45266e017e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-13 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Dengan mensubstitusikan titik garis ke persamaan bidang diperoleh pertidaksamaan, sehingga titik tersebut tidak memperhatikan persamaan bidang dan akibatnya garis dan bidang sejajar.<\/p>\n<p> Setelah kita mengetahui bahwa garis dan bidang sejajar, sekarang kita dapat menghitung jarak geometri antara keduanya. Caranya, kita ambil titik pada garis dan menghitung jarak dari titik tersebut ke bidang.<\/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-4413892ee4946e8c956d69a16a8d43ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2,1,-1) \\qquad \\qquad \\pi: \\ 4x+2y+z-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"353\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Jadi, kita menggunakan rumus jarak suatu titik ke bidang:<\/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-7fa5cd77905fa8723448de603b90b705_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{\\lvert A\\cdot x_0+B\\cdot y_0+C\\cdot z_0+D\\rvert}{\\sqrt{A^2+B^2+C^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"300\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Sekarang kita substitusikan nilai setiap yang tidak diketahui ke dalam rumus:<\/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-beb8f4629b0195d8ea0870d678585895_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{\\lvert 4\\cdot (-2)+2\\cdot 1+1\\cdot (-1)-6\\rvert}{\\sqrt{4^2+2^2+1^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"310\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Dan terakhir, kami melakukan operasi: <\/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-570743258f771c18a79b360b80b6a462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{\\lvert -8+2-1-6\\rvert}{\\sqrt{16+4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"204\" style=\"vertical-align: -16px;\"><\/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-3ea97bcf06817734423f2e6e662a1658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{\\lvert -13\\rvert}{\\sqrt{21}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"121\" style=\"vertical-align: -16px;\"><\/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-0a19bec182b20b9b1a5a08b41cd817c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{13}{\\sqrt{21}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"112\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Sehingga jarak garis ke bidang sama dengan jarak titik ke bidang yang dihitung:<\/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-82d2055df31134bca62040c367778b50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{d(r,\\pi)=d(P,\\pi) = }\\cfrac{\\bm{13}}{\\bm{\\sqrt{21}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"180\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Tentunya jarak harus selalu memberi kita nilai positif, karena jarak selalu positif. Jika kita mendapat hasil negatif, berarti kita melakukan kesalahan dalam mengambil langkah.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Di sini Anda akan menemukan cara menghitung jarak dari garis ke bidang dan, sebagai tambahan, Anda akan dapat melihat contoh dan latihan yang diselesaikan langkah demi langkah. Berapa jarak antara garis dan bidang? Dalam geometri analitik, jarak antara garis dan bidang dalam ruang bergantung pada posisi relatif antara dua elemen geometri berikut: Jika garis termasuk &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/\"> <span class=\"screen-reader-text\">Jarak garis ke bidang di ruang angkasa<\/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-273","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>Jarak garis ke bidang di ruang angkasa - 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\/rumus-jarak-garis-ke-bidang-dalam-ruang\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Jarak garis ke bidang di ruang angkasa - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di sini Anda akan menemukan cara menghitung jarak dari garis ke bidang dan, sebagai tambahan, Anda akan dapat melihat contoh dan latihan yang diselesaikan langkah demi langkah. Berapa jarak antara garis dan bidang? Dalam geometri analitik, jarak antara garis dan bidang dalam ruang bergantung pada posisi relatif antara dua elemen geometri berikut: Jika garis termasuk &hellip; Jarak garis ke bidang di ruang angkasa Selengkapnya &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-10T00:23:44+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/distance-entre-une-ligne-et-un-plan.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=\"2 menit\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Jarak garis ke bidang di ruang angkasa\",\"datePublished\":\"2023-07-10T00:23:44+00:00\",\"dateModified\":\"2023-07-10T00:23:44+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/\"},\"wordCount\":403,\"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\/rumus-jarak-garis-ke-bidang-dalam-ruang\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/\",\"url\":\"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/\",\"name\":\"Jarak garis ke bidang di ruang angkasa - 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Berapa jarak antara garis dan bidang? Dalam geometri analitik, jarak antara garis dan bidang dalam ruang bergantung pada posisi relatif antara dua elemen geometri berikut: Jika garis termasuk &hellip; Jarak garis ke bidang di ruang angkasa Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/","article_published_time":"2023-07-10T00:23:44+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/distance-entre-une-ligne-et-un-plan.webp"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"2 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Jarak garis ke bidang di ruang angkasa","datePublished":"2023-07-10T00:23:44+00:00","dateModified":"2023-07-10T00:23:44+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/"},"wordCount":403,"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\/rumus-jarak-garis-ke-bidang-dalam-ruang\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/","url":"https:\/\/mathority.org\/id\/rumus-jarak-garis-ke-bidang-dalam-ruang\/","name":"Jarak garis ke bidang di ruang angkasa - 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