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Langganan aktif akan dibatalkan, akses Nakafa Pro berakhir, dan sekolah yang kamu buat akan dialihkan ke anggota aktif lain.","delete-account-session-expired":"Sesi ini sudah terlalu lama. Masuk lagi untuk melanjutkan.","delete-account-school-member-required":"Tambahkan satu anggota aktif ke setiap sekolah yang kamu buat, lalu coba lagi.","delete-account-error":"Akunmu belum bisa dihapus. Tunggu sebentar, lalu coba lagi.","delete-account-sign-in-again":"Masuk lagi","teacher":"Guru","student":"Siswa","parent":"Orang tua","administrator":"Administrator","save":"Simpan","developers":"Developer","developers-description":"Akses alat developer, API key, dan sumber integrasi untuk membangun aplikasi dengan konten dan layanan edukasi Nakafa.","api-key":"API Key","api-key-description":"Kelola API keys untuk autentikasi dengan API Nakafa.","api-key-footer":"Buat API key baru di sini.","api-key-confirmation-description":"Simpan API key kamu baik-baik ya, jangan sampai hilang.","new-api-key":"API Key Baru","api-key-created":"Sip, API key berhasil dibuat!","no-api-keys":"Kamu belum punya API key.","create":"Buat","api-key-name":"Nama API Key","api-key-name-placeholder":"Masukkan nama API key kamu...","expires-in":"Kadaluarsa dalam","expires-in-placeholder":"Hari hingga kadaluarsa...","7-days":"7 hari","30-days":"30 hari","90-days":"90 hari","180-days":"180 hari","365-days":"365 hari","never":"Tidak pernah","expires":"Kadaluarsa","api-key-copied":"API key udah disalin!","delete-api-key":"Hapus API Key","delete-api-key-description":"Yakin mau hapus API key ini? 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Jadi $$k=2$$.\n\nMasukkan keduanya ke operasi terakhir:\n\n$$\\triangle(5,9)=5(2)+\\frac{9}{1}=19$$\n\nJadi $$\\triangle(5,9)=19$$.","projectile-title":"Gerak yang sulit dibayangkan jadi lebih mudah saat bisa kamu lihat sendiri","projectile-controls":"Pilih lintasan bola meriam","projectile-sixty-degree":"Enam Puluh Derajat","projectile-long-drive":"Jarak Jauh","projectile-high-arc":"Lengkung Tinggi","projectile-view-label":"Visual analisis gerak parabola bola meriam","projectile-horizontal-component":"Komponen mendatar","projectile-vertical-component":"Komponen vertikal","projectile-peak-time":"Waktu puncak","projectile-flight-time":"Waktu terbang","projectile-range":"Jangkauan","projectile-instantaneous-velocity":"Kecepatan pada dua detik"},"CurriculaSection":{"headline":"Kurikulummu boleh berbeda. Alur belajarmu tetap menyatu.","description":"Mulai dari jalur sekolahmu. 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Pada setiap fase penularan, jumlah orang yang tertular akan bertambah dengan pola yang menarik:","lesson-sequence-math":"1 = 2^0 \\quad 2 = 2^1 \\quad 4 = 2 \\times 2 = 2^2 \\quad 8 = 2 \\times 2 \\times 2 = 2^3 \\quad 16 = 2 \\times 2 \\times 2 \\times 2 = 2^4"},"About":{"meta-title":"Materi Belajar Terbuka Gratis dari SD sampai Universitas","title":"Belajar sampai paham.","description":"Pilih pelajaran, uji pemahaman lewat Tryout, lalu tanya Nina saat kamu buntu.","explore-subjects":"Jelajahi pelajaran","try-tryout":"Coba Tryout","contribute":"Kontribusi","start-learning":"Mulai belajar","join-community":"Bergabung komunitas","community-title":"Ketemu orang yang ngerti apa yang kamu pelajari","community-description":"Di balik tiap materi ada orang yang pernah bingung juga. Gabung Discord buat tanya jawab, bagi-bagi sumber belajar, atau bantu yang lain.","curriculum-title":"Belajar apa yang kamu butuhin, pas kamu butuhin","curriculum-description":"Pilih tingkatmu, pelajari materi apa saja, atau cari solusi masalahmu. Nggak perlu scroll lama-lama.","company":"Perusahaan","community":"Komunitas","about-us":"Tentang Kami","trusted-badge":"Belajar bersama","trusted-headline":"Siswa dan mahasiswa dari sekolah dan universitas ini belajar dengan Nakafa.","trusted-description":"Mereka mulai dari materi terbuka, lalu lanjut sesuai ritme masing-masing."},"File":{"exceeds-max-size":"{fileName} kegedean nih, maksimal {maxSizeFormatted} ya.","exceeds-max-size-multiple":"{fileName} kegedean nih, maksimal {maxSizeFormatted} ya.","not-accepted-file-type":"{fileName} tipenya nggak bisa diterima.","max-files-exceeded":"Maksimal {maxFiles} file yang diperbolehkan.","some-files-exceed-max-size":"Beberapa file melebihi ukuran maksimal {maxSizeFormatted}.","file-exceeds-max-size":"File melebihi ukuran maksimal {maxSizeFormatted}.","unknown-file":"File tidak dikenal","unknown-file-size":"Ukuran file tidak dikenal"},"Legal":{"terms-and-policies":"Syarat dan Kebijakan","terms-of-service":"Syarat dan Ketentuan","terms-of-service-description":"Baca Syarat dan Ketentuan Nakafa untuk memahami aturan dan panduan penggunaan platform dan layanan edukasi kami.","privacy-policy":"Kebijakan Privasi","privacy-policy-description":"Pelajari bagaimana Nakafa mengumpulkan, menggunakan, dan melindungi data pribadi dan privasi kamu saat menggunakan platform belajar kami.","security-policy":"Kebijakan Keamanan","security-policy-description":"Tinjau praktik dan kebijakan keamanan Nakafa untuk memahami bagaimana kami melindungi data dan menjamin keamanan platform.","legal-description":"Dengan menggunakan layanan kami, Anda setuju dengan Syarat dan Ketentuan dan Kebijakan Privasi."},"School":{"Common":{"home":"Beranda","learning":"Belajar","classes":"Kelas","nakafa-materials":"Materi Nakafa","yours":"Untukmu","notifications":"Notifikasi"},"Classes":{"classes":"Kelas","search-placeholder":"Cari kelas...","create-class":"Buat kelas","create-class-description":"Siapkan ruang belajar untuk materi, asesmen, dan diskusi siswa.","create":"Buat","name-label":"Nama","name-placeholder":"Kelas 10A","subject-label":"Mata pelajaran","subject-placeholder":"Matematika","year-label":"Tahun ajaran","year-placeholder":"2024/2025","create-class-failed":"Gagal bikin kelas nih. Coba lagi ya.","no-classes-found":"Belum ada kelas nih.","no-subjects-found":"Mata pelajaran nggak ketemu.","search-subjects-placeholder":"Cari mata pelajaran...","join-class":"Gabung kelas","join-class-description":"Masukkan kode undangan untuk masuk ke kelas yang sudah dibuat.","join":"Gabung","code":"Kode","code-placeholder":"Masukkan kode...","join-class-failed":"Gagal gabung kelas. 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Silakan coba lagi.","create-post-failed":"Gagal mengirim pesan. 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36:o33f,58:["$","path","0",{"d":"M3 9.3C3 7.5 5.06126 5.7 7.68468 5.7L16.9574 5.7C19.2 5.7 21 4.35 21 3","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"opacity":"$undefined"}] 59:["$","path","1",{"d":"M16.502 6L15.6072 18.3139C15.5177 19.7653 16.5097 20.9999 17.7654 20.9999C18.6967 20.9999 19.5236 20.3093 19.8181 19.2855L20.1016 18.2999","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"opacity":"$undefined"}] 5a:["$","path","2",{"d":"M10 6C9.86667 9.0625 9.6 14.75 9.2 16.5C8.8 18.25 8 20 6 20","stroke":"currentColor","strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"opacity":"$undefined"}] 5b:["$","h1",null,{"className":"font-medium text-3xl leading-tight tracking-tight","children":"Perkalian Skalar Bilangan Kompleks"}] 63:T166f,36:o166f, export const metadata = { title: "Perkalian Skalar Bilangan Kompleks", description: "Pelajari efek perkalian skalar pada vektor kompleks, termasuk perpanjangan, perpendekkan, dan pembalikan arah.", authors: [{ name: "Nabil Akbarazzima Fatih" }], date: "2025-05-01", subject: "Bilangan Kompleks", }; ## Apa itu Perkalian Skalar? Perkalian skalar adalah mengalikan bilangan kompleks dengan bilangan real (skalar). Perhatikan bagaimana vektor berubah ketika dikalikan skalar dan{" "} . } cameraPosition={[0, 0, 12]} showZAxis={false} data={[ // Vektor z asli { points: [ { x: 0, y: 0, z: 0 }, { x: 1, y: 2, z: 0 }, ], color: "#0284c7", labels: [{ text: "z", at: 1, offset: [0.5, 0.5, 0] }], cone: { position: "end" }, }, // Vektor 2z { points: [ { x: 0, y: 0, z: 0 }, { x: 2, y: 4, z: 0 }, // 2 * (1+2i) = 2+4i ], color: "#059669", labels: [{ text: "2z", at: 1, offset: [0.5, 0.5, 0] }], cone: { position: "end" }, }, // Vektor -0.5z { points: [ { x: 0, y: 0, z: 0 }, { x: -0.5, y: -1, z: 0 }, // -0.5 * (1+2i) = -0.5 - i ], color: "#e11d48", labels: [{ text: "-0.5z", at: 1, offset: [-0.5, -0.5, 0] }], cone: { position: "end" }, }, ]} /> Dari visualisasi di atas, kita bisa lihat: - Mengalikan dengan skalar (seperti ) akan **memperpanjang** vektor searah. - Mengalikan dengan skalar akan **memperpendek** vektor searah. - Mengalikan dengan skalar (seperti ) akan **membalikkan arah** vektor () dan mengubah panjangnya sesuai nilai . ## Definisi Matematis Jika adalah bilangan kompleks dan adalah skalar (bilangan real), maka perkalian skalarnya adalah: Artinya, kita cukup kalikan skalar ke bagian real () dan bagian imajiner () secara terpisah. ### Contoh Perhitungan Jika dan , maka: Jika dan , maka: ## Contoh Visualisasi Mari kita lihat beberapa contoh lagi untuk memperjelas efek perkalian skalar. ### Skalar Lebih dari Satu Perkalian dengan Skalar } description={ <> Vektor diperpanjang searah saat dikalikan{" "} , menjadi{" "} . } cameraPosition={[0, 0, 8]} showZAxis={false} data={[ { points: [ { x: 0, y: 0, z: 0 }, { x: -2, y: 1, z: 0 }, ], color: "#7c3aed", labels: [{ text: "z", at: 1, offset: [0.5, 0.5, 0] }], cone: { position: "end" }, }, { points: [ { x: 0, y: 0, z: 0 }, { x: -3, y: 1.5, z: 0 }, // 1.5 * (-2+i) ], color: "#65a30d", labels: [{ text: "1.5z", at: 1, offset: [-0.5, 0.5, 0] }], cone: { position: "end" }, }, ]} /> ### Skalar di antara Nol dan Satu Perkalian dengan Skalar } description={ <> Vektor diperpendek searah saat dikalikan{" "} , menjadi{" "} . } cameraPosition={[0, 0, 8]} showZAxis={false} data={[ { points: [ { x: 0, y: 0, z: 0 }, { x: 3, y: -2, z: 0 }, ], color: "#ea580c", labels: [{ text: "z", at: 1, offset: [-0.5, -0.5, 0] }], cone: { position: "end" }, }, { points: [ { x: 0, y: 0, z: 0 }, { x: 2.25, y: -1.5, z: 0 }, // 0.75 * (3-2i) ], color: "#0d9488", labels: [{ text: "0.75z", at: 1, offset: [0.5, 1, 0] }], cone: { position: "end" }, }, ]} /> ### Skalar Negatif Perkalian dengan menghasilkan invers aditif (negatif) dari bilangan kompleks. Perkalian dengan Skalar (Invers Aditif) } description={ <> Vektor dibalik arahnya () saat dikalikan , menjadi{" "} . } cameraPosition={[0, 0, 10]} showZAxis={false} data={[ { points: [ { x: 0, y: 0, z: 0 }, { x: -1, y: -3, z: 0 }, ], color: "#c026d3", labels: [{ text: "z", at: 1, offset: [-0.5, -0.5, 0] }], cone: { position: "end" }, }, { points: [ { x: 0, y: 0, z: 0 }, { x: 1, y: 3, z: 0 }, // -1 * (-1-3i) ], color: "#ca8a04", labels: [{ text: "-z", at: 1, offset: [0.5, 0.5, 0] }], cone: { position: "end" }, }, ]} /> 36:o555,5c:["$","$L62",null,{"content":"$63","slug":"/id/materi/matematika/bilangan-kompleks/perkalian-skalar-bilangan-kompleks","sourceUrl":"https://github.com/nakafaai/aksara/blob/a48d644c809419c93dc247239eabba9ced519051/packages/corpus/material/lesson/mathematics/complex-number/scalar-multiplication-complex-numbers/id.mdx"}] 64:T40b,

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2

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y

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z = -3 + 4i

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c = 3

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cz = 3(-3 + 4i) = 3(-3) + i(3 \times 4) = -9 + 12i

36:o178,6f:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L92",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$96"}}]}],"$L97"]}]}] 36:o8d1,70:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Jika ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

z = 5 - i

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c = -2

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cz = -2(5 - i) = -2(5) + i(-2 \times -1) = -10 + 2i

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z = -2 + i

36:o9fa,75:["$","div",null,{"className":"flex flex-col gap-6 rounded-xl border bg-card py-6 text-card-foreground shadow-sm content-auto-card","data-slot":"card","children":[["$","div",null,{"className":"@container/card-header grid auto-rows-min grid-rows-[auto_auto] items-start gap-2 px-6 has-data-[slot=card-action]:grid-cols-[1fr_auto] [.border-b]:pb-6","data-slot":"card-header","children":[["$","div",null,{"className":"font-medium leading-tight","data-slot":"card-title","children":["Perkalian dengan Skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

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c = 1.5

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z = 3 - 2i

36:o9ff,77:["$","div",null,{"className":"flex flex-col gap-6 rounded-xl border bg-card py-6 text-card-foreground shadow-sm content-auto-card","data-slot":"card","children":[["$","div",null,{"className":"@container/card-header grid auto-rows-min grid-rows-[auto_auto] items-start gap-2 px-6 has-data-[slot=card-action]:grid-cols-[1fr_auto] [.border-b]:pb-6","data-slot":"card-header","children":[["$","div",null,{"className":"font-medium leading-tight","data-slot":"card-title","children":["Perkalian dengan Skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

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c = 0.75

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z = -1 - 3i

36:oadc,7a:["$","div",null,{"className":"flex flex-col gap-6 rounded-xl border bg-card py-6 text-card-foreground shadow-sm content-auto-card","data-slot":"card","children":[["$","div",null,{"className":"@container/card-header grid auto-rows-min grid-rows-[auto_auto] items-start gap-2 px-6 has-data-[slot=card-action]:grid-cols-[1fr_auto] [.border-b]:pb-6","data-slot":"card-header","children":[["$","div",null,{"className":"font-medium leading-tight","data-slot":"card-title","children":["Perkalian dengan Skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

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"}}]}]," (",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

180^\\circ

"}}]}],") dan mengubah panjangnya sesuai nilai ","$La5","."]}] 36:o1000,97:["$","$L94",null,{"orientation":"horizontal"}] 99:["$","$L94",null,{"orientation":"horizontal"}] a6:T469,

1.5z = -3 + 1.5i

9b:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a6"}}]}] 9c:["$","div",null,{"className":"px-6","data-slot":"card-content","children":["$","$L8c",null,{"cameraPosition":[0,0,8],"showZAxis":false,"children":[["$","$L8d","line-0,0,0;-2,1,0",{"points":[{"x":0,"y":0,"z":0},{"x":-2,"y":1,"z":0}],"color":"#7c3aed","labels":[{"text":"z","at":1,"offset":[0.5,0.5,0]}],"cone":{"position":"end"}}],["$","$L8d","line-0,0,0;-3,1.5,0",{"points":[{"x":0,"y":0,"z":0},{"x":-3,"y":1.5,"z":0}],"color":"#65a30d","labels":[{"text":"1.5z","at":1,"offset":[-0.5,0.5,0]}],"cone":{"position":"end"}}]]}]}] a7:T44f,

0.75z = 2.25 - 1.5i

9e:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a7"}}]}] 9f:["$","div",null,{"className":"px-6","data-slot":"card-content","children":["$","$L8c",null,{"cameraPosition":[0,0,8],"showZAxis":false,"children":[["$","$L8d","line-0,0,0;3,-2,0",{"points":[{"x":0,"y":0,"z":0},{"x":3,"y":-2,"z":0}],"color":"#ea580c","labels":[{"text":"z","at":1,"offset":[-0.5,-0.5,0]}],"cone":{"position":"end"}}],["$","$L8d","line-0,0,0;2.25,-1.5,0",{"points":[{"x":0,"y":0,"z":0},{"x":2.25,"y":-1.5,"z":0}],"color":"#0d9488","labels":[{"text":"0.75z","at":1,"offset":[0.5,1,0]}],"cone":{"position":"end"}}]]}]}] a1:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c = -1

"}}]}] a8:T450,

-z = 1 + 3i

a2:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a8"}}]}] a3:["$","div",null,{"className":"px-6","data-slot":"card-content","children":["$","$L8c",null,{"cameraPosition":[0,0,10],"showZAxis":false,"children":[["$","$L8d","line-0,0,0;-1,-3,0",{"points":[{"x":0,"y":0,"z":0},{"x":-1,"y":-3,"z":0}],"color":"#c026d3","labels":[{"text":"z","at":1,"offset":[-0.5,-0.5,0]}],"cone":{"position":"end"}}],["$","$L8d","line-0,0,0;1,3,0",{"points":[{"x":0,"y":0,"z":0},{"x":1,"y":3,"z":0}],"color":"#ca8a04","labels":[{"text":"-z","at":1,"offset":[0.5,0.5,0]}],"cone":{"position":"end"}}]]}]}] 36:o2c1,a5:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

|c|

"}}]}] 36:o3a,2c:C 2a:C 2b:C 23:C 25:C 26:C 27:C 28:C 2e:false 2d:77709 36:C 72:["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c=-0.5

"}}]}] 73:["$","div",null,{"className":"px-6","data-slot":"card-content","children":["$","$L99",null,{"cameraPosition":[0,0,12],"showZAxis":false,"children":[["$","$L9a","line-0,0,0;1,2,0",{"points":[{"x":0,"y":0,"z":0},{"x":1,"y":2,"z":0}],"color":"#0284c7","labels":[{"text":"z","at":1,"offset":[0.5,0.5,0]}],"cone":{"position":"end"}}],["$","$L9a","line-0,0,0;2,4,0",{"points":[{"x":0,"y":0,"z":0},{"x":2,"y":4,"z":0}],"color":"#059669","labels":[{"text":"2z","at":1,"offset":[0.5,0.5,0]}],"cone":{"position":"end"}}],["$","$L9a","line-0,0,0;-0.5,-1,0",{"points":[{"x":0,"y":0,"z":0},{"x":-0.5,"y":-1,"z":0}],"color":"#e11d48","labels":[{"text":"-0.5z","at":1,"offset":[-0.5,-0.5,0]}],"cone":{"position":"end"}}]]}]}] 74:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Dari visualisasi di atas, kita bisa lihat:"}] 75:["$","ul",null,{"className":"my-4 list-disc space-y-4 pl-4.5 last:mb-0","children":["\n",["$","li",null,{"className":"space-y-4 text-pretty pl-1 leading-relaxed","children":["Mengalikan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

z

"}}]}]," dengan skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c > 1

"}}]}]," (seperti ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

2

z

z

"}}]}]," dengan skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

0 < c < 1

"}}]}]," akan ","$L9b"," vektor ","$L9c"," searah."]}],"\n","$L9d","\n"]}] 76:["$","h2",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-2xl","id":"definisi-matematis","children":["$","a",null,{"aria-label":"Link to Definisi Matematis","className":"group/heading inline-flex items-center gap-4","href":"#definisi-matematis","title":"Definisi Matematis","children":[["$","span",null,{"className":"text-pretty","children":"Definisi Matematis"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] 9e:T45a,

z = x + iy

77:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Jika ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$9e"}}]}]," adalah bilangan kompleks dan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c

"}}]}]," adalah skalar (bilangan real), maka perkalian skalarnya adalah:"]}] a0:T8fb,

cz = c(x + iy) = (cx) + i(cy)

78:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a0"}}]}],["$","$La1",null,{"orientation":"horizontal"}]]}]}] 79:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Artinya, kita cukup kalikan skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c

"}}]}]," ke bagian real (",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

x

"}}]}],") dan bagian imajiner (",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

y

"}}]}],") secara terpisah."]}] 7a:["$","h3",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-xl","id":"contoh-perhitungan","children":["$","a",null,{"aria-label":"Link to Contoh Perhitungan","className":"group/heading inline-flex items-center gap-4","href":"#contoh-perhitungan","title":"Contoh Perhitungan","children":[["$","span",null,{"className":"text-pretty","children":"Contoh Perhitungan"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] a2:T437,

z = -3 + 4i

7b:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Jika ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a2"}}]}]," dan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c = 3

"}}]}],", maka:"]}] a3:Tc89,

cz = 3(-3 + 4i) = 3(-3) + i(3 \times 4) = -9 + 12i

7c:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a3"}}]}],"$La4"]}]}] 7d:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Jika ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

z = 5 - i

"}}]}]," dan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c = -2

"}}]}],", maka:"]}] a5:Tcbb,

cz = -2(5 - i) = -2(5) + i(-2 \times -1) = -10 + 2i

7e:["$","div",null,{"className":"overflow-hidden rounded-xl border bg-card text-card-foreground content-auto-formula","data-markdown-ignore":"","data-math-block":"","children":["$","$L9f",null,{"className":"grid","children":[["$","div",null,{"className":"px-4","children":["$","div",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a5"}}]}],"$La6"]}]}] 7f:["$","h2",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-2xl","id":"contoh-visualisasi","children":["$","a",null,{"aria-label":"Link to Contoh Visualisasi","className":"group/heading inline-flex items-center gap-4","href":"#contoh-visualisasi","title":"Contoh Visualisasi","children":[["$","span",null,{"className":"text-pretty","children":"Contoh Visualisasi"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] 80:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":"Mari kita lihat beberapa contoh lagi untuk memperjelas efek perkalian skalar."}] 81:["$","h3",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-xl","id":"skalar-lebih-dari-satu","children":["$","a",null,{"aria-label":"Link to Skalar Lebih dari Satu","className":"group/heading inline-flex items-center gap-4","href":"#skalar-lebih-dari-satu","title":"Skalar Lebih dari Satu","children":[["$","span",null,{"className":"text-pretty","children":"Skalar Lebih dari Satu"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] a7:T411,

z = -2 + i

82:["$","div",null,{"className":"flex flex-col gap-6 rounded-xl border bg-card py-6 text-card-foreground shadow-sm content-auto-card","data-slot":"card","children":[["$","div",null,{"className":"@container/card-header grid auto-rows-min grid-rows-[auto_auto] items-start gap-2 px-6 has-data-[slot=card-action]:grid-cols-[1fr_auto] [.border-b]:pb-6","data-slot":"card-header","children":[["$","div",null,{"className":"font-medium leading-tight","data-slot":"card-title","children":["Perkalian dengan Skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c = 1.5

"}}]}]]}],["$","div",null,{"className":"text-muted-foreground text-sm","data-slot":"card-description","children":["Vektor ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$a7"}}]}]," diperpanjang searah saat dikalikan"," ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c = 1.5

"}}]}],", menjadi"," ","$La8","."]}]]}],"$La9"]}] 83:["$","h3",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-xl","id":"skalar-di-antara-nol-dan-satu","children":["$","a",null,{"aria-label":"Link to Skalar di antara Nol dan Satu","className":"group/heading inline-flex items-center gap-4","href":"#skalar-di-antara-nol-dan-satu","title":"Skalar di antara Nol dan Satu","children":[["$","span",null,{"className":"text-pretty","children":"Skalar di antara Nol dan Satu"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] aa:T411,

z = 3 - 2i

84:["$","div",null,{"className":"flex flex-col gap-6 rounded-xl border bg-card py-6 text-card-foreground shadow-sm content-auto-card","data-slot":"card","children":[["$","div",null,{"className":"@container/card-header grid auto-rows-min grid-rows-[auto_auto] items-start gap-2 px-6 has-data-[slot=card-action]:grid-cols-[1fr_auto] [.border-b]:pb-6","data-slot":"card-header","children":[["$","div",null,{"className":"font-medium leading-tight","data-slot":"card-title","children":["Perkalian dengan Skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c = 0.75

"}}]}]]}],["$","div",null,{"className":"text-muted-foreground text-sm","data-slot":"card-description","children":["Vektor ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$aa"}}]}]," diperpendek searah saat dikalikan"," ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c = 0.75

"}}]}],", menjadi"," ","$Lab","."]}]]}],"$Lac"]}] 85:["$","h3",null,{"className":"mt-10 mb-6 flex scroll-mt-44 items-center font-medium tracking-tight first:mt-0 last:mb-0 text-xl","id":"skalar-negatif","children":["$","a",null,{"aria-label":"Link to Skalar Negatif","className":"group/heading inline-flex items-center gap-4","href":"#skalar-negatif","title":"Skalar Negatif","children":[["$","span",null,{"className":"text-pretty","children":"Skalar Negatif"}],["$","div",null,{"className":"rounded-sm border p-2 opacity-0 transition-opacity ease-out group-hover/heading:opacity-100","children":["$","svg",null,{"xmlns":"http://www.w3.org/2000/svg","width":24,"height":24,"viewBox":"0 0 24 24","fill":"none","color":"currentColor","className":"size-4 shrink-0 text-muted-foreground","strokeWidth":2,"stroke":"currentColor","children":[["$","path","0",{"d":"M13.5 17H17C19.7614 17 22 14.7614 22 12C22 9.23858 19.7614 7 17 7H13.5M10.5 17H7C4.23858 17 2 14.7614 2 12C2 9.23858 4.23858 7 7 7H10.5","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}],["$","path","1",{"d":"M9 12H15","stroke":"currentColor","strokeLinecap":"round","strokeWidth":2}]]}]}]]}]}] 86:["$","p",null,{"className":"my-4 text-pretty tabular-nums leading-relaxed first:mt-0 last:mb-0","children":["Perkalian dengan ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

-1

"}}]}]," menghasilkan invers aditif (negatif) dari bilangan kompleks."]}] ad:T43b,

z = -1 - 3i

87:["$","div",null,{"className":"flex flex-col gap-6 rounded-xl border bg-card py-6 text-card-foreground shadow-sm content-auto-card","data-slot":"card","children":[["$","div",null,{"className":"@container/card-header grid auto-rows-min grid-rows-[auto_auto] items-start gap-2 px-6 has-data-[slot=card-action]:grid-cols-[1fr_auto] [.border-b]:pb-6","data-slot":"card-header","children":[["$","div",null,{"className":"font-medium leading-tight","data-slot":"card-title","children":["Perkalian dengan Skalar ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

c = -1

"}}]}]," (Invers Aditif)"]}],["$","div",null,{"className":"text-muted-foreground text-sm","data-slot":"card-description","children":["Vektor ",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"$ad"}}]}]," dibalik arahnya (",["$","span",null,{"data-markdown-ignore":"","children":["$","span",null,{"data-testid":"react-katex","dangerouslySetInnerHTML":{"__html":"

180^\\circ

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