[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"public-settings":3,"content:materi:vektor-3-dimensi-dot-product":11,"public-catalog":217},{"features.tutoring.enabled":4,"sections.subjects.enabled":5,"sections.pricing.enabled":5,"sections.teachers.enabled":5,"sections.testimonials.enabled":4,"sections.faqs.enabled":5,"bankSoal.enabled":5,"site.maintenance":4,"whatsapp.number":6,"brand.name":7,"brand.tagline":8,"contact.email":9,"contact.address":10},false,true,"6285156084435","MyFismat","Platform belajar Matematika, Fisika, dan Pemrograman untuk SD, SMP, SMA, dan Kuliah","support@myfismat.com","Indonesia",{"id":12,"type":13,"slug":14,"title":15,"excerpt":16,"coverUrl":17,"status":18,"publishedAt":19,"featured":4,"subjectId":20,"blocks":21,"seoTitle":17,"seoDescription":17,"readingMinutes":106,"order":107,"createdAt":108,"updatedAt":108,"subject":109,"levels":114,"topics":119,"tags":123,"author":17,"resolved":124,"related":144},"demo-v2-m-matematika-kuliah-3","MATERIAL","vektor-3-dimensi-dot-product","Vektor 3 Dimensi dan Dot Product (Perkalian Titik)","Vektor 3 dimensi: komponen, panjang vektor, vektor satuan, dot product atau perkalian titik, sudut antara dua vektor, proyeksi, dan contoh soal.",null,"PUBLISHED","2026-10-08T23:58:23.070Z","cmuzbyhdg000471r9l54yc9ia",[22,27,34,38,42,46,50,54,58,63,70,74,78,84,88,92,96,100],{"id":23,"data":24,"type":26},"s1-1",{"markdown":25},"**Vektor 3 dimensi** adalah vektor dengan tiga komponen, $\\mathbf u=\\langle u_x,u_y,u_z\\rangle$, yang panjangnya $|\\mathbf u|=\\sqrt{u_x^2+u_y^2+u_z^2}$. **Dot product** atau **perkalian titik** dua vektor adalah $\\mathbf u\\cdot\\mathbf v=u_xv_x+u_yv_y+u_zv_z=|\\mathbf u||\\mathbf v|\\cos\\theta$, hasilnya berupa skalar. Dot product nol menandakan dua vektor bukan nol saling tegak lurus.","text",{"id":28,"data":29,"type":33},"s1-2",{"title":30,"variant":31,"markdown":32},"Rumus vektor 3 dimensi","info","Panjang vektor: $|\\mathbf u|=\\sqrt{u_x^2+u_y^2+u_z^2}$\n\nDot product: $\\mathbf u\\cdot\\mathbf v=u_xv_x+u_yv_y+u_zv_z$\n\nSudut: $\\cos\\theta=\\dfrac{\\mathbf u\\cdot\\mathbf v}{|\\mathbf u||\\mathbf v|}$","callout",{"id":35,"data":36,"type":26},"s2-1",{"markdown":37},"## Apa Itu Vektor 3 Dimensi?\n\nVektor adalah besaran yang mempunyai **besar** dan **arah**. Di ruang tiga dimensi, vektor ditulis dengan tiga komponen terhadap sumbu $x$, $y$, dan $z$ yang saling tegak lurus. Ada beberapa notasi yang setara:\n\n$$\\mathbf u=\\langle1,2,2\\rangle=\\begin{pmatrix}1\\\\2\\\\2\\end{pmatrix}=\\mathbf i+2\\mathbf j+2\\mathbf k$$\n\nVektor $\\langle1,2,2\\rangle$ berarti bergerak 1 satuan ke arah $x$ positif, 2 ke arah $y$ positif, dan 2 ke arah $z$ positif. Vektor dari titik $A$ ke $B$ dihitung dengan $\\overrightarrow{AB}=B-A$.\n\n**Penjumlahan** vektor dilakukan per komponen, dan **perkalian skalar** mengalikan setiap komponen: $2\\langle1,2,2\\rangle=\\langle2,4,4\\rangle$.",{"id":39,"data":40,"type":26},"s3-1",{"markdown":41},"## Panjang Vektor 3 Dimensi dan Vektor Satuan\n\nPanjang (norma) vektor diperoleh dengan teorema Pythagoras dua kali: pertama pada bidang $xy$, lalu dengan komponen $z$:\n\n$$|\\mathbf u|=\\sqrt{u_x^2+u_y^2+u_z^2}$$\n\n**Vektor satuan** searah $\\mathbf u$ adalah $\\hat{\\mathbf u}=\\frac{\\mathbf u}{|\\mathbf u|}$, panjangnya 1. Vektor nol tidak punya arah, jadi tidak dapat dinormalkan.\n\nSudut vektor terhadap sumbu-sumbu (**kosinus arah**) adalah $\\cos\\alpha=\\frac{u_x}{|\\mathbf u|}$, $\\cos\\beta=\\frac{u_y}{|\\mathbf u|}$, $\\cos\\gamma=\\frac{u_z}{|\\mathbf u|}$, dan selalu berlaku $\\cos^2\\alpha+\\cos^2\\beta+\\cos^2\\gamma=1$.",{"id":43,"data":44,"type":26},"s4-1",{"markdown":45},"## Dot Product (Perkalian Titik): Rumus dan Makna\n\nDot product didefinisikan secara aljabar sebagai jumlah hasil kali komponen yang bersesuaian:\n\n$$\\mathbf u\\cdot\\mathbf v=u_xv_x+u_yv_y+u_zv_z$$\n\nSecara geometri, $\\mathbf u\\cdot\\mathbf v=|\\mathbf u||\\mathbf v|\\cos\\theta$, dengan $\\theta$ sudut antara kedua vektor ($0\\le\\theta\\le180^\\circ$). Tanda dot product memberi informasi arah:\n\n| Dot product | Sudut $\\theta$ | Arti |\n|---|---|---|\n| positif | lancip ($\u003C90^\\circ$) | searah sebagian |\n| nol | $90^\\circ$ | tegak lurus (untuk vektor bukan nol) |\n| negatif | tumpul ($>90^\\circ$) | berlawanan sebagian |\n\nSifat penting: $\\mathbf u\\cdot\\mathbf u=|\\mathbf u|^2$, komutatif, dan distributif terhadap penjumlahan. Dalam fisika, dot product gaya dan perpindahan memberi **usaha**; materi itu dibahas di [mapel Fisika](\u002Fmapel\u002Ffisika).",{"id":47,"data":48,"type":26},"s5-1",{"markdown":49},"## Sudut antara Dua Vektor dan Syarat Tegak Lurus\n\nDari rumus geometri, sudut antara dua vektor bukan nol adalah\n\n$$\\theta=\\arccos\\frac{\\mathbf u\\cdot\\mathbf v}{|\\mathbf u||\\mathbf v|}.$$\n\nDua vektor bukan nol **tegak lurus** jika dan hanya jika $\\mathbf u\\cdot\\mathbf v=0$. Syarat ini sering dipakai untuk mencari komponen yang belum diketahui.",{"id":51,"data":52,"type":26},"s6-1",{"markdown":53},"## Proyeksi Vektor\n\nProyeksi $\\mathbf u$ pada $\\mathbf v$ menjawab: \"seberapa besar bagian $\\mathbf u$ yang searah $\\mathbf v$?\"\n\n- **Proyeksi skalar** (panjang proyeksi): $\\dfrac{\\mathbf u\\cdot\\mathbf v}{|\\mathbf v|}$\n- **Proyeksi vektor**: $\\dfrac{\\mathbf u\\cdot\\mathbf v}{|\\mathbf v|^2}\\,\\mathbf v$\n\nTombol **Proyeksi** pada visual memperlihatkan proyeksi vektor ke bidang dan sumbu koordinat, yaitu kasus khusus ketika $\\mathbf v$ adalah vektor satuan sumbu.",{"id":55,"data":56,"type":26},"s7-1",{"markdown":57},"## Perkalian Titik dan Perkalian Silang: Apa Bedanya?\n\nSelain dot product, vektor 3 dimensi juga mempunyai **perkalian silang** (cross product) $\\mathbf u\\times\\mathbf v$. Perbedaannya:\n\n| | Perkalian titik $\\mathbf u\\cdot\\mathbf v$ | Perkalian silang $\\mathbf u\\times\\mathbf v$ |\n|---|---|---|\n| Hasil | skalar (bilangan) | vektor |\n| Besar | $\\vert\\mathbf u\\vert\\,\\vert\\mathbf v\\vert\\cos\\theta$ | $\\vert\\mathbf u\\vert\\,\\vert\\mathbf v\\vert\\sin\\theta$ |\n| Bernilai nol jika | tegak lurus | sejajar |\n| Kegunaan | sudut, proyeksi, usaha | vektor normal, luas jajargenjang |\n\nPerkalian silang hanya didefinisikan di ruang tiga dimensi dan hasilnya tegak lurus terhadap kedua vektor.",{"id":59,"data":60,"type":33},"visual-guide",{"title":61,"variant":31,"markdown":62},"Eksplorasi visual","- Vektor awal adalah $\\langle1,2,2\\rangle$. Baca panjangnya di panel info dan cocokkan dengan $\\sqrt{1+4+4}=3$.\n- Tekan **Proyeksi** untuk menampilkan garis bantu ke bidang dan sumbu. Garis-garis itu memperlihatkan komponen $x$, $y$, dan $z$.\n- Lihat sudut terhadap sumbu $z$ di panel info, lalu cek dengan $\\cos\\gamma=\\frac23$, sehingga $\\gamma\\approx48{,}2^\\circ$.\n- Ubah komponen menjadi $\\langle0,3,4\\rangle$ dengan penggeser $x$, $y$, $z$. Panjangnya harus 5.\n- Putar pandangan dengan menyeret. Apakah panjang atau sudut di panel ikut berubah? Mengapa tidak?",{"id":64,"data":65,"type":69},"visual",{"height":66,"assetId":67,"caption":68},520,"demo-v2-asset-vector","Vektor 3D interaktif: komponen dan panjang","interactive",{"id":71,"data":72,"type":26},"s8-1",{"markdown":73},"## Contoh Soal Vektor 3 Dimensi dan Dot Product\n\n### Contoh 1: Panjang, vektor satuan, dan sudut terhadap sumbu\n\nUntuk $\\mathbf u=\\langle1,2,2\\rangle$: $|\\mathbf u|=\\sqrt{1+4+4}=3$, vektor satuannya $\\langle\\frac13,\\frac23,\\frac23\\rangle$. Sudut terhadap sumbu $z$: $\\cos\\gamma=\\frac23$, jadi $\\gamma\\approx48{,}2^\\circ$.\n\n### Contoh 2: Menguji tegak lurus\n\n$\\mathbf v=\\langle2,1,-2\\rangle$. $\\mathbf u\\cdot\\mathbf v=2+2-4=0$, dan keduanya bukan vektor nol. Jadi $\\mathbf u\\perp\\mathbf v$. Cek: $|\\mathbf u+\\mathbf v|^2=|\\langle3,3,0\\rangle|^2=18=3^2+3^2$, sesuai Pythagoras.\n\n### Contoh 3: Sudut antara dua vektor\n\n$\\mathbf a=\\langle1,1,0\\rangle$ dan $\\mathbf b=\\langle1,0,1\\rangle$. $\\mathbf a\\cdot\\mathbf b=1$, $|\\mathbf a|=|\\mathbf b|=\\sqrt2$. Maka $\\cos\\theta=\\frac1{2}$ dan $\\theta=60^\\circ$.\n\n### Contoh 4: Proyeksi\n\nProyeksi $\\mathbf u=\\langle2,3,1\\rangle$ pada $\\mathbf v=\\langle1,1,1\\rangle$: $\\mathbf u\\cdot\\mathbf v=6$ dan $|\\mathbf v|^2=3$. Proyeksi vektornya $2\\langle1,1,1\\rangle=\\langle2,2,2\\rangle$, dan proyeksi skalarnya $\\frac6{\\sqrt3}=2\\sqrt3$.\n\n### Contoh 5: Mencari komponen\n\nTentukan $p$ agar $\\langle p,2,-1\\rangle$ tegak lurus $\\langle3,1,4\\rangle$. Syarat: $3p+2-4=0$, jadi $p=\\frac23$.",{"id":75,"data":76,"type":26},"s9-1",{"markdown":77},"## Kesalahan yang Sering Terjadi",{"id":79,"data":80,"type":33},"s9-2",{"title":81,"variant":82,"markdown":83},"Dot product adalah skalar","warning","Setelah mengalikan komponen, **jumlahkan** ketiganya. Menulis $\\langle2,2,-4\\rangle$ sebagai hasil dot product adalah keliru.",{"id":85,"data":86,"type":26},"s9-3",{"markdown":87},"- Menjumlahkan komponen untuk mencari panjang: $|\\langle2,-3,6\\rangle|=7$, bukan $5$ atau $11$.\n- Menaksir sudut dari gambar perspektif. Hitung dengan rumus kosinus.\n- Menyimpulkan tegak lurus dari dot product nol padahal salah satu vektor adalah vektor nol.",{"id":89,"data":90,"type":26},"s10-1",{"markdown":91},"## Ringkasan\n\n- $|\\mathbf u|=\\sqrt{u_x^2+u_y^2+u_z^2}$; vektor satuan $\\frac{\\mathbf u}{|\\mathbf u|}$.\n- $\\mathbf u\\cdot\\mathbf v=u_xv_x+u_yv_y+u_zv_z=|\\mathbf u||\\mathbf v|\\cos\\theta$.\n- Dot product nol ⇔ tegak lurus (vektor bukan nol).\n- Proyeksi vektor $\\frac{\\mathbf u\\cdot\\mathbf v}{|\\mathbf v|^2}\\mathbf v$.",{"id":93,"data":94,"type":26},"s11-1",{"markdown":95},"## Materi Terkait\n\nPanjang vektor adalah [teorema Pythagoras](\u002Fmateri\u002Fteorema-pythagoras) dalam tiga dimensi, sama seperti diagonal ruang [balok](\u002Fmateri\u002Frumus-volume-balok). Konsep gradien dan perubahan dari [definisi turunan](\u002Fmateri\u002Fdefinisi-turunan-limit) diperluas ke banyak variabel memakai vektor.",{"id":97,"data":98,"type":26},"sources",{"markdown":99},"## Sumber belajar\n\n- [OpenStax Calculus Volume 3 — Vectors in Three Dimensions](https:\u002F\u002Fopenstax.org\u002Fbooks\u002Fcalculus-volume-3\u002Fpages\u002F2-2-vectors-in-three-dimensions)\n- [OpenStax Calculus Volume 3 — The Dot Product](https:\u002F\u002Fopenstax.org\u002Fbooks\u002Fcalculus-volume-3\u002Fpages\u002F2-3-the-dot-product)\n\nPenjelasan dan contoh disusun ulang untuk materi ini; sumber dapat dibuka untuk belajar lebih lanjut.",{"id":101,"data":102,"type":105},"practice",{"title":103,"questionSetId":104},"Latihan soal Vektor 3D dan Dot 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11.","2026-10-08T09:24:41.119Z",{"id":395,"subjectId":223,"levelId":166,"parentId":17,"slug":1665,"name":1666,"description":1667,"order":1668,"createdAt":1669,"updatedAt":1669},"gelombang-mekanik","Gelombang Mekanik","Gelombang Mekanik — Kelas 11.",16,"2026-10-08T09:24:41.132Z",{"id":461,"subjectId":20,"levelId":166,"parentId":17,"slug":1426,"name":1427,"description":1671,"order":1668,"createdAt":1672,"updatedAt":1672},"Vektor — Kelas 11 Lanjut.","2026-10-08T09:24:41.485Z",{"id":317,"subjectId":223,"levelId":166,"parentId":17,"slug":1674,"name":1675,"description":1676,"order":1677,"createdAt":1678,"updatedAt":1678},"gelombang-cahaya","Gelombang Cahaya","Gelombang Cahaya — Kelas 11.",17,"2026-10-08T09:24:41.150Z",{"id":401,"subjectId":20,"levelId":166,"parentId":17,"slug":1680,"name":1681,"description":1682,"order":1677,"createdAt":1683,"updatedAt":1683},"transformasi-geometri","Transformasi Geometri","Transformasi Geometri — Kelas 11 Lanjut.","2026-10-08T09:24:41.498Z",{"id":347,"subjectId":20,"levelId":166,"parentId":17,"slug":1685,"name":1686,"description":1687,"order":1688,"createdAt":1689,"updatedAt":1689},"fungsi-dan-pemodelannya","Fungsi dan Pemodelannya","Fungsi dan Pemodelannya — Kelas 11 Lanjut.",18,"2026-10-08T09:24:41.504Z",{"id":383,"subjectId":223,"levelId":166,"parentId":17,"slug":1691,"name":1692,"description":1693,"order":1688,"createdAt":1694,"updatedAt":1694},"optik","Optik","Optik — Kelas 11.","2026-10-08T09:24:41.161Z",{"id":259,"subjectId":20,"levelId":166,"parentId":17,"slug":1696,"name":267,"description":1697,"order":1698,"createdAt":1699,"updatedAt":1699},"barisan-dan-deret-kelas-12-wajib","Barisan dan Deret — Kelas 12 Wajib.",19,"2026-10-08T09:24:41.512Z",{"id":556,"subjectId":223,"levelId":166,"parentId":17,"slug":1701,"name":1702,"description":1703,"order":1698,"createdAt":1704,"updatedAt":1704},"suhu-dan-kalor","Suhu dan Kalor","Suhu dan Kalor — Kelas 11.","2026-10-08T09:24:41.169Z",{"id":288,"subjectId":20,"levelId":166,"parentId":17,"slug":1706,"name":1707,"description":1708,"order":1709,"createdAt":1710,"updatedAt":1710},"bunga","Bunga","Bunga — Kelas 12 Wajib.",20,"2026-10-08T09:24:41.518Z",{"id":353,"subjectId":223,"levelId":166,"parentId":17,"slug":1712,"name":1713,"description":1714,"order":1709,"createdAt":1715,"updatedAt":1715},"teori-kinetik-gas","Teori Kinetik Gas","Teori Kinetik Gas — Kelas 11.","2026-10-08T09:24:41.188Z",{"id":562,"subjectId":20,"levelId":166,"parentId":17,"slug":1717,"name":1718,"description":1719,"order":1720,"createdAt":1721,"updatedAt":1721},"investasi-dan-pinjaman","Investasi dan Pinjaman","Investasi dan Pinjaman — Kelas 12 Wajib.",21,"2026-10-08T09:24:41.525Z",{"id":580,"subjectId":223,"levelId":166,"parentId":17,"slug":1723,"name":1724,"description":1725,"order":1720,"createdAt":1726,"updatedAt":1726},"termodinamika","Termodinamika","Termodinamika — Kelas 11.","2026-10-08T09:24:41.202Z",{"id":508,"subjectId":20,"levelId":166,"parentId":17,"slug":1728,"name":1729,"description":1730,"order":1731,"createdAt":1732,"updatedAt":1732},"kaidah-pencacahan","Kaidah Pencacahan","Kaidah Pencacahan — Kelas 12 Wajib.",22,"2026-10-08T09:24:41.540Z",{"id":449,"subjectId":223,"levelId":166,"parentId":17,"slug":1734,"name":1735,"description":1736,"order":1731,"createdAt":1737,"updatedAt":1737},"listrik-statis","Listrik Statis","Listrik Statis — Kelas 12.","2026-10-08T09:24:41.215Z",{"id":419,"subjectId":223,"levelId":166,"parentId":17,"slug":1739,"name":1740,"description":1741,"order":1742,"createdAt":1743,"updatedAt":1743},"listrik-arus-searah","Listrik Arus Searah","Listrik Arus Searah — Kelas 12.",23,"2026-10-08T09:24:41.227Z",{"id":467,"subjectId":20,"levelId":166,"parentId":17,"slug":1745,"name":1594,"description":1746,"order":1742,"createdAt":1747,"updatedAt":1747},"peluang-kelas-12-wajib","Peluang — Kelas 12 Wajib.","2026-10-08T09:24:41.547Z",{"id":431,"subjectId":20,"levelId":166,"parentId":17,"slug":1749,"name":1750,"description":1751,"order":1752,"createdAt":1753,"updatedAt":1753},"geometri-analitik","Geometri Analitik","Geometri Analitik — Kelas 12 Lanjut.",24,"2026-10-08T09:24:41.553Z",{"id":425,"subjectId":223,"levelId":166,"parentId":17,"slug":1755,"name":1756,"description":1757,"order":1752,"createdAt":1758,"updatedAt":1758},"kemagnetan","Kemagnetan","Kemagnetan — Kelas 12.","2026-10-08T09:24:41.245Z",{"id":300,"subjectId":20,"levelId":166,"parentId":17,"slug":1760,"name":1761,"description":1762,"order":1763,"createdAt":1764,"updatedAt":1764},"limit","Limit","Limit — Kelas 12 Lanjut.",25,"2026-10-08T09:24:41.562Z",{"id":252,"subjectId":223,"levelId":166,"parentId":17,"slug":1766,"name":1767,"description":1768,"order":1763,"createdAt":1769,"updatedAt":1769},"listrik-ac-arus-bolak-balik","Listrik AC \u002F Arus Bolak Balik","Listrik AC \u002F Arus Bolak Balik — Kelas 12.","2026-10-08T09:24:41.263Z",{"id":490,"subjectId":223,"levelId":166,"parentId":17,"slug":1771,"name":1772,"description":1773,"order":1774,"createdAt":1775,"updatedAt":1775},"gelombang-elektromagnetik","Gelombang Elektromagnetik","Gelombang Elektromagnetik — Kelas 12.",26,"2026-10-08T09:24:41.273Z",{"id":306,"subjectId":20,"levelId":166,"parentId":17,"slug":1777,"name":1778,"description":1779,"order":1774,"createdAt":1780,"updatedAt":1780},"turunan-fungsi","Turunan Fungsi","Turunan Fungsi — Kelas 12 Lanjut.","2026-10-08T09:24:41.575Z",{"id":514,"subjectId":20,"levelId":166,"parentId":17,"slug":1782,"name":1783,"description":1784,"order":1785,"createdAt":1786,"updatedAt":1786},"aplikasi-turunan","Aplikasi Turunan","Aplikasi Turunan — Kelas 12 Lanjut.",27,"2026-10-08T09:24:41.587Z",{"id":544,"subjectId":223,"levelId":166,"parentId":17,"slug":1788,"name":1789,"description":1790,"order":1785,"createdAt":1791,"updatedAt":1791},"pengantar-instrumental-digital","Pengantar Instrumental Digital","Pengantar Instrumental Digital — Kelas 12.","2026-10-08T09:24:41.283Z",{"id":389,"subjectId":20,"levelId":166,"parentId":17,"slug":1793,"name":1794,"description":1795,"order":1796,"createdAt":1797,"updatedAt":1797},"integral","Integral","Integral — Kelas 12 Lanjut.",28,"2026-10-08T09:24:41.599Z",{"id":532,"subjectId":223,"levelId":166,"parentId":17,"slug":1799,"name":1800,"description":1801,"order":1796,"createdAt":1802,"updatedAt":1802},"relativitas-khusus","Relativitas Khusus","Relativitas Khusus — Kelas 12.","2026-10-08T09:24:41.291Z",{"id":538,"subjectId":223,"levelId":166,"parentId":17,"slug":1804,"name":1805,"description":1806,"order":1807,"createdAt":1808,"updatedAt":1808},"gejala-kuantum","Gejala Kuantum","Gejala Kuantum — Kelas 12.",29,"2026-10-08T09:24:41.303Z",{"id":437,"subjectId":20,"levelId":166,"parentId":17,"slug":1810,"name":1811,"description":1812,"order":1807,"createdAt":1813,"updatedAt":1813},"penerapan-integral","Penerapan Integral","Penerapan Integral — Kelas 12 Lanjut.","2026-10-08T09:24:41.607Z",{"id":329,"subjectId":20,"levelId":166,"parentId":17,"slug":1815,"name":1816,"description":1817,"order":1818,"createdAt":1819,"updatedAt":1819},"analisis-data-peluang","Analisis Data & Peluang","Analisis Data & Peluang — Kelas 12 Lanjut.",30,"2026-10-08T09:24:41.611Z",{"id":473,"subjectId":223,"levelId":166,"parentId":17,"slug":1821,"name":1822,"description":1823,"order":1818,"createdAt":1824,"updatedAt":1824},"fisika-inti-dan-radioaktivitas","Fisika Inti dan Radioaktivitas","Fisika Inti dan Radioaktivitas — Kelas 12.","2026-10-08T09:24:41.318Z",{"id":1826,"subjectId":20,"levelId":166,"parentId":17,"slug":1827,"name":1828,"description":1829,"order":1830,"createdAt":1831,"updatedAt":1831},"cmuzbyhw700gw71r9o87lq7vd","logaritma","Logaritma","Logaritma — Kelas 12 Lanjut.",31,"2026-10-08T09:24:41.623Z",{"id":1833,"subjectId":20,"levelId":166,"parentId":17,"slug":1834,"name":1835,"description":1836,"order":1837,"createdAt":1838,"updatedAt":1838},"cmuzbyhw900gy71r9d9sjh9ie","program-linier","Program Linier","Program Linier — Kelas 12 Lanjut.",32,"2026-10-08T09:24:41.625Z",{"id":1840,"subjectId":20,"levelId":166,"parentId":17,"slug":1841,"name":1842,"description":1843,"order":1844,"createdAt":1845,"updatedAt":1845},"cmuzbyhwb00h071r97764r7hr","bangun-ruang","Bangun Ruang","Bangun Ruang — Kelas 12 Lanjut.",33,"2026-10-08T09:24:41.627Z",{"id":199,"subjectId":20,"levelId":116,"parentId":17,"slug":189,"name":200,"description":191,"order":1847,"createdAt":1848,"updatedAt":1848},100,"2026-10-08T23:58:22.977Z",{"id":1850,"subjectId":20,"levelId":166,"parentId":17,"slug":1851,"name":1852,"description":1853,"order":1847,"createdAt":1854,"updatedAt":1854},"demo-v2-topic-matematika-sma-1","fungsi-kuadrat","Fungsi Kuadrat","Fungsi kuadrat f(x) = ax² + bx + c: pengaruh a, b, c pada grafik, rumus titik puncak, sumbu simetri, diskriminan, dan contoh soal nilai maksimum.","2026-10-08T23:58:22.868Z",{"id":1856,"subjectId":231,"levelId":166,"parentId":17,"slug":1857,"name":1858,"description":1859,"order":1847,"createdAt":1860,"updatedAt":1860},"demo-v2-topic-pemrograman-sma-1","fungsi-python-def-return","Fungsi Python","Pelajari cara membuat fungsi Python dengan def, parameter, nilai default, dan return, plus perbedaan return dan print lewat contoh program yang jelas.","2026-10-08T23:58:23.661Z",{"id":1862,"subjectId":223,"levelId":162,"parentId":17,"slug":1863,"name":1864,"description":1865,"order":1847,"createdAt":1866,"updatedAt":1866},"demo-v2-topic-fisika-smp-1","glb-dan-glbb","GLB dan GLBB","Rumus GLB dan GLBB lengkap: jarak, kecepatan, percepatan, grafik posisi–waktu, perbedaan GLB dan GLBB, plus contoh soal gerak lurus dan pembahasannya.","2026-10-08T23:58:23.189Z",{"id":1868,"subjectId":223,"levelId":116,"parentId":17,"slug":1869,"name":1870,"description":1871,"order":1847,"createdAt":1872,"updatedAt":1872},"demo-v2-topic-fisika-kuliah-1","gerak-harmonik-sederhana","Gerak Harmonik Sederhana","Gerak harmonik sederhana untuk fisika dasar: persamaan x = A cos(ωt + φ), kecepatan, percepatan, periode pegas dan bandul, energi, serta contoh soal.","2026-10-08T23:58:23.364Z",{"id":1874,"subjectId":223,"levelId":166,"parentId":17,"slug":1875,"name":1876,"description":1877,"order":1847,"createdAt":1878,"updatedAt":1878},"demo-v2-topic-fisika-sma-1","hukum-newton-2","Hukum Newton 2","Hukum Newton 2: bunyi, rumus ΣF = m·a, diagram gaya, gaya gesek statis dan kinetis, serta contoh soal mencari percepatan dan massa beserta pembahasan.","2026-10-08T23:58:23.280Z",{"id":1880,"subjectId":231,"levelId":116,"parentId":17,"slug":1881,"name":1882,"description":1883,"order":1847,"createdAt":1884,"updatedAt":1884},"demo-v2-topic-pemrograman-kuliah-1","notasi-big-o-kompleksitas-waktu","Notasi Big O","Pahami notasi Big O dan kompleksitas waktu algoritma: O(1), O(log n), O(n), O(n²), cara menghitungnya dari kode Python, dan contoh soalnya.","2026-10-08T23:58:23.797Z",{"id":1886,"subjectId":231,"levelId":158,"parentId":17,"slug":1887,"name":1888,"description":1889,"order":1847,"createdAt":1890,"updatedAt":1890},"demo-v2-topic-pemrograman-sd-1","pengertian-algoritma","Pengertian Algoritma","Algoritma adalah langkah-langkah urut untuk menyelesaikan masalah. Pelajari pengertian, ciri, dan contoh algoritma sehari-hari untuk siswa SD.","2026-10-08T23:58:23.477Z",{"id":1892,"subjectId":223,"levelId":158,"parentId":17,"slug":1893,"name":1894,"description":1895,"order":1847,"createdAt":1896,"updatedAt":1896},"demo-v2-topic-fisika-sd-1","pengukuran-panjang","Pengukuran Panjang","Belajar pengukuran panjang untuk SD: satuan baku, alat ukur panjang, cara memakai penggaris, dan tangga satuan panjang km sampai mm + contoh soal.","2026-10-08T23:58:23.095Z",{"id":1898,"subjectId":20,"levelId":158,"parentId":17,"slug":1899,"name":1900,"description":1901,"order":1847,"createdAt":1902,"updatedAt":1902},"demo-v2-topic-matematika-sd-1","penjumlahan-pecahan","Penjumlahan Pecahan","Cara penjumlahan pecahan berpenyebut sama dan beda penyebut memakai KPK, pecahan senilai, dan pecahan campuran, lengkap dengan contoh soal SD.","2026-10-08T23:58:22.508Z",{"id":1904,"subjectId":20,"levelId":162,"parentId":17,"slug":1905,"name":1906,"description":1907,"order":1847,"createdAt":1908,"updatedAt":1908},"demo-v2-topic-matematika-smp-1","teorema-pythagoras","Teorema Pythagoras","Teorema Pythagoras a² + b² = c² untuk mencari sisi miring dan sisi siku-siku, tripel Pythagoras, diagonal ruang balok, dan contoh soal kelas 8.","2026-10-08T23:58:22.690Z",{"id":1910,"subjectId":231,"levelId":162,"parentId":17,"slug":1911,"name":1912,"description":1913,"order":1847,"createdAt":1914,"updatedAt":1914},"demo-v2-topic-pemrograman-smp-1","variabel-dan-tipe-data-python","Variabel dan Tipe Data Python","Belajar variabel dan tipe data Python dari nol: int, float, string, boolean, operator, print, dan input, lengkap dengan contoh program dan latihan.","2026-10-08T23:58:23.562Z",{"id":1916,"subjectId":231,"levelId":116,"parentId":17,"slug":1917,"name":1918,"description":1919,"order":1920,"createdAt":1921,"updatedAt":1921},"demo-v2-topic-pemrograman-kuliah-2","class-dan-object-python","Class dan Object Python","Pelajari class dan object Python untuk OOP: __init__, self, atribut, method, inheritance, dan invarian objek lewat contoh program dompet digital.",101,"2026-10-08T23:58:23.830Z",{"id":1923,"subjectId":20,"levelId":166,"parentId":17,"slug":1924,"name":1925,"description":1926,"order":1920,"createdAt":1927,"updatedAt":1927},"demo-v2-topic-matematika-sma-2","eksponen-dan-logaritma","Eksponen dan Logaritma","Materi eksponen dan logaritma kelas 10: sifat-sifat eksponen, definisi dan sifat logaritma, persamaan eksponen dan logaritma, serta contoh soal.","2026-10-08T23:58:22.902Z",{"id":1929,"subjectId":223,"levelId":158,"parentId":17,"slug":1930,"name":1931,"description":1932,"order":1920,"createdAt":1933,"updatedAt":1933},"demo-v2-topic-fisika-sd-2","macam-macam-gaya","Gaya dan Macam-Macamnya","Gaya adalah tarikan atau dorongan. Pelajari macam-macam gaya (otot, gravitasi, magnet, gesek, pegas), pengaruhnya pada benda, dan contoh soalnya.","2026-10-08T23:58:23.126Z",{"id":1935,"subjectId":20,"levelId":158,"parentId":17,"slug":1936,"name":1937,"description":1938,"order":1920,"createdAt":1939,"updatedAt":1939},"demo-v2-topic-matematika-sd-2","rumus-keliling-dan-luas-bangun-datar","Keliling dan Luas Bangun Datar","Rumus keliling dan luas bangun datar: persegi, persegi panjang, segitiga, jajargenjang, trapesium, dan lingkaran, dengan cara mudah dan contoh soal.","2026-10-08T23:58:22.588Z",{"id":1941,"subjectId":231,"levelId":166,"parentId":17,"slug":1942,"name":1943,"description":1944,"order":1920,"createdAt":1945,"updatedAt":1945},"demo-v2-topic-pemrograman-sma-2","list-dan-dictionary-python","List dan Dictionary Python","Pahami list dan dictionary Python: indeks, append, key-value, get, dan perulangan, plus perbedaan keduanya lewat contoh pengolahan data nilai siswa.","2026-10-08T23:58:23.694Z",{"id":1947,"subjectId":223,"levelId":162,"parentId":17,"slug":1948,"name":1949,"description":1950,"order":1920,"createdAt":1951,"updatedAt":1951},"demo-v2-topic-fisika-smp-2","rumus-massa-jenis","Massa Jenis","Rumus massa jenis ρ = m\u002FV lengkap dengan satuan kg\u002Fm³ dan g\u002Fcm³, tabel massa jenis air, besi, dan minyak, cara menghitung, serta contoh soal pembahasan.","2026-10-08T23:58:23.217Z",{"id":1953,"subjectId":231,"levelId":162,"parentId":17,"slug":1954,"name":1955,"description":1956,"order":1920,"createdAt":1957,"updatedAt":1957},"demo-v2-topic-pemrograman-smp-2","percabangan-python-if-else","Percabangan Python","Pelajari percabangan Python dengan if, elif, dan else: operator perbandingan, operator logika, dan contoh program nilai rapor lengkap dengan outputnya.","2026-10-08T23:58:23.592Z",{"id":1959,"subjectId":231,"levelId":158,"parentId":17,"slug":1960,"name":1961,"description":1962,"order":1920,"createdAt":1963,"updatedAt":1963},"demo-v2-topic-pemrograman-sd-2","percabangan-dalam-algoritma","Percabangan dalam Algoritma","Percabangan dalam algoritma memilih tindakan berdasarkan syarat jika-maka. Pelajari pengertian, contoh sehari-hari, dan simbol flowchart-nya.","2026-10-08T23:58:23.503Z",{"id":1965,"subjectId":20,"levelId":162,"parentId":17,"slug":1966,"name":1967,"description":1968,"order":1920,"createdAt":1969,"updatedAt":1969},"demo-v2-topic-matematika-smp-2","persamaan-garis-lurus","Persamaan Garis Lurus","Persamaan garis lurus y = mx + c, rumus gradien dari dua titik, garis sejajar dan tegak lurus, cara menggambar grafik, serta contoh soal kelas 8.","2026-10-08T23:58:22.730Z",{"id":1971,"subjectId":223,"levelId":116,"parentId":17,"slug":1972,"name":1973,"description":1974,"order":1920,"createdAt":1975,"updatedAt":1975},"demo-v2-topic-fisika-kuliah-2","usaha-oleh-gaya-berubah","Usaha Gaya Berubah","Usaha oleh gaya berubah dengan integral W = ∫F dx, rumus usaha W = F·s sebagai dot product, usaha pegas, dan teorema usaha–energi, plus contoh soal.","2026-10-08T23:58:23.394Z",{"id":214,"subjectId":20,"levelId":166,"parentId":17,"slug":204,"name":215,"description":206,"order":1977,"createdAt":1978,"updatedAt":1978},102,"2026-10-08T23:58:22.934Z",{"id":1980,"subjectId":231,"levelId":166,"parentId":17,"slug":1981,"name":1982,"description":1983,"order":1977,"createdAt":1984,"updatedAt":1984},"demo-v2-topic-pemrograman-sma-3","pencarian-linear-dan-biner","Pencarian Linear dan Biner","Pelajari pencarian linear dan pencarian biner (binary search): cara kerja, syarat data terurut, kode Python, dan perbandingan jumlah langkahnya.","2026-10-08T23:58:23.724Z",{"id":1986,"subjectId":231,"levelId":162,"parentId":17,"slug":1987,"name":1988,"description":1989,"order":1977,"createdAt":1990,"updatedAt":1990},"demo-v2-topic-pemrograman-smp-3","perulangan-python-for-while","Perulangan Python","Pelajari perulangan Python dengan for, range(), dan while, lengkap dengan contoh program menghitung total, break, continue, dan latihan soal.","2026-10-08T23:58:23.624Z",{"id":1992,"subjectId":231,"levelId":158,"parentId":17,"slug":1993,"name":1994,"description":1995,"order":1977,"createdAt":1996,"updatedAt":1996},"demo-v2-topic-pemrograman-sd-3","perulangan-dalam-algoritma","Perulangan dalam Algoritma","Perulangan dalam algoritma mengulang langkah yang sama beberapa kali. Pelajari pengertian, contoh perulangan sehari-hari, dan cara menghitung hasilnya.","2026-10-08T23:58:23.529Z",{"id":1998,"subjectId":20,"levelId":162,"parentId":17,"slug":1999,"name":2000,"description":2001,"order":1977,"createdAt":2002,"updatedAt":2002},"demo-v2-topic-matematika-smp-3","rumus-prisma-dan-limas","Prisma dan Limas","Rumus volume prisma dan limas, luas permukaan, sifat sisi-rusuk-titik sudut, jaring-jaring, limas 3D interaktif, dan contoh soal bangun ruang sisi datar.","2026-10-08T23:58:22.780Z",{"id":2004,"subjectId":231,"levelId":116,"parentId":17,"slug":2005,"name":2006,"description":2007,"order":1977,"createdAt":2008,"updatedAt":2008},"demo-v2-topic-pemrograman-kuliah-3","unit-test-python","Unit Test Python","Pelajari unit test Python dengan modul unittest: assertEqual, assertRaises, nilai batas, dan cara menjalankan tes, lengkap dengan contoh dan versi pytest.","2026-10-08T23:58:23.864Z",{"id":121,"subjectId":20,"levelId":116,"parentId":17,"slug":14,"name":122,"description":16,"order":1977,"createdAt":2010,"updatedAt":2010},"2026-10-08T23:58:23.045Z",{"id":2012,"subjectId":20,"levelId":158,"parentId":17,"slug":2013,"name":2014,"description":2015,"order":1977,"createdAt":2016,"updatedAt":2016},"demo-v2-topic-matematika-sd-3","rumus-volume-kubus","Volume dan Luas Permukaan Kubus","Rumus volume kubus V = s³ dan luas permukaan kubus L = 6s², sifat kubus, jaring-jaring, kubus 3D interaktif, dan contoh soal bertingkat.","2026-10-08T23:58:22.619Z",{"id":2018,"subjectId":20,"levelId":162,"parentId":17,"slug":2019,"name":2020,"description":2021,"order":2022,"createdAt":2023,"updatedAt":2023},"demo-v2-topic-matematika-smp-4","rumus-tabung-dan-kerucut","Tabung dan Kerucut","Rumus volume tabung dan kerucut, luas permukaan, garis pelukis, jaring-jaring berupa persegi panjang dan juring, kerucut 3D interaktif, dan contoh soal.",103,"2026-10-08T23:58:22.828Z",{"id":2025,"subjectId":231,"levelId":166,"parentId":17,"slug":2026,"name":2027,"description":2028,"order":2022,"createdAt":2029,"updatedAt":2029},"demo-v2-topic-pemrograman-sma-4","try-except-python","Try Except dan Membaca File","Pelajari try except Python untuk menangani error seperti ValueError dan FileNotFoundError, serta cara membaca dan menulis file txt dengan with open.","2026-10-08T23:58:23.755Z",{"id":2031,"subjectId":20,"levelId":158,"parentId":17,"slug":2032,"name":2033,"description":2034,"order":2022,"createdAt":2035,"updatedAt":2035},"demo-v2-topic-matematika-sd-4","rumus-volume-balok","Volume dan Luas Permukaan Balok","Rumus volume balok V = p × l × t dan luas permukaan balok L = 2(pl + pt + lt), sifat balok, jaring-jaring, balok 3D interaktif, dan contoh soal.","2026-10-08T23:58:22.651Z",[]]