{"id":72,"date":"2023-09-16T13:03:21","date_gmt":"2023-09-16T13:03:21","guid":{"rendered":"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/"},"modified":"2023-09-16T13:03:21","modified_gmt":"2023-09-16T13:03:21","slug":"perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/","title":{"rendered":"Perkalian silang dua vektor (atau perkalian silang)"},"content":{"rendered":"<p>Halaman ini menjelaskan apa itu perkalian silang dua vektor dan cara menghitungnya. Anda juga akan melihat cara mencari arah dan arah perkalian silang menggunakan aturan tangan kanan (atau pembuka botol). Terlebih lagi, Anda akan menemukan kegunaan operasi jenis ini, serta contoh, latihan, dan masalah yang diselesaikan langkah demi langkah. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-el-producto-vectorial-de-dos-vectores\"><\/span> Berapakah perkalian silang dua vektor?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dalam matematika, <strong>perkalian silang<\/strong> adalah operasi antara dua vektor dalam ruang tiga dimensi (dalam R3). Hasil operasi vektor ini adalah sebuah vektor yang arahnya tegak lurus terhadap kedua vektor yang dikalikan, dan modulnya sama dengan hasil kali modul vektor-vektor pengalinya dengan sinus sudut yang dibentuknya. Dengan kata lain rumusnya 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-625267c9d98347748a771c7cec9bfcec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\times \\vv{\\text{v}}\\rvert = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}}\\rvert \\cdot \\text{sen}(\\alpha)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"186\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Seperti yang Anda lihat pada rumus sebelumnya, perkalian silang dilambangkan<\/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-850177c0489097bd9409ba9b13b07506_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> , oleh karena itu disebut juga <strong>perkalian silang.<\/strong> Kadang-kadang juga disebut produk vektor Gibbs, karena ia menemukannya. <\/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\/quel-est-le-produit-vectoriel-de-deux-vecteurs-.webp\" alt=\"\" class=\"wp-image-3883\" width=\"307\" height=\"310\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Seperti yang dapat Anda lihat pada representasi grafis sebelumnya, perkalian silang tegak lurus terhadap dua vektor yang dikalikan keduanya, sehingga normal terhadap bidang yang menampung vektor-vektor tersebut. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-para-calcular-el-producto-vectorial-de-dos-vectores\"><\/span> Rumus untuk menghitung perkalian silang dua vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Jika kita mengetahui koordinat Cartesian dari vektor-vektor tersebut, cara paling sederhana untuk menghitung perkalian silang vektor-vektor tersebut adalah dengan mencari determinan 3&#215;3. Perhatikan cara melakukannya: <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Pertimbangkan dua vektor apa pun:<\/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-581394386a4c68ca2bfa92fb4e2445ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (\\text{u}_x,\\text{u}_y,\\text{u}_z) \\qquad \\vv{\\text{v}}= (\\text{v}_x,\\text{v}_y,\\text{v}_z)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"267\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Produk vektornya 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-56551111a4f5a18a4609772ebaeaf919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] \\text{u}_x &amp; \\text{u}_y &amp; \\text{u}_z \\\\[1.1ex] \\text{v}_x &amp;\\text{v}_y&amp;\\text{v}_z \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Dimana 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-899f7cb82c85508ac2129e2393976f80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{i}, \\vv{j},\\vv{k}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"38\" style=\"vertical-align: -4px;\"><\/p>\n<p> Ini adalah vektor satuan pada arah sumbu X, Y dan Z.<\/p>\n<\/div>\n<p> Mari kita lihat contoh cara menghitung perkalian silang 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-89a657062237b32001dad723a07ad2ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (3,1,0) \\qquad \\vv{\\text{v}}= (2,1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"227\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Untuk menentukan hasil kali vektor antar vektor, kita harus membuat determinan orde 3 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-abc77b698bf6f4fddec1ab2dcc8b07f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 3&amp; 1 &amp; 0 \\\\[1.1ex] 2 &amp;1&amp;-1 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dalam hal ini, kita akan menyelesaikan determinan dengan bahan pembantu atau kofaktor (aturan Sarrus juga dapat digunakan):<\/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-eeeac04b3f0edd64e5413629051551fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 3&amp; 1 &amp; 0 \\\\[1.1ex] 2 &amp;1&amp;-1 \\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} 1 &amp; 0 \\\\[1.1ex] 1&amp;-1 \\end{vmatrix} -\\vv{j}\\begin{vmatrix}  3&amp;  0 \\\\[1.1ex] 2 &amp;-1 \\end{vmatrix}+\\vv{z}\\begin{vmatrix}3&amp; 1 \\\\[1.1ex] 2 &amp;1 \\end{vmatrix}  \\\\[2ex] &amp; = -\\vv{i}+3\\vv{j}+\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"411\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Maka hasil perkalian vektor kedua vektor tersebut adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-625f3af33d3cd3b9991682014c911024_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\bm{(-1,3,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"determinar-la-direccion-y-el-sentido-del-producto-vectorial\"><\/span> Menentukan arah dan arah perkalian silang<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Terkadang kita tidak perlu mengetahui komponen-komponen vektor hasil perkalian silang, tetapi cukup mencari modulusnya, arahnya, dan arahnya. Hal ini sering terjadi dalam fisika, khususnya dalam perhitungan gaya.<\/p>\n<p> Jadi, ada beberapa aturan untuk mencari arah dan arah perkalian vektor, yang paling terkenal adalah <strong>aturan tangan kanan<\/strong> , baik dengan tiga jari atau dengan seluruh tangan, dan <strong>aturan pembuka botol (atau sekrup)<\/strong> . Anda dapat menggunakan salah satunya, jadi Anda tidak perlu mengetahui semuanya, kami akan tetap menjelaskan ketiga aturan tersebut kepada Anda sehingga Anda dapat tetap menggunakan yang paling Anda sukai. \ud83d\ude09<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"regla-de-la-mano-derecha-3-dedos\"><\/span> Aturan tangan kanan (3 jari)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Aturan atau hukum tangan kanan versi 3 jari melibatkan langkah-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;\">Letakkan jari telunjuk tangan kanan Anda ke arah vektor pertama perkalian silang\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a13b5c3a9f833efc0ddf4caf790f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"27\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Letakkan jari tengah (atau jari tengah) tangan kanan Anda ke arah vektor kedua perkalian silang\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acab6198e5d0337e7d0e9ed7814c16d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"26\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">Posisi ibu jari yang dihasilkan menunjukkan arah dan arah perkalian silang\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fce88450a631ec24e37f45befae0675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}\\times\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> <\/li>\n<\/ol>\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\/regle-ou-loi-de-la-main-droite.webp\" alt=\"\" class=\"wp-image-892\" width=\"392\" height=\"353\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"regla-de-la-mano-derecha-palma-de-la-mano\"><\/span> Aturan tangan kanan (telapak tangan)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Aturan atau hukum tangan kanan versi palmar sangat mirip dengan aturan sebelumnya. Untuk menerapkannya, Anda harus mengikuti langkah-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;\">Letakkan tangan kanan Anda sambil menunjuk dengan jari-jari Anda searah dengan vektor pertama perkalian silang\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a13b5c3a9f833efc0ddf4caf790f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"27\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Tutup tangan kanan Anda dengan menggerakkan jari-jari Anda ke arah vektor kedua perkalian silang\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acab6198e5d0337e7d0e9ed7814c16d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"26\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> Anda harus menutup tangan Anda pada sisi yang sudut (atau jarak) antar vektornya lebih kecil.<\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">Posisi ibu jari yang dihasilkan menentukan arah perkalian silang\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fce88450a631ec24e37f45befae0675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}\\times\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> <\/li>\n<\/ol>\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\/vecteur-produit-regle-de-droite.webp\" alt=\"produk vektor penggaris tangan kanan\" class=\"wp-image-898\" width=\"373\" height=\"342\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"regla-del-sacacorchos\"><\/span> aturan pembuka botol<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> <strong>Aturan pembuka botol atau sekrup<\/strong> mirip dengan aturan tangan kanan yang menggunakan seluruh telapak tangan. Prosedurnya adalah sebagai 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;\">Dengan menggunakan imajinasi Anda, letakkan pembuka botol (atau sekrup) dengan pegangan mengarah ke arah yang sama dengan vektor pertama perkalian silang.\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a13b5c3a9f833efc0ddf4caf790f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"27\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Kemudian putar pembuka botol ke arah vektor kedua perkalian silang\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4c87e248254db96e4d2c996a62911e87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{v}})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"21\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> seolah-olah Anda akan memasukkannya ke dalam gabus. Anda perlu memutar pembuka botol ke sisi yang jarak antar vektornya paling pendek.<\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">Arah titik spiral pembuka botol akan menjadi arah dan arah perkalian vektor\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fce88450a631ec24e37f45befae0675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}\\times\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> <\/li>\n<\/ol>\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\/regle-du-tire-bouchon-ou-de-la-vis.webp\" alt=\"pembuka botol atau penggaris sekrup\" class=\"wp-image-902\" width=\"327\" height=\"500\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-del-producto-vectorial-de-dos-vectores\"><\/span> Sifat-sifat perkalian silang dua vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Perkalian silang dua buah vektor mempunyai ciri-ciri sebagai berikut:<\/p>\n<ul>\n<li> <strong>Sifat antikomutatif:<\/strong> urutan vektor-vektor yang terlibat dalam perkalian vektor tidak tetap, karena tandanya bervariasi menurutnya.<\/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-da6aeea2768e41a7e3630dc83ff1e31b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\\times\\vv{\\text{v}} = - \\vv{\\text{v}}\\times\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"119\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Sifat distributif<\/strong> 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-bfddca49d7ded207f392f54341fff56d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\\times(\\vv{\\text{v}}+\\vv{\\text{w}}) = \\vv{\\text{u}}\\times\\text{v}}+\\vv{\\text{u}}\\times \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"220\" 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-68e08abafcbf4b05614819ca1a364a47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\\times(\\vv{\\text{v}}-\\vv{\\text{w}}) = \\vv{\\text{u}}\\times\\text{v}}-\\vv{\\text{u}}\\times \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"220\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Sifat homogen<\/strong> : mengalikan vektor perkalian silang dengan skalar (bilangan real) sama dengan mengalikan hasil perkalian silang dengan skalar tersebut.<\/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-2ee3552bb6b72db4d30b4aa9f73e99a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k \\cdot (\\vv{\\text{u}}\\times\\vv{\\text{v}}) =  (k\\cdot \\vv{\\text{u}})\\times\\vv{\\text{v}}=\\vv{\\text{u}}\\times(k\\cdot\\vv{\\text{v}})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"278\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Vektor yang dihasilkan dari perkalian vektor <strong>tegak lurus<\/strong> terhadap dua vektor yang terlibat dalam operasi tersebut.<\/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-6a96345e09a0fdb952557c9138c72ac4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c} \\vv{\\text{u}} \\perp (\\vv{\\text{u}}\\times\\vv{\\text{v}}) \\\\[2ex] \\vv{\\text{v}} \\perp (\\vv{\\text{u}}\\times\\vv{\\text{v}}) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"87\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Selanjutnya jika kedua vektor ortogonal maka persamaan berikut terpenuhi:<\/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-d49d463798c6381c9a8c065417ee3dbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\perp \\vv{\\text{v}} \\ \\longrightarrow \\ \\begin{cases} \\vv{\\text{u}} \\cdot (\\vv{\\text{u}}\\times\\vv{\\text{v}})=0 \\\\[2ex] \\vv{\\text{v}} \\cdot (\\vv{\\text{u}}\\times\\vv{\\text{v}})=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"220\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Perkalian silang dua <strong>vektor sejajar<\/strong> sama dengan vektor nol (atau nol).<\/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-5a2e4e335c132d024b2932139a51f101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\ || \\ \\vv{\\text{v}} \\ \\longrightarrow \\ \\vv{\\text{u}}\\times\\vv{\\text{v}}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Jika kita tidak mengetahui sudut yang dibentuk oleh dua vektor, modulus perkalian vektornya juga dapat dihitung menggunakan 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-f27e24e2a7310fb1e727891c17de911a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}}\\times\\vv{\\text{v}} \\rvert = \\sqrt{ \\lvert \\vv{\\text{u}}\\rvert ^2 \\cdot \\lvert \\vv{\\text{v}} \\rvert ^2 - (\\vv{\\text{u}}\\cdot \\vv{\\text{v}})^2 \\vphantom{\\frac{1}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"233\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calcular-el-area-de-un-paralelogramo-o-un-triangulo-mediante-el-producto-vectorial\"><\/span> Hitung luas jajar genjang atau segitiga menggunakan perkalian silang<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Secara geometris, modulus perkalian silang dua vektor berimpit dengan luas jajar genjang yang kedua vektor tersebut sebagai sisinya. Oleh karena itu, <strong>perkalian silang dapat digunakan untuk menghitung luas jajar genjang.<\/strong> <\/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\/produit-vectoriel-de-deux-vecteurs-dans-lespace.webp\" alt=\"perkalian silang dua vektor dalam ruang\" class=\"wp-image-929\" width=\"300\" height=\"173\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Selain itu, diagonal jajar genjang membaginya menjadi dua segitiga, atau dengan kata lain segitiga adalah setengah jajar genjang. Jadi, <strong>luas segitiga<\/strong> adalah setengah modulus hasil kali silang yang mengambil dua sisinya sebagai 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\/produit-vectoriel-de-deux-vecteurs-dans-r2.webp\" alt=\"perkalian silang dua vektor di r2\" class=\"wp-image-927\" width=\"300\" height=\"173\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Ingatlah bahwa modulus suatu vektor dalam ruang 3 dimensi adalah akar dari jumlah kuadrat 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-8cf7995798482007ea32b809e80a4062_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{v}} \\rvert =  \\sqrt{\\vphantom{\\frac{1}{2}} \\text{v}_x^2+\\text{v}_y^2+\\text{v}_z^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"155\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p> Inilah dua penerapan perkalian silang dua vektor dalam bidang matematika. Namun masih memiliki kegunaan lain, misalnya dalam fisika digunakan untuk menghitung medan magnet. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-productos-vectoriales-de-vectores\"><\/span> Latihan soal perkalian vektor dari vektor<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Hitunglah perkalian silang 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-a1c06e6d75f54661416056d31d409d8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (-1,4,2) \\qquad \\vv{\\text{v}}= (0,-2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" 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 solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Untuk menentukan hasil kali vektor antar vektor, kita harus menyelesaikan determinan dimensi 3\u00d73 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-44db63ee02936f6e5f21891c3e412fb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] -1&amp; 4 &amp; 2 \\\\[1.1ex] 0 &amp;-2&amp;1  \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"160\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini, kita akan menyelesaikan determinan dengan bahan pembantu atau kofaktor (tetapi aturan Sarrus juga dapat digunakan):<\/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-fe298c37814c92498e4fd8ade0620951_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\begin{vmatrix}\\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] -1&amp; 4 &amp; 2 \\\\[1.1ex] 0 &amp;-2&amp;1\\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} 4 &amp; 2 \\\\[1.1ex]-2&amp;1\\end{vmatrix} -\\vv{j}\\begin{vmatrix}  -1&amp; 2 \\\\[1.1ex] 0 &amp;1\\end{vmatrix}+\\vv{z}\\begin{vmatrix}-1&amp; 4 \\\\[1.1ex] 0 &amp;-2\\end{vmatrix}  \\\\[2ex] &amp; = 8\\vv{i}+\\vv{j}+2\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"387\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maka hasil perkalian vektor kedua vektor tersebut adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a1efe3e9fce5a83e193da24a5ff60835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\bm{(8,1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" 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> Tentukan perkalian silang 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-210c7ec7b29e06198c491e83e7825a42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (3,-2,4) \\qquad \\vv{\\text{v}}= (1,5,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" 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 solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Untuk mencari hasil kali vektor antara kedua vektor tersebut, kita harus menyelesaikan determinan 3\u00d73 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-6a23d8e45f9065f70c576e6b8db02465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 3&amp; -2 &amp; 4 \\\\[1.1ex] 1 &amp;5&amp;-3  \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini, kita akan menyelesaikan determinan dengan adjoint atau kofaktor (walaupun aturan Sarrus dapat digunakan secara bergantian):<\/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-02ffb40666893faa7677234065f3f85f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\begin{vmatrix}\\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 3&amp; -2 &amp; 4 \\\\[1.1ex] 1 &amp;5&amp;-3\\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} -2 &amp; 4 \\\\[1.1ex] 5&amp;-3\\end{vmatrix} -\\vv{j}\\begin{vmatrix}  3&amp; 4 \\\\[1.1ex]  1&amp;-3\\end{vmatrix}+\\vv{z}\\begin{vmatrix}3&amp; -2  \\\\[1.1ex] 1 &amp;5\\end{vmatrix}  \\\\[2ex] &amp; = -14\\vv{i}+13\\vv{j}+17\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"386\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maka hasil perkalian vektor antara kedua vektor tersebut adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a8f7c253fdd7c5be3635a42a3826acfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\bm{(-14,13,17)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"160\" 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 3<\/h3>\n<p> Mengetahui modulus dua vektor dan 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-b632ab1324e7ece2d4c1f5c54249b425_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"|\\vv{\\text{u}}|= 5 \\qquad |\\vv{\\text{v}}|= 6 \\qquad \\alpha = 30\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"226\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Tentukan besar perkalian silang kedua vektor tersebut. <\/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 solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Kita dapat dengan mudah menghitung modulus perkalian vektor antara kedua vektor dengan menerapkan 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-06dcff41e0dcf31152f0047507056f24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\lvert \\vv{\\text{u}} \\times \\vv{\\text{v}}\\rvert &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}}\\rvert \\cdot \\text{sen}(\\alpha) \\\\[2ex] &amp; = 5 \\cdot 6 \\cdot \\text{sen}(30\u00ba) \\\\[2ex] &amp;= 30 \\cdot 0,5 \\\\[2ex] &amp;= \\bm{15} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"140\" width=\"186\" 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 4<\/h3>\n<p> Dari vektor-vektor berikut yang terdapat pada bidang layar : <\/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\/produit-vectoriel-en-ligne-de-deux-vecteurs.webp\" alt=\"perkalian silang dua vektor sejajar\" class=\"wp-image-938\" width=\"151\" height=\"231\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Hitung besar, arah dan arti vektor yang dihasilkan dari operasi 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-facaf3cf3a645b725e61c9fcb195e53c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\times \\vv{\\text{v}}\\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" 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 solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Kedua vektor tersebut tegak lurus, sehingga norma hasil kali vektornya 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-d8f675fe7eb44c050c508c4771c0a439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\lvert \\vv{\\text{u}} \\times \\vv{\\text{v}}\\rvert &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}}\\rvert \\cdot \\text{sen}(\\alpha) \\\\[2ex] &amp; = 3 \\cdot 4 \\cdot \\text{sen}(90\u00ba) \\\\[2ex] &amp;= 12 \\cdot 1 \\\\[2ex] &amp;= \\bm{12} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"140\" width=\"186\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sebaliknya, vektor yang dihasilkan dari perkalian vektor tegak lurus terhadap dua vektor yang ikut serta dalam operasi, <strong>sehingga arahnya akan tegak lurus terhadap layar.<\/strong><\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, dengan menggunakan aturan garis lurus (atau pembuka botol), kita dapat menyimpulkan bahwa <strong>arah vektor yang dihasilkan akan mengarah ke bagian dalam layar.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 5<\/h3>\n<p> Hitunglah luas jajar genjang yang mempunyai vektor-vektor berikut pada kedua 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-43326a03cb829fb91d8c265bddf92b8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (2,3,-2) \\qquad \\vv{\\text{v}}= (5,0,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" 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 solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Luas jajar genjang bertepatan dengan modulus perkalian silang vektor-vektor yang membentuknya. Oleh karena itu kami menghitung produk vektor dari vektor-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-0e7c1825be82d94c4eae49c73f509858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\vv{\\text{u}}\\times \\vv{\\text{v}} = \\begin{vmatrix}\\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex]2&amp; 3 &amp; -2 \\\\[1.1ex] 5 &amp;0&amp;-1\\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} 3 &amp; -2 \\\\[1.1ex] 0&amp;-1\\end{vmatrix} -\\vv{j}\\begin{vmatrix}  2&amp; -2 \\\\[1.1ex] 5 &amp;-1\\end{vmatrix}+\\vv{z}\\begin{vmatrix}2&amp; 3  \\\\[1.1ex] 5 &amp;0\\end{vmatrix}  \\\\[2ex] &amp; = -3\\vv{i}-8\\vv{j}-15\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"411\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian modul Anda: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-164e46102f5d27babfec98f25d479fab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A=\\lvert\\vv{\\text{u}}\\times \\vv{\\text{v}} \\rvert = \\sqrt{\\vphantom{\\frac{1}{2}} (-3)^2+(-8)^2+(-15)^2}=\\bm{17,26} \\ \\mathbf{u}\\bm{^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"404\" style=\"vertical-align: -10px;\"><\/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> Temukan luas segitiga yang titik sudutnya 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-dc08f9ac339b7ac3787a449cf5558c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(2,1,0) \\qquad B(4,0,3)\\qquad C(-1,2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"294\" 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 solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita harus menghitung vektor-vektor yang membentuk sisi-sisi segitiga: <\/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-f7661fc5ae3b35c76e0fe98203258962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B- A = (4,0,3)-(2,1,0) = (2,-1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"351\" 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-5d3f55a240188777fdbb3d74cbf4f61f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC} =C- B =(-1,2,3)- (4,0,3) = (-5,2,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"366\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Luas suatu segitiga adalah setengah besar hasil kali vektor dari vektor-vektor yang membentuknya. Oleh karena itu kami menghitung produk vektor dari vektor-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-42a0ae5858bcb681ee92ec1ed67424c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\vv{\\text{u}}\\times \\vv{\\text{v}} = \\begin{vmatrix}\\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex]2&amp; -1 &amp; 3 \\\\[1.1ex] -5 &amp;2&amp;0\\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} -1 &amp; 3 \\\\[1.1ex] 2&amp;0\\end{vmatrix} -\\vv{j}\\begin{vmatrix}  2&amp;  3 \\\\[1.1ex] -5 &amp;0\\end{vmatrix}+\\vv{z}\\begin{vmatrix}2&amp; -1  \\\\[1.1ex] -5 &amp;2\\end{vmatrix}  \\\\[2ex] &amp; = -6\\vv{i}-15\\vv{j}-\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"453\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah modul Anda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4a52c4a997a5b68d67b9ae52ca599661_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert\\vv{\\text{u}}\\times \\vv{\\text{v}} \\rvert = \\sqrt{\\vphantom{\\frac{1}{2}} (-6)^2+(-15)^2+(-1)^2}=16,19\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"342\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, luas segitiga akan menjadi setengah modul: <\/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-b56f201265ba609f4b7c75f486120fc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A=\\cfrac{1}{2}\\cdot  \\lvert\\vv{\\text{u}}\\times \\vv{\\text{v}} \\rvert = \\cfrac{1}{2}\\cdot 16,19=\\bm{8,09} \\ \\mathbf{u}\\bm{^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"284\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Halaman ini menjelaskan apa itu perkalian silang dua vektor dan cara menghitungnya. Anda juga akan melihat cara mencari arah dan arah perkalian silang menggunakan aturan tangan kanan (atau pembuka botol). Terlebih lagi, Anda akan menemukan kegunaan operasi jenis ini, serta contoh, latihan, dan masalah yang diselesaikan langkah demi langkah. Berapakah perkalian silang dua vektor? Dalam &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/\"> <span class=\"screen-reader-text\">Perkalian silang dua vektor (atau perkalian silang)<\/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-72","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>\u25b7 Hitung perkalian silang dua vektor (contoh)<\/title>\n<meta name=\"description\" content=\"Penjelasan cara menghitung perkalian silang dua vektor. Dengan menyelesaikan soal perkalian silang dua vektor.\" \/>\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-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u25b7 Hitung perkalian silang dua vektor (contoh)\" \/>\n<meta property=\"og:description\" content=\"Penjelasan cara menghitung perkalian silang dua vektor. Dengan menyelesaikan soal perkalian silang dua vektor.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-16T13:03:21+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-625267c9d98347748a771c7cec9bfcec_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\t<meta name=\"twitter:label2\" content=\"Estimasi waktu membaca\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 menit\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Perkalian silang dua vektor (atau perkalian silang)\",\"datePublished\":\"2023-09-16T13:03:21+00:00\",\"dateModified\":\"2023-09-16T13:03:21+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/\"},\"wordCount\":449,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/id\/#organization\"},\"articleSection\":[\"Vektor\"],\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/\",\"url\":\"https:\/\/mathority.org\/id\/perkalian-silang-dua-vektor-contoh-rumus-silang-latihan-yang-diselesaikan\/\",\"name\":\"\u25b7 Hitung perkalian silang dua vektor (contoh)\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/#website\"},\"datePublished\":\"2023-09-16T13:03:21+00:00\",\"dateModified\":\"2023-09-16T13:03:21+00:00\",\"description\":\"Penjelasan cara menghitung perkalian silang dua vektor. 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