{"id":211,"date":"2023-07-11T20:48:25","date_gmt":"2023-07-11T20:48:25","guid":{"rendered":"https:\/\/mathority.org\/id\/modul-contoh-rumus-vektor-latihan-yang-diselesaikan\/"},"modified":"2023-07-11T20:48:25","modified_gmt":"2023-07-11T20:48:25","slug":"modul-contoh-rumus-vektor-latihan-yang-diselesaikan","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/modul-contoh-rumus-vektor-latihan-yang-diselesaikan\/","title":{"rendered":"Cara menghitung modulus suatu vektor"},"content":{"rendered":"<p>Pada halaman ini Anda akan melihat penjelasan tentang besaran suatu vektor dan cara menghitungnya beserta rumusnya. Anda juga akan dapat melihat cara mencari modul dari dua titik: asal dan akhir. Selain itu, Anda akan menemukan cara menentukan komponen suatu vektor dari modulusnya dan sifat-sifat modulus suatu vektor. Anda bahkan dapat berlatih dengan contoh, latihan, dan soal langkah demi langkah. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-104\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-el-modulo-de-un-vector\"><\/span> Berapakah modulus suatu vektor?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Besarnya suatu vektor<\/strong> menyatakan jarak antara titik asal dan titik akhir. Oleh karena itu, besar suatu vektor sama dengan <strong>panjang<\/strong> vektor tersebut. <\/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\/module-dune-longueur-de-vecteur.webp\" alt=\"modulus vektor panjang\" class=\"wp-image-353\" width=\"182\" height=\"179\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<p> Seperti yang dapat Anda lihat pada representasi grafis di atas, besaran suatu vektor dilambangkan dengan garis vertikal di setiap sisi 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-e7513a2086faba37053531b9addea2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{AB}\\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Di sisi lain, modulus suatu vektor sama dengan <strong>norma suatu vektor<\/strong> , sehingga Anda dapat melihatnya ditulis seperti itu juga. Inilah sebabnya ada ahli matematika yang juga merepresentasikan modul vektor dengan dua garis vertikal di setiap sisinya: <\/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-fdcec36c9625381e65a49270cd8a2331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\lvert \\vv{AB}\\rvert\\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-modulo-de-un-vector\"><\/span> Rumus modulus suatu vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Untuk mencari besar suatu vektor pada bidang, kita harus menerapkan rumus berikut: <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Untuk menentukan <strong>besar suatu vektor,<\/strong> kita harus menghitung akar kuadrat (positif) dari jumlah kuadrat komponen-komponennya. Dengan kata lain, jika kita mempunyai vektor 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-d4b2f8c9cdb09377a66fbce8392c30ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (\\text{u}_x,\\text{u}_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"91\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Modulnya 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-f63fa0a6f4110553705d4e3d6cf23692_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ \\text{u}_x^2+\\text{u}_y^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"117\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<\/div>\n<p> Misalnya kita akan menghitung besar vektor berikut dengan menggunakan 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-3e1f083b8e9df80dc493a280f5c20cc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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-c93cbb567f755c6f6f5ed9ddd8fce245_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert =\\sqrt{4^2+3^2} = \\sqrt{16+9}=\\sqrt{25} = \\bm{5}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"285\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calcular-el-modulo-de-un-vector-con-las-coordenadas-de-su-origen-y-su-extremo\"><\/span> Hitung besarnya suatu vektor dengan koordinat asal dan ujungnya<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Kita baru saja melihat bagaimana besaran suatu vektor ditentukan jika kita mengetahui komponen-komponennya, tetapi apa jadinya jika kita hanya mengetahui titik awal dan akhir vektor?<\/p>\n<p> Jadi, untuk menghitung besaran suatu vektor dari koordinat asal dan akhirnya, Anda harus mengikuti dua langkah berikut:<\/p>\n<ol style=\"color:#ff6f00; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#000000;font-weight: normal;\">Pertama kita cari komponen vektornya. Untuk melakukan ini, kita perlu mengurangkan titik ekstrem dikurangi titik asal.<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">Dan kemudian kita menghitung modulus vektor yang diperoleh dengan rumus yang kita lihat di bagian sebelumnya.<\/span><\/li>\n<\/ol>\n<p> Mari kita lihat bagaimana hal ini dilakukan melalui sebuah contoh:<\/p>\n<ul>\n<li> Hitunglah besar vektor yang titik asalnya adalah\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a16d799fdc0fa3c371c35ba5f0f3a3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan sebagai poin terakhir<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a3a744084783890d8d12db98e82e348_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(-1,4).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<p> Pertama-tama kita perlu mencari komponen-komponen vektor, jadi kita kurangi titik akhirnya dikurangi titik asal:<\/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-2837e9a238c2d7143e91f36f1bdc953d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB}=B-A=(-1,4)-(2,1)=(-3,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"315\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Setelah kita mengetahui vektornya, kita menghitung besarnya menggunakan rumus besaran 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-32da93798b33cfd623c145783850b8b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} \\vv{AB} \\end{vmatrix} =\\sqrt{(-3)^2+3^2} = \\sqrt{9+9}=\\sqrt{18}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"295\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> Dan kita biarkan hasilnya sebagai akar kuadrat, karena tidak eksak. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-calcular-las-componentes-de-un-vector-a-partir-de-su-modulo\"><\/span> Cara menghitung komponen suatu vektor dari modulusnya <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<p> Kita telah melihat cara mengekstrak besaran suatu vektor dari komponen-komponennya, namun prosesnya juga dapat dibalik. Dengan kata lain, kita dapat menghitung komponen suatu vektor melalui modulusnya.<\/p>\n<p> Proses mencari komponen suatu vektor dari besarnya disebut <strong>penguraian vektor<\/strong> . Jadi, untuk menguraikan suatu vektor tentu saja kita membutuhkan besarnya dan sudut yang dibentuknya terhadap sumbu absis (sumbu X).<\/p>\n<p> Sehingga komponen X dan Y vektor dapat dihitung dengan perbandingan trigonometri: <\/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\/decomposition-dun-vecteur-dans-matab.webp\" alt=\"dekomposisi vektor di matlab\" class=\"wp-image-388\" width=\"390\" height=\"231\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Seperti terlihat pada gambar, besaran suatu vektor membentuk segitiga siku-siku dengan komponen-komponennya, sehingga rumus dasar trigonometri dapat diterapkan.<\/p>\n<p> Harus diingat bahwa, tidak seperti modulus suatu vektor, komponen-komponennya dapat bernilai negatif karena sinus dan kosinus dapat bernilai negatif.<\/p>\n<p> Sebagai contoh, kita akan menyelesaikan dekomposisi vektor dari vektor yang besar dan sudutnya terhadap sumbu OX 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-3899abb56397b041d612a1fb9d33a70a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert = 10 \\qquad \\alpha = 60\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Komponen horizontal suatu vektor sama dengan modulus dikalikan dengan kosinus sudut:<\/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-e3b237fcbcb6df7294c9b2dd5d7f06cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{u}_x= \\lvert \\vv{\\text{u}}\\rvert \\cdot \\text{cos}(60\u00ba)= 10 \\cdot 0,5 = 5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Dan komponen vertikal vektor sama dengan mengalikan modulus dengan sinus sudut:<\/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-b2f9805cff94727a43f3bf53e78e9133_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{u}_y= \\lvert \\vv{\\text{u}}\\rvert \\cdot \\text{sen}(60\u00ba)= 10 \\cdot 0,87 = 8,7\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"268\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Jadi vektornya 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-8137f59704bc3ee0eabf752d669ce25d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\mathbf{u}}\\bm{ = (5 \\ ,\\ 8,7)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-del-modulo-de-un-vector\"><\/span> Sifat modulus suatu vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Modulus merupakan salah satu jenis operasi vektor yang mempunyai ciri-ciri sebagai berikut:<\/p>\n<ul>\n<li> Besaran suatu vektor <strong>tidak pernah negatif<\/strong> , selalu sama dengan atau lebih besar dari 0.<\/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-923d73a359ab40f1ffaba643bff0ca98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert \\geq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Faktanya, satu-satunya vektor yang ada dengan besaran nol adalah vektor nol, yaitu 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-3fbfff66ff910ebae6196cf59b4251eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (0,0) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Besaran hasil kali suatu vektor dengan bilangan real (atau skalar) sama dengan mengalikan nilai absolut skalar dengan besaran vektor. Oleh karena itu, persamaan berikut berlaku:<\/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-c814e6a42f23a1c1ab2c413261fa3d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert k \\cdot \\vv{\\text{u}} \\rvert = \\lvert k  \\rvert \\cdot \\lvert \\vv{\\text{u}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Pertidaksamaan segitiga<\/strong> diverifikasi: modulus jumlah dua vektor lebih kecil atau sama dengan jumlah modulusnya secara terpisah.<\/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-63e1eae823666827bce2c51133a8a49b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}}+\\vv{\\text{v}} \\rvert \\leq \\lvert\\vv{\\text{u}} \\rvert+\\lvert\\vv{\\text{v}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Selain itu, besaran jumlah dua vektor dihubungkan dengan perkalian titik dengan persamaan berikut: <\/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-a16809d6f89f2053f5c732a7acd486ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}}+\\vv{\\text{v}} \\rvert = \\sqrt{\\lvert \\vv{\\text{u}} \\rvert ^2+\\lvert \\vv{\\text{v}} \\rvert ^2 +2\\cdot \\vv{\\text{u}}\\cdot \\vv{\\text{v}} \\vphantom{\\sqrt{x^2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"242\" style=\"vertical-align: -9px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"vector-unitario\"><\/span> vektor satuan<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dalam matematika, <strong>vektor satuan<\/strong> adalah vektor yang modulusnya sama dengan satu.<\/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-d377140c532b698d7cdb3b180f2b7e11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert=1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"49\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, panjang suatu vektor satuan adalah satu satuan.<\/p>\n<p> Tampaknya sangat sulit bagi sebuah vektor untuk memiliki modulus tepat 1, namun sebenarnya mudah untuk menemukan jenis vektor ini: <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Untuk mencari vektor satuan suatu vektor, cukup membaginya dengan modulusnya:<\/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-1b8ce39ff18883208f914f48d4463051_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}_u = \\cfrac{\\vv{\\text{v}}}{\\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"64\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> 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-62e58ce540d042ffd138cfec23ebac58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}_u\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\"><\/p>\n<p> adalah vektor satuan dari<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7d6a20023310ef9d6c49931c265af1ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}},\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\"><\/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-9a59cd4f2581db3318d38a2a77340a64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{v}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"15\" style=\"vertical-align: -5px;\"><\/p>\n<p> modul Anda.<\/p>\n<\/div>\n<p> Vektor satuan disebut juga versor atau vektor ternormalisasi.<\/p>\n<p> Selain itu, vektor satuan mempunyai arah dan arah yang sama dengan vektor aslinya.<\/p>\n<p> Misalnya, kita akan menghitung vektor satuan dari vektor 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-06fc528ca9d541ea032c50af916549a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Untuk menormalkan vektor, pertama-tama kita perlu menghitung besarnya:<\/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-86ef7a2aa2d0201e764e4868b473b3e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{v}} \\rvert=\\sqrt{1^2+(-1)^2} = \\sqrt{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"188\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Dan terakhir, kita menghitung vektor satuan dengan membagi vektor asli dengan modulusnya: <\/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-668d60d54811f2fd2be96dd0180563ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{v}}_u = \\cfrac{\\vv{\\text{v}}}{\\lvert \\vv{\\text{v}} \\rvert} = \\frac{(1,-1)}{\\sqrt{2}}= \\bm{\\left(\\frac{1}{\\sqrt{2}},-\\frac{1}{\\sqrt{2}} \\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"267\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-modulos-de-vectores\"><\/span> Latihan modul vektor terpecahkan<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Hitung besarnya vektor 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-78feb0de1149c92d8000673c3bd9c750_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}=(6,8)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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 modul vektor kita harus menerapkan rumusnya: <\/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-2223831dac0e96dc39b2c1f575a96656_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert\\vv{a} \\rvert= \\sqrt{6^2+8^2} =\\sqrt{36+64} = \\sqrt{100} = \\bm{10}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"313\" 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> Urutkan vektor-vektor berikut dari yang terpendek hingga terpanjang. <\/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-0cd5b4cce27e3e747335f1b83a27ea14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}=(4,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" 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-9528dc0e7c4fdec1f5ea0897cb5f1080_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{b}=(3,1)\" 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-d3d0c1b4ab12a00987c3821811108881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{c}=(5,12)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" 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-9611ddc072acda8752bf9cf38687c790_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{d}=(-6,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" 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\"> Panjang suatu vektor sama dengan besarnya. Oleh karena itu, kita perlu menghitung modulus semua vektor: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f774b05c948e7cc2648508e840bec8d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left|\\vv{a}\\right|= \\sqrt{4^2+(-2)^2} =\\sqrt{16+4} = \\sqrt{20}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"284\" style=\"vertical-align: -6px;\"><\/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-26f7996534e539a7c403a494276e9b43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"|\\vv{b}|= \\sqrt{3^2+1^2} =\\sqrt{9+1} = \\sqrt{10}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"243\" 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-41d80256e32169ae228c30631ed6a7c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix}\\vv{c}\\end{vmatrix}= \\sqrt{5^2+12^2} =\\sqrt{25+144} = \\sqrt{169} = 13\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"331\" style=\"vertical-align: -7px;\"><\/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-b8d50ea2580fbf8d44a19147ea34b730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"|\\vv{d}| = \\sqrt{(-6)^2+(-3)^2} =\\sqrt{36+9} = \\sqrt{45}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi, vektor-vektor yang diurutkan dari panjang (atau modul) terkecil hingga terbesar 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-8e3ca300d8666eb94fd4ddd22088e00a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"|\\vv{b}|< \\begin{vmatrix}\\vv{a}\\end{vmatrix} < |\\vv{d}| < \\begin{vmatrix}\\vv{c}\\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"144\" style=\"vertical-align: -7px;\"><\/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> Tentukan besar vektor yang titik asalnya 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-4fa493d9071a18cc176e19c8aeda71e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(-3,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan sebagai poin terakhir <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f7da7d69ba3de7c0a400c5739021b3ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(7,-4).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" 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 modulnya, Anda harus mencari vektornya terlebih dahulu. Untuk melakukan ini, kita kurangi titik ekstrem dikurangi titik asal:<\/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-237f500f255108fb2f6673bdf1ac0c88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB}=B-A=(7,-4)-(-3,2)=(10,-6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"337\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah kita mengetahui vektornya, modulusnya dihitung menggunakan rumus modulus: <\/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-4c92f7a1361051e25fa90e0ed878a676_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} \\vv{AB} \\end{vmatrix} =\\sqrt{10^2+(-6)^2} = \\sqrt{100+36}=\\bm{\\sqrt{136}}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"340\" style=\"vertical-align: -7px;\"><\/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> Uraikan vektor berikut dan temukan 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-992e1cdc6a75b8bdfe860c97dc9911e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{a} \\rvert =8 \\qquad \\alpha = 45\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" 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\"> Komponen horizontal suatu vektor sama dengan modulus dikalikan dengan kosinus sudut:<\/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-33aa199d1341f57e7a85abaa3c261a91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_x= \\lvert \\vv{a}\\rvert \\cdot \\text{cos}(45\u00ba)= 8 \\cdot 0,71 = 5,66\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"267\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan komponen vertikal vektor sama dengan mengalikan modulus dengan sinus sudut:<\/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-269cf4eed12832856838538f0a314aaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_y= \\lvert \\vv{a}\\rvert \\cdot \\text{sen}(45\u00ba)= 8 \\cdot 0,71 = 5,66\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"267\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi vektornya 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-b9a4b542b5e7893c3da383ffa65a133b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\mathbf{u}}\\bm{ = (5,66 \\ ,\\ 5,66)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini kedua komponennya identik, yaitu sudut kemiringan vektor adalah 45\u00ba.<\/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> Hitunglah vektor yang mempunyai arah dan arah yang sama dengan vektor berikut tetapi dengan modul 1. <\/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-764894932cc153ee326360c077c75ec9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a} = (-4,3)\" 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\"> Vektor yang mempunyai arah dan arah yang sama tetapi dengan modul 1 adalah vektor satuan. Untuk menghitungnya, pertama-tama kita cari modul vektornya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d067857f68bb43d3ee3693466272cc36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{a} \\rvert=\\sqrt{(-4)^2+3^2} = \\sqrt{16+9} = \\sqrt{25} = 5\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"315\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita menghitung vektor satuan dengan membagi vektor asli dengan modulusnya: <\/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-ff7322f3b35f78c5e151f4e7dc59eb98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{a}_u = \\cfrac{\\vv{a}}{\\lvert \\vv{a} \\rvert} = \\frac{(-4,3)}{5}= \\bm{\\left(-\\frac{4}{5},\\frac{3}{5} \\right)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"238\" style=\"vertical-align: -17px;\"><\/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<p> Dekomposisi vektor berikut secara vektor dan hitung vektor satuannya: <\/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-305f67adc42d0194ae1c8bbca09484a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{a} \\rvert =6 \\qquad \\alpha = 20\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" 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\"> Pertama, kita menguraikan vektornya dan mencari koordinatnya: <\/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-425294f0e8b14e63cf0d59c6fa95f367_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_x= \\lvert \\vv{a}\\rvert \\cdot \\text{cos}(20\u00ba)= 6 \\cdot 0,94 = 5,64\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"267\" 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-2a795bfcb22419526d23f6d0ff419fdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a_y= \\lvert \\vv{a}\\rvert \\cdot \\text{sen}(20\u00ba)= 6 \\cdot 0,34 = 2,05\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"266\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi vektornya 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-abc6fa01a5b562627b05dc37ad7f59d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}= (5,64 \\ ,\\ 2,05)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita menghitung vektor satuan dengan membagi vektor yang diperoleh dengan modulnya:<\/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-287411b8f47ca842a6db2cbe1c9a6ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{a}_u = \\cfrac{\\vv{a}}{\\lvert \\vv{a} \\rvert} = \\frac{(5,64 \\ ,\\ 2,05)}{6}= \\bm{(0,94 \\ , \\ 0,34) }\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"318\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Perhatikan bahwa komponen vektor satuan sama dengan kosinus dan sinus sudut yang dibentuknya terhadap sumbu X.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Pada halaman ini Anda akan melihat penjelasan tentang besaran suatu vektor dan cara menghitungnya beserta rumusnya. Anda juga akan dapat melihat cara mencari modul dari dua titik: asal dan akhir. Selain itu, Anda akan menemukan cara menentukan komponen suatu vektor dari modulusnya dan sifat-sifat modulus suatu vektor. Anda bahkan dapat berlatih dengan contoh, latihan, dan &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/modul-contoh-rumus-vektor-latihan-yang-diselesaikan\/\"> <span class=\"screen-reader-text\">Cara menghitung modulus suatu vektor<\/span> Selengkapnya &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[54],"tags":[],"class_list":["post-211","post","type-post","status-publish","format-standard","hentry","category-vektor"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Cara menghitung modulus suatu vektor - 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\/modul-contoh-rumus-vektor-latihan-yang-diselesaikan\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cara menghitung modulus suatu vektor - Mathority\" \/>\n<meta property=\"og:description\" content=\"Pada halaman ini Anda akan melihat penjelasan tentang besaran suatu vektor dan cara menghitungnya beserta rumusnya. 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