{"id":212,"date":"2023-07-11T19:16:54","date_gmt":"2023-07-11T19:16:54","guid":{"rendered":"https:\/\/mathority.org\/id\/menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan\/"},"modified":"2023-07-11T19:16:54","modified_gmt":"2023-07-11T19:16:54","slug":"menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan\/","title":{"rendered":"Hitung perkalian titik dua vektor"},"content":{"rendered":"<p>Di halaman ini Anda akan melihat apa itu dan bagaimana menghitung perkalian titik dua vektor. Anda juga akan mempelajari cara mencari sudut antara dua vektor menggunakan perkalian titik dan, sebagai tambahan, semua properti perkalian titik. Terakhir, Anda akan dapat berlatih dengan contoh dan latihan yang diselesaikan 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=\"como-calcular-el-producto-escalar-entre-dos-vectores\"><\/span> Cara menghitung perkalian titik antara dua vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dalam matematika, perkalian titik adalah operasi vektor yang mengalikan dua vektor dan mengubahnya menjadi bilangan real. Jadi, ada dua cara untuk menghitung perkalian titik dua vektor: <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/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\"> Jika kita mengetahui koordinat dua vektor, kita dapat mencari perkalian titiknya dengan mengalikan komponen X dan Y lalu menjumlahkan hasilnya. Dengan kata lain, jika kita mempunyai dua 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-c06f6c1238ab7803750f830e16891f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (\\text{u}_x,\\text{u}_y) \\qquad \\vv{\\text{v}} = (\\text{v}_x,\\text{v}_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"216\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Produk skalar di antara keduanya 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-c469402aae308ebcf911d56f71dadef2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\vv{\\text{u}} \\cdot \\vv{\\text{v}} = \\text{u}_x\\cdot \\text{v}_x + \\text{u}_y\\cdot \\text{v}_y\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"174\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<\/div>\n<p> Misalnya, hasil kali titik antara dua vektor berikut 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-78adae8ec882bbd01c5bb2704f3ffa0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (1,2) \\qquad \\vv{\\text{v}} = (-1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" 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-756e86b9b5b7a9f5df2cacf728d82855_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}}&amp;=(1,2)\\cdot (-1,3) \\\\[1.5ex]&amp;=1\\cdot (-1) + 2 \\cdot 3 \\\\[1.5ex] &amp; = -1+6  \\\\[1.5ex] &amp; =\\bm{5} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ini adalah cara mencari perkalian titik antara dua vektor. Namun, ada juga cara lain: <\/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\"> Sebaliknya, jika kita mengetahui modulus dan sudut antara dua vektor, hasil kali skalar antara kedua vektor tersebut dapat ditentukan dengan menghitung hasil kali modul keduanya dengan kosinus sudut yang dibentuknya:<\/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-46f454c7efa9e990e47b8fd3858fbade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\vv{\\text{u}} \\cdot \\vv{\\text{v}} = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha )\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -5px;\"><\/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-4501274336c637b37c6332eae5c6c229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" 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-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> adalah modul dari vektor<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" 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-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> masing-masing dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> sudut yang mereka buat. <\/p>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<p> Ingatlah bahwa besar suatu vektor adalah akar kuadrat 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-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<p> Sebagai contoh, kita akan menyelesaikan perkalian skalar dua buah vektor yang modulusnya dan sudut antara keduanya 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-46c26555360d66fd213087ee2432e68e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert =3 \\qquad \\lvert \\vv{\\text{v}} \\rvert = 4 \\qquad \\alpha=60\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"226\" 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-a363625d93f01339437d5ad065050025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha ) \\\\[1.5ex] &amp;= 3 \\cdot 4 \\cdot \\cos(60\u00ba)\\\\[1.5ex] &amp; = 3 \\cdot 4 \\cdot 0,5 \\\\[1.5ex] &amp;= \\bm{6} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"168\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Di sisi lain, perkalian titik disebut juga perkalian titik, perkalian skalar, atau perkalian titik.<\/p>\n<p> <strong>Catatan:<\/strong> Jangan bingung membedakan perkalian titik dengan perkalian silang karena walaupun namanya mirip, keduanya merupakan konsep yang berbeda. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"encontrar-el-angulo-entre-dos-vectores-utilizando-el-producto-escalar\"><\/span> Temukan sudut antara dua vektor menggunakan perkalian titik <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> Setelah kita melihat definisi perkalian titik, Anda mungkin bertanya-tanya apa tujuan mengalikan dua vektor? Nah, salah satu penerapan perkalian titik adalah untuk menghitung sudut yang dibentuk oleh dua buah vektor. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/angle-entre-deux-vecteurs-et-produit-scalaire.webp\" alt=\"sudut antara dua vektor perkalian titik\" class=\"wp-image-583\" width=\"172\" height=\"175\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Dengan menyelesaikan kosinus rumus perkalian titik, kita memperoleh:<\/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-0534ed8d223cf93e44493b80ebfa83d4_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*} \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mari kita lihat bagaimana hal ini dilakukan melalui sebuah contoh:<\/p>\n<ul>\n<li> Tentukan sudut antara dua vektor 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-d65d095d633d49431fcf3ee16757025e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,2) \\qquad \\vv{\\text{v}} = (-1,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Pertama kita perlu mencari besar kedua 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-b8435f99e0a124c838bc163cb3ac1c67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ 4^2+2^2}= \\sqrt{20}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"170\" 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-c8a9191c8cc87d5eeb9ba2adef3860f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{ (-1)^2+5^2}= \\sqrt{26}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"197\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sekarang kita gunakan rumus untuk menghitung kosinus sudut antara dua 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-808047a2b37f1b5534038383a4d4b111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}=\\cfrac{ 4\\cdot (-1) + 2\\cdot 5}{\\sqrt{20}\\cdot \\sqrt{26}} = \\cfrac{6}{\\sqrt{520}} = 0,26\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"387\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Terakhir, kita mencari sudut yang bersesuaian dengan melakukan invers cosinus menggunakan kalkulator:<\/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-36498fae23e4068e3d1c3735dafd2e64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(0,26) = \\bm{74,93\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"158\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Jadi vektor-vektor tersebut membentuk sudut 74,93\u00ba. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-del-producto-escalar-de-dos-vectores\"><\/span> Sifat-sifat perkalian titik dua vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Perkalian titik mempunyai ciri-ciri sebagai berikut:<\/p>\n<ul>\n<li> <strong>Sifat komutatif<\/strong> : Urutan perkalian vektor 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-fcc9e9f7ec3121774701301a70313b86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}} \\cdot \\vv{\\text{v}} =\\vv{\\text{v}} \\cdot \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"88\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Sifat distributif<\/strong> : Perkalian titik bersifat distributif 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-0853c6a7d518e218e60c0d047bde152e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}( \\vv{\\text{v}}+ \\vv{\\text{w}} )=\\vv{\\text{u}} \\cdot \\vv{\\text{v}}+ \\vv{\\text{u}} \\cdot \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" 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-585d05c6a06b6c57b9ab0e640a371a9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}( \\vv{\\text{v}}- \\vv{\\text{w}} )=\\vv{\\text{u}} \\cdot \\vv{\\text{v}}- \\vv{\\text{u}} \\cdot \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Properti asosiatif<\/strong> : Kita dapat mengalikan perkalian titik dengan konstanta sebelum atau sesudah melakukan operasi, karena hasilnya setara:<\/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-1d8c1312b87c767bb4439f6c6c693dad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle k\\cdot (\\vv{\\text{u}} \\cdot \\vv{\\text{v}}) = (k\\cdot\\vv{\\text{u}}) \\cdot \\vv{\\text{v}} =\\vv{\\text{u}} \\cdot (k\\cdot\\vv{\\text{v}})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"252\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Jika dua vektor <strong>ortogonal<\/strong> (atau tegak lurus), maka hasil kali titiknya adalah nol. Sifat ini dapat dengan mudah ditunjukkan karena dua vektor yang tegak lurus membentuk sudut 90\u00ba, dan kosinus 90\u00ba sama dengan 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-77e3b216b12f49a9dde99c0fbf626658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(90\u00ba ) \\\\[1.5ex] &amp;=\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot 0 \\\\[1.5ex] &amp;= 0 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"91\" width=\"175\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Sebaliknya, jika dua vektor <strong>sejajar<\/strong> maka hasil kali skalarnya sama dengan hasil kali modulnya. Sifat ini juga dapat dengan mudah diverifikasi karena dua vektor yang arahnya sama membentuk sudut 0\u00ba, yang kosinusnya sama dengan 1:<\/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-777ec2544cac64aa065df21872989b4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(0\u00ba) \\\\[1.5ex] &amp;=\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot 1 \\\\[1.5ex] &amp;= \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"96\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Akhirnya, perkalian titik suatu vektor dengan dirinya sendiri setara dengan kuadrat besarnya: <\/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-0161de621030dcfe2a1d5fccc94048bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\vv{\\text{u}} \\cdot \\vv{\\text{u}} &amp; = \\lvert \\vv{\\text{u}} \\rvert ^2\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-productos-escalares-entre-dos-vectores\"><\/span> Menyelesaikan masalah hasil kali skalar antara dua vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Hitung perkalian titik pada bidang 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-8c09eac43618120b3a2365fcd22278ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,-3) \\qquad \\vv{\\text{v}} = (5,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" 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 perkalian titik dua vektor, kita perlu mengalikan koordinat X dan koordinat Y, lalu menjumlahkan 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-f9cb0f372eee67ad149d5b2cff8d2f99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned}\\vv{\\text{u}} \\cdot \\vv{\\text{v}}  &amp; = (4,-3)\\cdot (5,2)  \\\\[1.5ex] &amp; = 4\\cdot 5 + (-3) \\cdot 2 \\\\[1.5ex] &amp; = 20-6\\\\[1.5ex] &amp; =\\bm{14} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"165\" 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 2<\/h3>\n<p> Tentukan hasil kali skalar dua buah vektor yang modulusnya dan sudut yang dibentuknya 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-55cc75d8aa625082e47ed6396449f550_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert =6 \\qquad \\lvert \\vv{\\text{v}} \\rvert = 3 \\qquad \\alpha=45\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" 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\"> Karena kita mengetahui modulnya dan sudut antar modulnya, kita dapat langsung menerapkan rumus perkalian 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-38999386ccbd92758c9968f025beff72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha ) \\\\[1.5ex] &amp;= 6 \\cdot 3 \\cdot \\cos(45\u00ba)\\\\[1.5ex] &amp; = 6 \\cdot 3 \\cdot 0,71 \\\\[1.5ex] &amp;= \\bm{12,73} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"133\" width=\"168\" 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> Berapakah sudut antara 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-256770a8f550eb60828f1027831ec423_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(3,8) \\qquad  \\vv{\\text{v}} =(-4,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" 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 perlu menghitung besar kedua 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-3a43677b59edab4031eae3cf64775e63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ 3^2+8^2}= \\sqrt{73}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"170\" 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-34ffe2db4622d64f22c5ec525dbc01ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{ (-4)^2+1^2}= \\sqrt{17}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"197\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menggunakan rumus untuk menghitung kosinus sudut yang dibentuk oleh 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-eecb349d0dcd0362cfd6a67b639e4edb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}=\\cfrac{ 3\\cdot (-4) + 8\\cdot 1}{\\sqrt{73}\\cdot \\sqrt{17}} = \\cfrac{-4}{\\sqrt{1241}} = -0,11\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"409\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, kita mencari sudut yang bersesuaian dengan melakukan invers kosinus menggunakan kalkulator: <\/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-b3e2f0498a5af3652b484888d36d1f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(-0,11) = \\bm{96,52\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"170\" 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 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-ae35d4962f496060eec23769832a6649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(5,2) \\qquad \\vv{\\text{v}} =(-1,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" 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-256d0ec6c6c20b5914f3ab0716119a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(\\vv{\\text{u}} \\cdot\\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -7px;\"><\/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 perlu menyelesaikan perkalian titik di dalam tanda kurung, kemudian melakukan perkalian dengan perkalian titik di luar: <\/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-256d0ec6c6c20b5914f3ab0716119a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(\\vv{\\text{u}} \\cdot\\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" 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-8652a87d0713db4774dae56ca4328e49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl((5,2) \\cdot (-1,6) \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"129\" 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-c6f9d3b4f48b49bdfdfc69a19dd13903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(5 \\cdot (-1) + 2 \\cdot 6 \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"133\" 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-afe85bb11fcdcc61ae43a2d7f6b7603c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(-5 + 12 \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" 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-64de0e6750bcabc711c6d73ec0d0d869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\cdot 7\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" 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-32d67549905fec64c67a14e7a8b694e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{28}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" 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 5<\/h3>\n<p> Diketahui tiga vektor dua dimensi 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-202792a7dea6027ec366cffa5ff30e22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(-2,6) \\qquad \\vv{\\text{v}} =(4,-3)\\qquad \\vv{\\text{w}} =(-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"333\" 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-8413dfd8859a9b5a549dbfd002683439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} \\cdot \\bigl( 5 \\vv{\\text{u}}- 2 \\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -7px;\"><\/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 mengalikan vektor dengan skalar di dalam tanda kurung: <\/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-8413dfd8859a9b5a549dbfd002683439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} \\cdot \\bigl( 5 \\vv{\\text{u}}- 2 \\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" 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-dc363692e0558fc697dee94018108f2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot \\bigl( 5 (-2,6)- 2(4,-3)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"225\" 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-c51b85108b7dc83bc81a75702db939bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot \\bigl( (-10,30)- (8,-6)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"225\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita melakukan pengurangan 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-ca704e156d875e9ca62b62452c77cc3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot  (-10 -8,30-(-6))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"224\" 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-28e80b50176016afcc5901693ef5426d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot  (-18,36)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, kita menyelesaikan perkalian 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-498a8066459db0849a08866445623dbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1)\\cdot (-18) + 2 \\cdot 36\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"155\" 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-f3c885a9e81d347e3914a736b32bb382_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18 + 72\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"55\" 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-56a66a32d168d33967dda62795778226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{90}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" 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 6<\/h3>\n<p> Hitung nilai dari<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> sehingga vektor-vektor berikut tegak 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-1b1c6281d6e6570fd94edcd837293e2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(-2,-3) \\qquad  \\vv{\\text{v}} =(k,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"208\" 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\"> Dua buah vektor tegak lurus membentuk sudut 90\u00ba. Jadi kosinus sudutnya harus nol, karena cos(90\u00ba)=0. Belum: <\/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-e83a6b694c8dfa0975854f1bffec44de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(90\u00ba) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"133\" 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-3ef11c2ecbf7bc8dff4217a761960387_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0=\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"86\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Penyebut pecahan membagi seluruh ruas kanan persamaan, sehingga kita dapat meneruskannya dengan mengalikannya di ruas lainnya: <\/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-797dd0ce47130f959c984510894f08b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 \\cdot \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert  =\\vv{\\text{u}} \\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" 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-b58e54d3d5fa6e123ca5e27a27d77ad1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0  =\\vv{\\text{u}} \\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami sekarang menyelesaikan produk 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-67bb2b17e1eb4d090327e96f5f3a8bcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =(-2,-3) \\cdot (k,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" 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-77ccd428939d3270f6feeef3ca9681e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-2 \\cdot k + (-3)\\cdot 6\" 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-4dacdcd81cad7dcb737b48de38e3b4a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-2 k -18\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"105\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan, akhirnya, kami mengklarifikasi hal yang tidak diketahui: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-35a0330d3e83b419577d9448ff01008f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2k =-18\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" 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-6abbb3c976dc4f8d66093a3fb3a40cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle k =\\cfrac{-18}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"75\" 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-67556bf37fde34c4177ff3f3c037f95c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{k =-9}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" 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 7<\/h3>\n<p> Hitung 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-28575fb8fa361427b255d8744e982cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha , \\beta\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"30\" 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-4de02fc502ed5dbd15f371728ea270a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\gamma\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p> yang membentuk sisi-sisi segitiga berikut: <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-angle-resolu-entre-vecteurs-produit-scalaire.webp\" alt=\"latihan dan soal diselesaikan selangkah demi selangkah dari perkalian skalar dua vektor\" class=\"wp-image-560\" width=\"290\" height=\"226\" srcset=\"\" sizes=\"auto, \"><\/figure>\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\"> Titik-titik sudut yang membentuk segitiga adalah titik-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-75a4919fae29190e3effdeedcec8eb6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(2,1) \\qquad B(4,4) \\qquad C(6,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"230\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung sudut dalam suatu segitiga, kita dapat menghitung vektor masing-masing sisinya, lalu mencari sudut yang dibentuknya menggunakan rumus perkalian titik.<\/p>\n<p class=\"has-text-align-left\"> Misalnya untuk mencari 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-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> Kami menghitung vektor sisi-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-6a9da14fa9cc4e50b06bdfa76801b083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (4,4)-(2,1)= (2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"287\" 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-8e4e2e72bee87bba3e7657a53935e660_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C - A = (6,2)-(2,1)= (4,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kita mencari sudut yang dibentuk oleh dua vektor menggunakan rumus perkalian 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-7b6aad49b300d421fc3bb486f051294c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{AB} \\rvert = \\sqrt{2^2+3^2} = \\sqrt{13}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" 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-f6657897e68d6b68f79277c89abe6868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{AC} \\rvert = \\sqrt{4^2+1^2} = \\sqrt{17}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" 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-a966db5753cbb53c424c0f962fb27102_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{AB} \\cdot \\vv{AC}}{\\lvert \\vv{AB} \\rvert \\cdot \\lvert \\vv{AC} \\rvert}=\\cfrac{ 2\\cdot 4 + 3\\cdot 1}{\\sqrt{13}\\cdot \\sqrt{17}} = \\cfrac{11}{\\sqrt{221}} =0,74\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"396\" 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-9fac783dc0113263dfb5c31b58231fae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\alpha = 42,27\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"79\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita ulangi prosedur yang sama untuk menentukan 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-ea160d5901518098e691e051e6efa4a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\beta:\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"20\" 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-398c0b2dc840abfc63700a084e9e2956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC} = C - B = (6,2)-(4,4)= (2,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"302\" 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-b7825d9e3b0ceee57e7ecd470e52a242_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{BC} \\rvert = \\sqrt{2^2+(-2)^2} = \\sqrt{8}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"207\" 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-e0e73fd58d6a5b487af9f971fdcdc97f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\beta) =\\cfrac{\\vv{AB} \\cdot \\vv{BC}}{\\lvert \\vv{AB} \\rvert \\cdot \\lvert \\vv{BC} \\rvert}=\\cfrac{ 2\\cdot 2 + 3\\cdot (-2)}{\\sqrt{13}\\cdot \\sqrt{8}} = \\cfrac{-2}{\\sqrt{104}} =-0,20\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"437\" 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-1b2f148d28b9679b8267886497e16518_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\beta = 101,31\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, untuk mencari sudut terakhir, kita bisa mengulangi prosedur yang sama. Namun, semua sudut dalam segitiga harus berjumlah 180 derajat, jadi: <\/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-662cae07e8d96d1164dad2b0358302fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\gamma = 180 -42,27-101,31 = \\bm{36,42\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"266\" style=\"vertical-align: -4px;\"><\/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 melihat apa itu dan bagaimana menghitung perkalian titik dua vektor. Anda juga akan mempelajari cara mencari sudut antara dua vektor menggunakan perkalian titik dan, sebagai tambahan, semua properti perkalian titik. Terakhir, Anda akan dapat berlatih dengan contoh dan latihan yang diselesaikan langkah demi langkah. Cara menghitung perkalian titik antara dua &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan\/\"> <span class=\"screen-reader-text\">Hitung perkalian titik dua 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":[56],"tags":[],"class_list":["post-212","post","type-post","status-publish","format-standard","hentry","category-kalkulator-ilmiah"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Hitung perkalian titik dua vektor -<\/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\/menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Hitung perkalian titik dua vektor -\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan melihat apa itu dan bagaimana menghitung perkalian titik dua vektor. Anda juga akan mempelajari cara mencari sudut antara dua vektor menggunakan perkalian titik dan, sebagai tambahan, semua properti perkalian titik. Terakhir, Anda akan dapat berlatih dengan contoh dan latihan yang diselesaikan langkah demi langkah. 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ini Anda akan melihat apa itu dan bagaimana menghitung perkalian titik dua vektor. Anda juga akan mempelajari cara mencari sudut antara dua vektor menggunakan perkalian titik dan, sebagai tambahan, semua properti perkalian titik. Terakhir, Anda akan dapat berlatih dengan contoh dan latihan yang diselesaikan langkah demi langkah. Cara menghitung perkalian titik antara dua &hellip; Hitung perkalian titik dua vektor Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan\/","article_published_time":"2023-07-11T19:16:54+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c06f6c1238ab7803750f830e16891f1f_l3.png"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"4 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/menghitung-hasil-kali-skalar-antara-dua-vektor-contoh-latihan-yang-diselesaikan\/"},"author":{"name":"Tim 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