{"id":210,"date":"2023-07-11T21:43:45","date_gmt":"2023-07-11T21:43:45","guid":{"rendered":"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/"},"modified":"2023-07-11T21:43:45","modified_gmt":"2023-07-11T21:43:45","slug":"perkalian-suatu-vektor-dengan-bilangan-real-skalar","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/","title":{"rendered":"Perkalian suatu vektor dengan suatu bilangan"},"content":{"rendered":"<p>Halaman ini menjelaskan cara mengalikan vektor dengan bilangan real (atau skalar) secara numerik dan grafis. Selain itu, Anda juga akan menemukan contoh dan latihan penyelesaian hasil kali vektor dengan skalar. Terakhir, sifat-sifat operasi vektor jenis ini juga dijelaskan. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcomo-se-multiplica-un-vector-por-un-numero-real\"><\/span> Bagaimana cara mengalikan vektor dengan bilangan real? <span class=\"ez-toc-section-end\"><\/span><\/h2>\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 menghitung hasil kali suatu vektor dan suatu bilangan (atau skalar) secara numerik, setiap komponen vektor harus dikalikan dengan bilangan 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-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 class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-abb3b5391be9c0e4d871ff45367ca838_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\\cdot \\vv{\\text{u}} =(k\\cdot \\text{u}_x \\ , \\ k\\cdot \\text{u}_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"170\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<\/div>\n<p> Jadi, hasil perkalian suatu vektor dengan suatu bilangan akan menghasilkan vektor baru yang mempunyai ciri-ciri sebagai berikut:<\/p>\n<ul 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;\">Hasil perkalian suatu vektor dengan skalar menghasilkan vektor baru yang arahnya sama dengan vektor aslinya.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Selain itu, vektor baru akan mempunyai arah yang sama jika bilangannya positif.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Atau sebaliknya jika bilangannya negatif.<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">Besarnya vektor yang dihasilkan sama dengan besar vektor asal dikalikan skalar.<\/span><\/li>\n<\/ul>\n<p> Pada grafik berikut Anda dapat melihat bagaimana arah vektor dipertahankan terlepas dari tanda skalarnya. Sebaliknya, arah vektor bergantung pada tanda bilangan yang dikalikan. <\/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\/multiplication-ou-produit-dun-nombre-ou-dun-scalaire-par-un-vecteur.webp\" alt=\"perkalian atau hasil kali suatu bilangan atau skalar dengan suatu vektor\" class=\"wp-image-283\" width=\"305\" height=\"206\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Selanjutnya pada grafik berikut terlihat jelas bahwa besar vektor hasil kali sama dengan besar vektor asal dikalikan skalar. <\/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\/numero-de-produit-de-multiplication-de-module-par-vecteur.webp\" alt=\"Bagaimana modul perkalian atau hasil kali suatu bilangan dengan vektor berkembang?\" class=\"wp-image-290\" width=\"255\" height=\"313\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Jelasnya, jika kita mengalikan vektor dengan angka yang lebih besar dari 1, hasilnya adalah vektor yang panjangnya lebih besar (modulusnya lebih besar). Sebaliknya jika kita mengalikan vektor dengan bilangan yang kurang dari 1, maka hasilnya adalah vektor yang panjangnya lebih pendek (modulusnya lebih kecil).<\/p>\n<p> <strong>Catatan:<\/strong> Jangan bingung antara hasil kali vektor dan skalar dengan <a href=\"https:\/\/mathority.org\/id\/menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan\/\">hasil kali titik vektor<\/a> . Meskipun memiliki nama yang mirip, keduanya merupakan dua konsep yang sangat berbeda. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-del-producto-de-un-vector-por-un-escalar\"><\/span> Contoh perkalian vektor dengan skalar<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Selanjutnya, kita akan melihat contoh numerik tentang cara menghitung hasil kali vektor dan bilangan:<\/p>\n<ul>\n<li> Kalikan vektor berikut dengan 4:<\/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-0a3b02659b94a332ef3cd2cc62769d94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} =(2,-3)\" 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-998ae86a47fde44f6006a0744f03c562_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot \\vv{\\text{u}} =(4 \\cdot 2 \\ , \\ 4 \\cdot (-3)) = \\bm{(8,-12)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"263\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Seperti yang Anda lihat, operasi vektor jenis ini tidak terlalu rumit, karena Anda tidak perlu melakukan banyak perhitungan.<\/p>\n<p> Namun, ada operasi vektor yang lebih rumit, seperti penjumlahan vektor dan pengurangan vektor. Jika Anda sudah memahami cara menghitung hasil kali vektor dan skalar, kami sarankan Anda melanjutkan ke tingkat berikutnya dan melihat cara menyelesaikan <a href=\"https:\/\/mathority.org\/id\/jumlah-vektor,-contoh-latihan-yang-diselesaikan-secara-grafis-dan-numerik\/\">penjumlahan vektor<\/a> dan <a href=\"https:\/\/mathority.org\/id\/kurangi-vektor-secara-numerik,-contoh-grafis,-latihan-yang-diselesaikan,-kurangi\/\">pengurangan vektor<\/a> , karena ini adalah operasi yang lebih sulit dan, pada kenyataannya, mereka lebih banyak digunakan (mereka lebih penting). <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-de-la-multiplicacion-de-un-vector-por-un-numero\"><\/span> Sifat mengalikan vektor dengan bilangan<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Hasil kali suatu vektor dan suatu bilangan mempunyai sifat-sifat sebagai berikut:<\/p>\n<ul>\n<li> <strong>Sifat asosiatif<\/strong> : Jika suatu vektor dikalikan dengan lebih dari satu bilangan, maka urutan perkaliannya tidak menjadi masalah.<\/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-a7cc3b0e34695b9960f32199046e868e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k \\cdot (k' \\cdot \\vv{\\text{u}}) =(k\\cdot k') \\cdot \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"171\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Sifat distributif<\/strong> terhadap penjumlahan dan pengurangan vektor:<\/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-a8b55a3b25ad66154d6e8bb18fc70da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k \\cdot (\\vv{\\text{u}}+  \\vv{\\text{v}})=k\\cdot \\vv{\\text{u}} + k\\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"187\" 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-086f60151d160ac3b6d4f5add59434b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k \\cdot (\\vv{\\text{u}}- \\vv{\\text{v}})=k\\cdot \\vv{\\text{u}} - k\\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"187\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Sifat distributif<\/strong> terhadap penjumlahan skalar:<\/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-7354955effb7b59f41793537f4517851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(k+k') \\cdot \\vv{\\text{u}} =k\\cdot \\vv{\\text{u}} + k'\\cdot \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"196\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Elemen netral<\/strong> : Jelas, vektor apa pun dikalikan dengan 1 menghasilkan vektor itu sendiri: <\/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-154c7987de1e55e762dcf229eae9fd23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1\\cdot \\vv{\\text{u}} = \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-multiplicacion-de-un-vector-por-un-escalar\"><\/span> Menyelesaikan masalah perkalian vektor dengan skalar<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Hitung secara analitis hasil perkalian vektor berikut dengan 3: <\/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-8b5a56f394f2b178a0175a53e6a55a1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}=(-2,5)\" 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\"> Untuk mencari hasil kali, Anda harus mengalikan setiap koordinat vektor dengan 3: <\/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-4e0b8a6167489c27e91bef24f222e3af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3\\cdot \\vv{a}=(3\\cdot (-2), 3 \\cdot 5) = \\bm{(-6,15)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"250\" 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> Kalikan vektor berikut dengan 6 dan temukan 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-85db994a57a1325a456909a04efa2e23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}=(3,-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" 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-tama kita mengalikan 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-7c6c6b2d9291528aff357e7e94daf221_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6\\cdot \\vv{a}=(6\\cdot 3, 6 \\cdot (-4)) = (18,-24)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"259\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Saat ini ada dua cara untuk menghitung besarnya vektor yang diperoleh. Yang pertama cari besar vektor aslinya, lalu kalikan dengan 6: <\/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-7a7bc760b63c8efa2c36d2d49802000c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{a} \\rvert =\\sqrt{3^2+(-4)^2} = \\sqrt{25} =5\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"229\" 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-32f0e3a2cb32142aa26a1fbb0f3bf9df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6\\cdot \\lvert \\vv{a} \\rvert =6\\cdot 5 =\\bm{30}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan cara kedua adalah dengan menghitung langsung besaran vektor yang diperoleh dari perkalian:<\/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-06f0793bfdae917904c47247b9b1f08f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert 6\\cdot \\vv{a} \\rvert =\\sqrt{18^2+(-24)^2} = \\sqrt{900} =\\bm{30}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"287\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sehingga dengan kedua prosedur tersebut terlihat bahwa hasilnya tidak bergantung pada metode penghitungan modulus.<\/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> 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-cf8d63b14f91ee738fe5107ab2a05779_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}=(4,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Hitung operasi berikut secara aljabar: <\/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-5557b66d322653674e2266800c73c958_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot \\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" 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-a2f511cc4014aae8cf90bef442f30bdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\cdot \\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" 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-d09a188710cecb56283c4a2dea7a97e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{2} \\cdot \\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"33\" 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-7c4742e8a37f45e358d2e7740c5aadef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot \\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Selanjutnya tentukan apakah vektor-vektor yang dihasilkan mempunyai arah dan arah yang sama dengan vektor aslinya, dan urutkan dari yang terpendek ke terpanjang. <\/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\"> Kita hitung dulu perkaliannya: <\/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-6b92c621a30ebf15a0628cba3e0f6be5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot \\vv{a}=(2\\cdot 4, 2 \\cdot (-1)) = (8,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" 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-71e1eb36b88b2ca47d69f0bfe4d1a33b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\cdot \\vv{a}=(-3\\cdot 4, -3 \\cdot (-1)) = (-12,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"290\" 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-d9866b27859b66475a36d182919ec4e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{2}\\cdot \\vv{a}=\\left(\\frac{1}{2}\\cdot 4, \\frac{1}{2} \\cdot (-1)\\right) = \\left(2,-\\frac{1}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"277\" style=\"vertical-align: -17px;\"><\/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-6904a22660dc2e21ab3f775e1453535c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot \\vv{a}=(4\\cdot 4, 4 \\cdot (-1)) = (16,-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"250\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, vektor yang dikalikan dengan bilangan positif mempunyai arah dan arah yang sama dengan vektor aslinya. Dan vektor-vektor yang dikalikan dengan bilangan negatif mempunyai arah yang sama tetapi berlawanan arah dengan vektor aslinya.<\/p>\n<p class=\"has-text-align-center\"> Vektor-vektor yang searah dan searah:<\/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-50dd29950a1b957a78493d3de5b4bf5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2\\vv{a}, \\ 4\\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" 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-9fac7295c812fa36c4fcba97e316b5f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{2}\\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"20\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> Vektor yang arahnya sama tetapi maknanya berbeda:<\/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-d1b95a4d066d0dd0da8cdf0373d19c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, kita harus mengurutkan vektor-vektor menurut panjangnya, atau modulusnya. Vektor dengan panjang terpanjang (atau modul terbesar) adalah vektor yang dikalikan dengan bilangan lebih besar (dalam nilai absolut), dan vektor dengan panjang terpendek (atau modul terkecil) adalah vektor yang dikalikan dengan bilangan lebih kecil angka (dalam nilai absolut). Jadi urutan panjangnya 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-b1e7cf58774569c9f128f49948c9fead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} \\displaystyle\\frac{1}{2}\\vv{a}\\end{vmatrix} <\\lvert 2\\vv{a}\\rvert < \\lvert -3\\vv{a}\\rvert < \\lvert 4\\vv{a}\\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"199\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Perhatikan bahwa panjang atau modulus tidak bergantung pada tanda skalar yang dikalikan, karena arah vektor tidak mengubah modulusnya.<\/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> Perhatikan dua 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-740cd5f2540cb57342f82c58e1076cbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a} =(7,-2) \\qquad \\vv{b} =(-3,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"205\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Hitung operasi 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-5abd316200fbcf062072dfdf51266cda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2 \\vv{a} - 3 \\vv{b}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" 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\"> Pertama-tama kita selesaikan perkalian vektor dengan angka: <\/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-08a07c35741b9e7427990abb5bb89ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2 \\cdot \\vv{a} - 3 \\cdot \\vv{b}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" 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-5fa31543d7c2a129c47b030cfa5fc4e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2 \\cdot (7,-2) - 3 \\cdot(-3,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"171\" 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-d7624e801962ac397ca316243c4e3420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(14,-4) - (-9,15)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita kurangi 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-72660b240eab9459d0b0aefb1d10d728_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(14-(-9),-4-15)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" 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-fe8b5f018f4a24a925bfce0c2c997dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(23,-19)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" 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 5<\/h3>\n<p> Lakukan perkalian vektor dengan skalar berikut dan buat grafik hasilnya: <\/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-b1c2fa6e2c114b3bbce76ca987020ad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot (1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" 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-1adddd551d57bc95b951c6573080090a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\cdot (-1,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" 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-33be3689306562b8a4ae74d0970fc1f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot (2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" 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-bd3d3bba47ccc84d7616e38ca863b22d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{3}{2}\\cdot (-2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"76\" 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-8fe6e18de5ed93f74fa207a3f9ecc2c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -\\frac{1}{2}\\cdot (8,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"77\" style=\"vertical-align: -12px;\"><\/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-tama kita mengalikan vektor dengan skalar real: <\/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-6846258bc5f69aaea3f86991095d3199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot (1,2) = (2\\cdot 1,2\\cdot 2)=\\bm{(2,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"230\" 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-605e263cab7e34f01290372f0e2666c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\cdot (-1,1) = (-3\\cdot (-1),-3\\cdot 1)=\\bm{(3,-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"325\" 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-b8b4cf7b606673d72ae9cdd0004f0757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot (2,1) = (4\\cdot 2,4\\cdot 1)=\\bm{(8,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"230\" 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-51454cc5c3239bb4552918b1a693b499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{3}{2}\\cdot (-2,4)= \\left(\\frac{3}{2}\\cdot (-2), \\frac{3}{2}\\cdot 4 \\right) = \\bm{(-3,6)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"307\" style=\"vertical-align: -17px;\"><\/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-8fbbfff835499e861d11de5da6714d10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -\\frac{1}{2}\\cdot (8,6)= \\left(-\\frac{1}{2}\\cdot 8, -\\frac{1}{2}\\cdot 6 \\right) = \\bm{(-4,-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"322\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, setelah kita menghitung vektornya, kita merepresentasikannya dalam grafik: <\/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\/exercices-et-problemes-resolus-de-multiplication-dun-vecteur-par-un-nombre.webp\" alt=\"latihan dan menyelesaikan masalah perkalian vektor dengan bilangan\" class=\"wp-image-327\" width=\"469\" height=\"370\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Halaman ini menjelaskan cara mengalikan vektor dengan bilangan real (atau skalar) secara numerik dan grafis. Selain itu, Anda juga akan menemukan contoh dan latihan penyelesaian hasil kali vektor dengan skalar. Terakhir, sifat-sifat operasi vektor jenis ini juga dijelaskan. Bagaimana cara mengalikan vektor dengan bilangan real? Untuk menghitung hasil kali suatu vektor dan suatu bilangan (atau &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/\"> <span class=\"screen-reader-text\">Perkalian suatu vektor dengan suatu bilangan<\/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-210","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>Perkalian vektor dengan angka - 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\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Perkalian vektor dengan angka - Mathority\" \/>\n<meta property=\"og:description\" content=\"Halaman ini menjelaskan cara mengalikan vektor dengan bilangan real (atau skalar) secara numerik dan grafis. Selain itu, Anda juga akan menemukan contoh dan latihan penyelesaian hasil kali vektor dengan skalar. Terakhir, sifat-sifat operasi vektor jenis ini juga dijelaskan. Bagaimana cara mengalikan vektor dengan bilangan real? Untuk menghitung hasil kali suatu vektor dan suatu bilangan (atau &hellip; Perkalian suatu vektor dengan suatu bilangan Selengkapnya &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-11T21:43:45+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4b2f8c9cdb09377a66fbce8392c30ec_l3.png\" \/>\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<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Perkalian suatu vektor dengan suatu bilangan\",\"datePublished\":\"2023-07-11T21:43:45+00:00\",\"dateModified\":\"2023-07-11T21:43:45+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/\"},\"wordCount\":89,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/id\/#organization\"},\"articleSection\":[\"Vektor\"],\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/\",\"url\":\"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/\",\"name\":\"Perkalian vektor dengan angka - 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Selain itu, Anda juga akan menemukan contoh dan latihan penyelesaian hasil kali vektor dengan skalar. Terakhir, sifat-sifat operasi vektor jenis ini juga dijelaskan. Bagaimana cara mengalikan vektor dengan bilangan real? Untuk menghitung hasil kali suatu vektor dan suatu bilangan (atau &hellip; Perkalian suatu vektor dengan suatu bilangan Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/","article_published_time":"2023-07-11T21:43:45+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4b2f8c9cdb09377a66fbce8392c30ec_l3.png"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Perkalian suatu vektor dengan suatu bilangan","datePublished":"2023-07-11T21:43:45+00:00","dateModified":"2023-07-11T21:43:45+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/"},"wordCount":89,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"articleSection":["Vektor"],"inLanguage":"id","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/","url":"https:\/\/mathority.org\/id\/perkalian-suatu-vektor-dengan-bilangan-real-skalar\/","name":"Perkalian vektor dengan angka - 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