{"id":265,"date":"2023-07-10T04:25:24","date_gmt":"2023-07-10T04:25:24","guid":{"rendered":"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/"},"modified":"2023-07-10T04:25:24","modified_gmt":"2023-07-10T04:25:24","slug":"persamaan-bidang-dalam-ruang","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/","title":{"rendered":"Persamaan bidang di luar angkasa"},"content":{"rendered":"<p>Di halaman ini Anda akan menemukan rumus untuk semua persamaan dalam denah dan cara menghitungnya. Anda juga akan menemukan cara mencari persamaan bidang apa pun dengan vektor normalnya. Selain itu, Anda akan dapat melihat contoh dan latihan dengan latihan persamaan denah yang diselesaikan. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-ecuacion-del-plano\"><\/span> Apa persamaan bidangnya?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dalam geometri analitik, <strong>persamaan bidang<\/strong> adalah persamaan yang memungkinkan bidang apa pun dinyatakan secara matematis. Jadi, untuk mencari persamaan suatu bidang, Anda hanya memerlukan sebuah titik dan dua vektor bebas linier yang dimiliki bidang tersebut.<\/p>\n<p> Sebelum melanjutkan ke penjelasan persamaan bidang, ada baiknya anda memahami apa itu <a href=\"https:\/\/mathority.org\/id\/ilmu-ukur-bidang\/\">bidang (geometri)<\/a> , karena jika tidak maka akan ada hal-hal yang tidak anda pahami. Jika Anda belum sepenuhnya paham, Anda dapat memeriksanya di tautan ini, tempat kami memusatkan semua yang perlu Anda ketahui tentang rencana tersebut. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuales-son-las-ecuaciones-del-plano\"><\/span> Apa persamaan rencana tersebut?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Seperti yang kita lihat pada definisi persamaan bidang, setiap titik pada bidang datar dapat dinyatakan sebagai kombinasi linier dari 1 titik dan 2 vektor. <\/p>\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\/equations-planes.webp\" alt=\"persamaan bidang xy online\" class=\"wp-image-2443\" width=\"404\" height=\"142\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Namun, syarat yang diperlukan agar persamaan tersebut dapat berkorespondensi dengan suatu bidang adalah bahwa kedua vektor pada bidang tersebut mempunyai independensi linier, yaitu kedua vektor tersebut tidak boleh sejajar satu sama lain.<\/p>\n<p> Jadi, semua jenis persamaan bidang adalah: <strong>persamaan vektor<\/strong> , <strong>persamaan parametrik<\/strong> , <strong>persamaan implisit (atau umum)<\/strong> , dan <strong>persamaan bidang kanonik (atau segmental)<\/strong> .<\/p>\n<p> Selanjutnya kita akan melihat secara detail penjelasan dan rumus semua persamaan denahnya. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-vectorial-del-plano\"><\/span> Persamaan vektor bidang<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Perhatikan sebuah titik dan dua vektor arah pada sebuah 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-cf5d4130501bb01b15aa80f8f80caf1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c} P(P_x,P_y,P_z) \\\\[2ex] \\vv{\\text{u}}=(\\text{u}_x,\\text{u}_y,\\text{u}_z)\\\\[2ex] \\vv{\\text{v}}=(\\text{v}_x,\\text{v}_y,\\text{v}_z)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"116\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <strong>Rumus persamaan vektor suatu bidang<\/strong> 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-d9227901692832cb0c176a896d35e896_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      (x,y,z)=P+\\lambda \\vv{\\text{u}} + \\mu \\vv{\\text{v}} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Atau setara:<\/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-78b41d21b63c22ec05d3f93576a897e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y,z)=(P_x,P_y,P_z)+\\lambda (\\text{u}_x,\\text{u}_y,\\text{u}_z) + \\mu (\\text{v}_x,\\text{v}_y,\\text{v}_z)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"398\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Emas<\/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-2b5c45836864531b8e37025dabadd24a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" 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-461fe1a58a75801541487ddf10d32abd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"><\/p>\n<p> adalah dua skalar, yaitu dua bilangan real. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuaciones-parametricas-del-plano\"><\/span> Persamaan parametrik bidang<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Persamaan parametrik suatu bidang dapat ditentukan dari persamaan vektornya. Di bawah ini Anda dapat melihat demonya.<\/p>\n<p> Biarkan persamaan vektor bidang apa pun menjadi:<\/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-78b41d21b63c22ec05d3f93576a897e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y,z)=(P_x,P_y,P_z)+\\lambda (\\text{u}_x,\\text{u}_y,\\text{u}_z) + \\mu (\\text{v}_x,\\text{v}_y,\\text{v}_z)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"398\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Kami mengoperasikan dan pertama-tama melakukan perkalian vektor dengan skalar:<\/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-9eb7c00ddf8ba235e3698c85a0f23db0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y,z)=(P_x,P_y,P_z)+ (\\lambda\\text{u}_x,\\lambda\\text{u}_y,\\lambda\\text{u}_z) +(\\mu\\text{v}_x,\\mu\\text{v}_y,\\mu\\text{v}_z)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"440\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Selanjutnya kita tambahkan komponen-komponennya:<\/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-e3ff52d13a3d4400800b0f72148f99c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y,z)=(P_x+\\lambda \\text{u}_x + \\mu \\text{v}_x,P_y+\\lambda \\text{u}_y + \\mu \\text{v}_y,P_z+\\lambda \\text{u}_z + \\mu \\text{v}_z)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"467\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Dan terakhir, kita memperoleh persamaan parametrik denah tersebut dengan mengasimilasi koordinat yang bersesuaian dengan setiap variabel secara terpisah:<\/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-e1791802331aa9973126b3d7c7f1b716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\displaystyle \\begin{cases}x=P_x + \\lambda \\text{u}_x + \\mu \\text{v}_x \\\\[1.7ex] y=P_y + \\lambda \\text{u}_y + \\mu \\text{v}_y\\\\[1.7ex] z=P_z + \\lambda\\text{u}_z + \\mu \\text{v}_z \\end{cases} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Emas:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b5c45836864531b8e37025dabadd24a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" 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-461fe1a58a75801541487ddf10d32abd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"><\/p>\n<p> adalah dua skalar, yaitu dua bilangan real.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b77e9af6839a6bc60da39dd1798dd6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{u}_x,\\text{u}_y,\\text{u}_z\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"69\" style=\"vertical-align: -6px;\"><\/p>\n<p> adalah komponen dari salah satu dari dua vektor pemandu denah tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8a3eef2109d6a80a337c88337a1443e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(\\text{u}_x,\\text{u}_y,\\text{u}_z).\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" style=\"vertical-align: -6px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f61b32275ccdbca7f8d5e0b3c750dd35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{v}_x,\\text{v}_y,\\text{v}_z\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -6px;\"><\/p>\n<p> adalah komponen vektor pengarah denah lainnya <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-75c6a57a037206319f16dec389993ded_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(\\text{v}_x,\\text{v}_y,\\text{v}_z).\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"119\" style=\"vertical-align: -6px;\"><\/p>\n<\/li>\n<\/ul>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-implicita-o-general-del-plano\"><\/span> Persamaan bidang implisit atau umum<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Perhatikan sebuah titik dan dua vektor arah pada sebuah 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-cf5d4130501bb01b15aa80f8f80caf1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c} P(P_x,P_y,P_z) \\\\[2ex] \\vv{\\text{u}}=(\\text{u}_x,\\text{u}_y,\\text{u}_z)\\\\[2ex] \\vv{\\text{v}}=(\\text{v}_x,\\text{v}_y,\\text{v}_z)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"116\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Persamaan implisit, umum, atau Cartesian suatu bidang diperoleh dengan menyelesaikan determinan berikut dan menetapkan hasilnya sama dengan 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-68d67612dfa54d76666aa37b702a472f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dengan demikian, <strong>persamaan implisit atau umum dari rencana yang dihasilkan<\/strong> adalah sebagai berikut:<\/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-7dcacf16123986ecd33dace4f4411914_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\displaystyle Ax+By+Cz+D=0 \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Persamaan bidang jenis ini disebut juga persamaan bidang kartesius. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-canonica-o-segmentaria-del-plano\"><\/span> Persamaan bidang kanonik atau segmental<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> <strong>Rumus persamaan kanonik atau segmental suatu bidang<\/strong> adalah sebagai berikut:<\/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-7c19853d465a703aa398bde04fa3222c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\displaystyle \\cfrac{x}{a}+\\cfrac{y}{b} + \\cfrac{z}{c} = 1  \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Emas:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah titik potong antara bidang dan sumbu X.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah titik potong antara bidang dan sumbu Y.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"c\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> Di sinilah bidang memotong sumbu Z. <\/li>\n<\/ul>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p> Persamaan kanonik (atau persamaan segmental) bidang juga dapat diperoleh dari persamaan umumnya:<\/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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Pertama, kita selesaikan koefisien D dari persamaan:<\/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-7e6829185741f883a29bf004cbf570a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz=-D\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"166\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Kemudian kita membagi seluruh persamaan denah dengan nilai parameter D yang diubah tandanya:<\/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-04843e22b4176c0ce921483f93dffeab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{Ax+By+Cz}{-D}=\\cfrac{-D}{-D}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"176\" style=\"vertical-align: -12px;\"><\/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-278ce62f85ca44612254f48e96154726_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{Ax}{-D}+\\cfrac{By}{-D}+\\cfrac{Cz}{-D}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"166\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Dan, dengan menggunakan sifat-sifat pecahan, kita sampai pada ekspresi berikut:<\/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-a58254e773c7c14b5b337a4330997125_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{-\\frac{D}{A}}+\\cfrac{y}{-\\frac{D}{A}}+\\cfrac{z}{-\\frac{D}{A}}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"167\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, dari ungkapan ini kami menyimpulkan rumus yang memungkinkan suku-suku persamaan kanonik atau segmental suatu bidang dihitung secara langsung:<\/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-86975df14352b1b0c2ca05d2daaf40f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a=-\\cfrac{D}{A} \\qquad b=-\\cfrac{D}{B} \\qquad c=-\\cfrac{D}{C}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"260\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, untuk dapat membentuk varian persamaan denah tersebut, koefisien A, B, dan C harus berbeda dari nol, sehingga menghindari ketidakpastian pecahan. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-calcular-la-ecuacion-de-un-plano-a-partir-de-su-vector-normal\"><\/span> Cara menghitung persamaan bidang dari vektor normalnya<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Masalah yang sangat umum dalam persamaan bidang adalah mencari persamaan bidang tertentu jika diberi sebuah titik dan vektor normalnya (atau tegak lurus). Jadi, mari kita lihat cara kerjanya.<\/p>\n<p> Namun perlu diketahui terlebih dahulu bahwa <strong>komponen X, Y, Z dari vektor tegak lurus bidang tersebut <strong>masing-masing<\/strong> berimpit<\/strong> <strong>dengan koefisien A, B, C dari persamaan implisit (atau umum) bidang tersebut.<\/strong><\/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-27f3ee5d7e81864550f3b86fdd53e89d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\color{orange} \\boxed{ \\color{black} \\quad \\pi : \\ Ax+By+C+D = 0 \\quad \\iff \\quad \\vv{n} = (A,B,C) \\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"540\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Emas<\/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-10affe1faee06a5faa4ef6d9c0473b1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah vektor ortogonal terhadap bidang<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26622dd58bf71cd1b543c3d83233c561_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Setelah kita mengetahui hubungan sebelumnya, mari kita lihat contoh penyelesaian soal persamaan bidang jenis ini:<\/p>\n<ul>\n<li> Tentukan persamaan implisit atau umum bidang yang melalui titik tersebut\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6df0e548515bb2b24f352853a2614015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan salah satu vektor normalnya adalah<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-44298f830c420011a4326017f5fd7cfb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(3,-1,2) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<p> Rumus persamaan bidang implisit, umum, atau kartesius 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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Jadi, dari vektor normal kita dapat mencari koefisien A, B dan C karena ekuivalen dengan komponen-komponen vektor normalnya:<\/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-596b3fdc65160234c06b0d28aebea74f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(3,-1,2) \\ \\longrightarrow \\ 3x-1y+2z+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"322\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sedangkan kita hanya perlu mencari parameter D. Caranya, kita substitusikan koordinat titik milik bidang tersebut ke dalam persamaan: <\/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-6df0e548515bb2b24f352853a2614015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,0,-2)\" 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-09a0b099394ba4634fb6d7aad3a627e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3\\cdot 1-0+2\\cdot (-2)+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"210\" 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-b90d57a1708925626a300c7e5db673ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3-4+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"109\" 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-d2dfb99a77bfcee1b72540db9cf579a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"91\" 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-16ace3f6683252e5630a1091bbc0404e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Jadi persamaan implisit atau umum dari rencana tersebut 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-cde6db2f935eb7d0ad39141f86aab013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{3x-y+2z+1 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"153\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-ecuacion-del-plano\"><\/span> Memecahkan Masalah Persamaan Bidang<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Tentukan persamaan vektor bidang yang memuat 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-967feca863a9a4d3f1e7f0267f5e75e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(0,-2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan melewati dua poin berikut:<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b080bdb2f119700090341574b7bbf489_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(1,3,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/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-3ad5d13132fc818eac77b60b1ac15e13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(2,-1,5).\" 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 persamaan suatu bidang, diperlukan sebuah titik dan dua vektor dan dalam hal ini kita hanya mempunyai satu vektor, oleh karena itu kita harus mencari vektor pengarah bidang lainnya. Untuk melakukan ini, kita dapat menghitung vektor yang mendefinisikan dua titik pada 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-b85ba6303c469618bd0d69f560c11535_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (2,-1,5) - (1,3,-1) = (1,-4,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"379\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita telah mengetahui dua vektor arah bidang dan sebuah titik, maka kita menggunakan rumus persamaan vektor 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-d80b8a9d79088c24cb1940b2abeb18bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y,z)=P+\\lambda \\vv{\\text{u}} + \\mu \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"178\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kita substitusikan dua vektor dan salah satu dari dua titik pada bidang tersebut ke dalam persamaan: <\/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-941c8d43b51f4c2f838a0ef55b1f87fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(x,y,z)=(1,3,-1)+\\lambda (0,-2,3) + \\mu (1,-4,6)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"354\" style=\"vertical-align: -5px;\"><\/p>\n<\/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> Temukan persamaan parametrik bidang yang memuat tiga titik berikut: <\/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-df564519d92ccbe87c5500460231d2b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(4,1,0) \\qquad B(2,-3,-1) \\qquad C(1,5,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"308\" 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 mencari persamaan parametrik bidang tersebut, kita perlu mencari dua vektor bebas linier yang terhubung dalam bidang tersebut. Dan untuk ini, kita dapat menghitung dua vektor yang ditentukan oleh 3 titik: <\/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-14d7f4f29cbdc14a69357bf1b8f29b4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (2,-3,-1) - (4,1,0) = (-2,-4,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"406\" 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-0a33c0974226c3e3c0a51de22c0b8b38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C - A = (1,5,3) - (4,1,0) = (-3,4,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"350\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Koordinat kedua vektor yang ditemukan tidak proporsional, sehingga saling bebas linier.<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita telah mengetahui dua vektor arah dan sebuah titik pada bidang, kita menerapkan rumus persamaan parametrik 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-f5adabb85c9285653d6b638f7c48ba50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}x=P_x + \\lambda \\text{u}_x + \\mu \\text{v}_x \\\\[1.7ex] y=P_y + \\lambda \\text{u}_y + \\mu \\text{v}_y \\\\[1.7ex] z=P_z + \\lambda\\text{u}_z + \\mu \\text{v}_z \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"167\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kita substitusikan dua vektor dan salah satu dari tiga titik pada bidang tersebut ke dalam persamaan: <\/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-f57edaf8a85108cffb796470ffca8484_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}x=4 + \\lambda \\cdot (-2)+ \\mu \\cdot (-3) \\\\[1.7ex] y=1 + \\lambda \\cdot (-4) + \\mu \\cdot 4 \\\\[1.7ex] z=0 + \\lambda\\cdot (-1) + \\mu \\cdot 3 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"218\" 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-4cab5ddc074bd7df6849d71854207cf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}\\bm{x=4 -2 \\lambda-3\\mu } \\\\[1.7ex] \\bm{y=1-4 \\lambda+4 \\mu } \\\\[1.7ex] \\bm{z=-\\lambda + 3\\mu } \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"138\" style=\"vertical-align: 0px;\"><\/p>\n<\/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> Temukan persamaan implisit atau umum dari bidang yang melalui titik 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-b61f338575699a918d594595ddc6fb02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2,1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan berisi vektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5150cc543174c7c079a38b016685eb3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(4,1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -5px;\"><\/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-0664398c6f1938e4ec5efaae48ab1c70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(5,3,-1).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" 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 menghitung persamaan umum atau implisit bidang, perlu diselesaikan determinan yang dibentuk oleh dua vektor, ketiga variabel, dan koordinat titik berikut:<\/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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi, kita substitusikan vektor dan titik 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-02e103601cd9992a8a8c087d016a08c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}4 &amp; 5 &amp; x+2 \\\\[1.1ex]1 &amp; 3 &amp; y-1 \\\\[1.1ex]3&amp; 1 &amp; z+1 \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"133\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita selesaikan determinan matriks 3\u00d73 dengan metode pilihan Anda:<\/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-ec35e71b9cca25aa9907c97da2ea2e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"12(z+1)+15(y-1)+1(x+2)-9(x+2)-4(y-1)-5(z+1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"534\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, kami melakukan operasi dan mengelompokkan istilah serupa: <\/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-fea5513ce57c88cac89a695b47b7a0c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8(x+2)+11(y-1)+7(z+1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" 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-b965060b2baad7f1b7fa66e0555a23f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8x-16+11y-11+7z+7=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"262\" style=\"vertical-align: -4px;\"><\/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-52635b24c2d396ff9a47fbd210c56bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8x+11y+7z-20= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi persamaan implisit atau umum dari rencana tersebut 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-fc97887dad7e25ff823eee155fb58358_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{-8x+11y+7z-20 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/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> Tentukan apakah intinya<\/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-10de27107c1cf63dc889433e271d4a78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,5,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<p> milik rencana berikut: <\/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-54a545de55c14f27d77bfda0188789a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi : \\ 2x+y+6z-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"184\" style=\"vertical-align: -4px;\"><\/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 suatu titik berada pada bidang, persamaannya harus diverifikasi. Oleh karena itu, kita perlu mensubstitusikan koordinat Cartesius titik tersebut ke dalam persamaan bidang dan memeriksa apakah persamaan tersebut terpenuhi: <\/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-25ef56f6218537e6592c6ce17e0c3cb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+y+6z-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"153\" style=\"vertical-align: -4px;\"><\/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-10de27107c1cf63dc889433e271d4a78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,5,-3)\" 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-b30e787cb2dfac5e87b0758b866e10a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot (-1)+5+6\\cdot (-3)-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"232\" 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-66acebc4d56a2fb70fe23b2ead99dcc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2+5-18-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"155\" 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-64706704bc6f3ef930293722159a8861_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-20\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Intinya tidak memperhatikan persamaan bidang tersebut, <strong>sehingga bukan bagian dari bidang tersebut.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 5<\/h3>\n<p> Temukan persamaan segmental bidang yang persamaan umum (atau implisitnya) 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-d3bae7c36a177b1d78c043d5cb96e89a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x-2y-6z+6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"162\" style=\"vertical-align: -4px;\"><\/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\"> Pertama, kita hapus suku bebas dari persamaan:<\/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-82645f1698125207e2f86dc92d0d19d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x-2y-6z=-6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"145\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kemudian kita membagi seluruh persamaan denah dengan nilai koefisien D yang diubah tandanya: <\/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-b0e67dcc4d588043160bee40c086a1bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3x-2y-6z}{-6}=\\cfrac{-6}{-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"155\" style=\"vertical-align: -12px;\"><\/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-8dcb648b6776e874b07b18a16175d7a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3x}{-6}+\\cfrac{-2y}{-6}+\\cfrac{-6z}{-6}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"181\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan, dengan menggunakan sifat-sifat pecahan, kita sampai pada ekspresi berikut:<\/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-cd67201799ce9fe131bb81beb6050b3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{\\frac{-6}{3}}+\\cfrac{y}{\\frac{-6}{-2}}+\\cfrac{z}{\\frac{-6}{-6}}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"143\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi persamaan segmental (atau kanonik) bidang tersebut 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-130fe442b594fbe32ced55d35b52fb9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\bm{x}}{\\bm{-2}}+\\cfrac{\\bm{y}}{\\bm{3}}+\\cfrac{\\bm{z}}{\\bm{1}}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"120\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 6<\/h3>\n<div class=\"wp-block-group\">\n<div class=\"wp-block-group__inner-container is-layout-flow\">\n<p> Menghitung persamaan implisit atau umum bidang dalam ruang yang melalui suatu titik<\/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-e42bc8fa114f50a19858a526eabb6e30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,4,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan salah satu vektor normalnya adalah <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b007924d3ec5c7cd5de5d1d46cc86711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(5,-2,-3) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" 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\"> Rumus persamaan bidang implisit, umum, atau kartesius 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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nah, dari vektor normal kita dapat mencari koefisien A, B dan C, karena masing-masing sama dengan komponen vektor normalnya:<\/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-8f4b71cc8d8c1610a5d5706ac44d1ad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(5,-2,-3) \\ \\longrightarrow \\ 5x-2y-3z+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"336\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi kita hanya perlu mencari parameter D. Caranya, kita substitusikan koordinat titik milik bidang tersebut ke dalam persamaan: <\/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-e42bc8fa114f50a19858a526eabb6e30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,4,-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-720c689bc073e6c0f865bf406d92cbba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5\\cdot 3-2\\cdot 4-3\\cdot (-3)+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"232\" 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-3edf41b26a3b9c8bdead74d05767ec60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"15-8+9+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"147\" 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-0db457697a3637eb92bff90460d8e98f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"16+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"86\" 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-a2efb3d90e15e4474268d6de0570da4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D=-16\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"71\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kesimpulannya, persamaan implisit atau umum dari rencana tersebut 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-ee0c114a24764dd24df9d58aab7155ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{5x-2y-3z-16 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Latihan 7<\/h3>\n<p> Temukan persamaan parametrik bidang yang memuat garis<\/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-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> dan sejajar dengan kanan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23a7daa116b8874af1538c91f8d239de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\"><\/p>\n<p> menjadi garis: <\/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-624f315685b292c4bb05e9cb4b931a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ \\begin{cases} x=1+t \\\\[1.7ex] y=2-3t\\\\[1.7ex] z=4+2t \\end{cases} \\qquad \\qquad s: \\ \\frac{x-4}{2} = \\frac{y+3}{2}= \\frac{z-2}{-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"424\" style=\"vertical-align: 0px;\"><\/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 mencari persamaan parametrik bidang, kita perlu mengetahui dua vektor arah dan sebuah titik pada bidang tersebut. Deklarasi tersebut memberi tahu kita bahwa deklarasi tersebut berisi garis<\/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-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> Oleh karena itu, kita dapat mengambil vektor arah dan sebuah titik pada garis ini untuk mendefinisikan bidang tersebut. Lebih lanjut pernyataan tersebut menyatakan bahwa bidang tersebut sejajar dengan garis<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4cc36ef269909ae645021a09d5e91016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s,\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: -4px;\"><\/p>\n<p> jadi kita juga bisa menggunakan vektor arah garis ini untuk persamaan bidangnya.<\/p>\n<p class=\"has-text-align-left\"> hak<\/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-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> dinyatakan dalam bentuk persamaan parametrik, sehingga komponen vektor arahnya adalah koefisien suku parameter<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40f8b062c79839dcf7f2885a9e1469e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" 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-6b40bdea86f773b36fb40078fb4ddf23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{r} =(1,-3,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan koordinat Cartesian suatu titik pada garis yang sama adalah suku-suku bebas dari persamaan parametrik:<\/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-d565383a925076ae118032f7b9b62f7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sebaliknya garis lurus<\/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-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> berbentuk persamaan kontinu sehingga komponen vektor arahnya adalah penyebut pecahan:<\/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-a8f571b217df7a6bbe833d706091457a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{s} =(2,2,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, persamaan parametrik dari denah tersebut 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-f5adabb85c9285653d6b638f7c48ba50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}x=P_x + \\lambda \\text{u}_x + \\mu \\text{v}_x \\\\[1.7ex] y=P_y + \\lambda \\text{u}_y + \\mu \\text{v}_y \\\\[1.7ex] z=P_z + \\lambda\\text{u}_z + \\mu \\text{v}_z \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"167\" 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-c81f4d8e5aa907f111b3389d5137736e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}x=1 + \\lambda \\cdot 1+ \\mu \\cdot 2 \\\\[1.7ex] y=2 + \\lambda \\cdot (-3) + \\mu \\cdot 2 \\\\[1.7ex] z=4 + \\lambda\\cdot 2 + \\mu \\cdot (-3) \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"189\" 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-fccd86ac9a3e4084e324d8e5b1071e59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}\\bm{x=1 + \\lambda+2\\mu } \\\\[1.7ex] \\bm{y=2-3 \\lambda+2 \\mu } \\\\[1.7ex] \\bm{z=4+2\\lambda -3\\mu } \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"137\" style=\"vertical-align: 0px;\"><\/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 menemukan rumus untuk semua persamaan dalam denah dan cara menghitungnya. Anda juga akan menemukan cara mencari persamaan bidang apa pun dengan vektor normalnya. Selain itu, Anda akan dapat melihat contoh dan latihan dengan latihan persamaan denah yang diselesaikan. Apa persamaan bidangnya? Dalam geometri analitik, persamaan bidang adalah persamaan yang memungkinkan &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/\"> <span class=\"screen-reader-text\">Persamaan bidang di luar 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-265","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>Persamaan bidang dalam ruang - 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\/persamaan-bidang-dalam-ruang\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Persamaan bidang dalam ruang - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan menemukan rumus untuk semua persamaan dalam denah dan cara menghitungnya. 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Dalam geometri analitik, persamaan bidang adalah persamaan yang memungkinkan &hellip; Persamaan bidang di luar angkasa Selengkapnya &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-10T04:25:24+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equations-planes.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=\"6 menit\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Persamaan bidang di luar angkasa\",\"datePublished\":\"2023-07-10T04:25:24+00:00\",\"dateModified\":\"2023-07-10T04:25:24+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/\"},\"wordCount\":1242,\"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\/persamaan-bidang-dalam-ruang\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/\",\"url\":\"https:\/\/mathority.org\/id\/persamaan-bidang-dalam-ruang\/\",\"name\":\"Persamaan bidang dalam ruang - 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