Pada artikel Fisika kelas 10 kali ini, kamu akan mengetahui cara menjumlahkan vektor menggunakan tiga metode, yaitu metode grafis, analisis, dan uraian.
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Siapa di antara kamu yang suka lari? Eits! Bukan lari dari masalah kehidupan loh, ya hehe. Tapi, olahraga lari, jogging gitu misalnya.
Kamu tahu nggak nih, kalau jogging itu banyak manfaatnya, lho! Mulai dari meningkatkan kekebalan tubuh, fisik menjadi lebih fit dan segar, sampai menghilangkan stres. Wah, boleh juga tuh! Hitung-hitung, menghilangkan penat akibat banyaknya tugas di sekolah atau menyegarkan pikiran sebelum menghadapi ujian.
Ngomong-ngomong masalah jogging, Rogu juga rutin melakukan jogging setiap Minggu pagi, lho. Biasanya, Rogu jogging di sekitar komplek tempat ia tinggal. Nah, berikut ini merupakan gambaran rute jogging yang biasa Rogu lewati.
Kira-kira nih, kamu bisa nggak menghitung berapa jarak yang ditempuh Rogu dari titik A ke titik D?
Wah, kalau itu sih caranya mudah sekali, ya. Kita hanya tinggal menjumlahkan jarak dari titik AB ke titik BC, lalu ke titik CD. Sehingga,
AB + BC + CD = 550 m + 650 m + 700 m = 1.900 m
Simpel banget, kan? Tapi, bagaimana dengan perpindahan Rogu dari titik A ke titik D?
Nah, jika kamu ingat, perpindahan itu termasuk besaran vektor. Perpindahan ditentukan oleh kedudukan awal dan kedudukan akhir, serta dapat bertanda positif maupun negatif, bergantung pada arah perpindahannya.
Baca Juga: Apa Perbedaan antara Kelajuan dan Kecepatan?
Gambar rute jogging Rogu di atas bisa kita analogikan sebagai vektor nih, dengan memisalkan F1 merupakan vektor di titik AB, F2 merupakan vektor di titik BC, dan F3 merupakan vektor di titik CD.
Kemudian, perpindahan dari titik A ke titik D dapat ditentukan dengan mencari besar resultan vektornya saja. Apa itu resultan vektor? Resultan vektor adalah hasil dari penjumlahan dua atau lebih vektor.
Terdapat beberapa metode yang bisa kita gunakan untuk mencari resultan vektor nih, di antaranya metode grafis, metode analisis vektor, atau metode uraian.
So, kalau kamu mau tahu metode apa yang tepat untuk mencari besar perpindahan Rogu dari titik A ke titik D, yuk simak baik-baik artikel ini!
1. Metode Grafis
Metode yang pertama adalah metode grafis. Metode grafis adalah metode yang digunakan untuk menentukan besar resultan vektor dengan cara mengukurnya.
Panjang resultan vektor dapat diukur menggunakan mistar (penggaris), sedangkan besar sudut vektor (arah vektor) diukur menggunakan busur derajat. Perlu kamu ingat, pengukuran besar resultan vektor menggunakan metode grafis harus berdasarkan skala dan besar sudut yang tepat, ya.
Nah, jika kamu menyimak cerita Rogu di atas, metode grafis ini merupakan metode yang tepat untuk mencari besar perpindahan Rogu dari titik A ke titik D.
Langkah pertama yang bisa kamu lakukan adalah menetapkan skala dari masing-masing besaran vektor. Ingat! skala yang kita tentukan harus tepat dan juga sesuai ya.
Berdasarkan cerita Rogu, besar vektor F1 = 550 m, besar vektor F2 = 650 m, dan besar vektor F3 = 700 m. Misalkan, untuk ketiga vektor, kita menetapkan skala 100 m = 1 cm.
Artinya, setiap panjang 100 m kita gambar dengan 1 cm di kertas. Jadi, vektor F1 dapat digambar sepanjang 5,5 cm, vektor F2 digambar sepanjang 6,5 cm, dan vektor F3 digambar sepanjang 7 cm.
Paham sampai di sini? Kita lanjut, ya.
Kemudian, langkah kedua adalah menggambar besar dan arah masing-masing vektor seperti pada gambar di bawah ini. Panjang vektor R = F1 + F2 + F3 dapat dihitung menggunakan penggaris. Sementara itu, sudut arah vektor R dihitung menggunakan busur derajat.
Sebelumnya, kita sudah tahu ya kalau untuk mencari perpindahan dari satu titik ke titik lain kita hanya tinggal menghitung besar resultan vektornya saja, jadi sudah dapat kita ketahui nih kalau perpindahan Rogu dari titik A ke titik D adalah sebesar 1.300 m.
Jelas ya? Bagi yang belum paham, tulis saja pertanyaanmu di kolom komentar, oke?
Oh iya, penggunaan metode grafis dalam menghitung jumlah dua atau lebih vektor ternyata memiliki kelemahan lho, yaitu dapat menimbulkan kesalahan sistematis. Nah, untuk menghindari kesalahan tersebut, kita dapat menggunakan metode yang akan kita bahas selanjutnya, yaitu metode analitis.
Baca Juga: Pengertian Besaran, Satuan & Dimensi dalam Pengukuran Fisika
2. Metode Analitis
Metode analitis adalah metode yang digunakan untuk menentukan besar resultan vektor secara matematis dengan menggunakan rumus. Adapun rumus yang digunakan merupakan rumus kosinus (cos) untuk menentukan besar resultan vektor dan rumus sinus (sin) untuk menentukan arah resultan vektor.
Sekarang, supaya kamu lebih mudah untuk memahami cara mencari besar dan arah resultan vektor menggunakan metode ini, yuk, langsung saja kita simak contoh soal berikut ini.
Contoh Soal
Hitunglah besar dan arah vektor resultannya terhadap sumbu x positif!
Penyelesaian:
a. Besar resultan vektor
b. Arah resultan vektor
Jadi, besar resultan vektornya adalah 4√19 N dan arah resultan vektornya adalah 22,3o terhadap sumbu x positif. Gimana, mudah, kan? Oke, selanjutnya, kita masuk ke metode penjumlahan vektor yang terakhir, nih. Apakah itu? Yap! Metode uraian.
3. Metode Uraian
Metode penjumlahan vektor yang terakhir adalah metode uraian. Pada materi sebelumnya, kamu telah mempelajari cara mencari komponen-komponen dari suatu vektor kan.
Nah, pada metode uraian ini, sebelum kita mencari besar resultan vektor, kita uraikan terlebih dahulu vektor-vektor tersebut menjadi komponen vektor pada sumbu x dan komponen vektor pada sumbu y di koordinat kartesius.
Kamu masih ingat kan cara mencari komponen vektor pada sumbu x dan y? Hayo, bagi yang sudah lupa, dipahami lagi ya materi sebelumnya.
Setelah kita menguraikan vektor-vektor tersebut menjadi komponen vektor, barulah kita bisa mencari besar resultan vektornya, yaitu dengan menggunakan rumus dan arah resultan vektornya dengan rumus .
Nah, ∑Ax ini artinya jumlah komponen-komponen vektor pada sumbu x dan ∑Ay artinya jumlah komponen-komponen vektor pada sumbu y.
Baca Juga: Mengenal Vektor Bidang 2 Dimensi & Vektor Ruang 3 Dimensi
Perlu kamu perhatikan, besar suatu vektor akan selalu bernilai positif. Selain itu, dalam menentukan arah vektor, kita harus memperhatikan tanda Ax dan Ay yang nantinya akan menentukan kuadran dari vektor dalam sistem koordinat seperti pada tabel berikut ini:
Bingung? Tenang, nggak usah bingung-bingung, kita langsung coba kerjakan contoh soal di bawah ini saja, yuk! Let’s go!
Contoh Soal
Apabila F1 = 2 N, F2 = 10 N, dan F3 = 6 N, maka tentukan resultan dari ketiga vektor tersebut!
Pembahasan:
Hal pertama yang bisa kita lakukan untuk mengerjakan soal di atas adalah dengan menguraikan vektor F1, F2, dan F3 terhadap sumbu x dan sumbu y.
Pada sumbu x:
- F1x → -F1x = -2 N (tanda negatif menandakan arah vektor ke kiri).
- F2x = F2 sin θ → F2x = (10) sin 53°= (10)(0,8) = 8 N (tanda positif menandakan arah vektor ke kanan).
- F3x = 0 N (angka nol (0) menandakan F3 tidak memiliki proyeksi vektor/komponen vektor pada sumbu x karena F3 tegak lurus terhadap sumbu x).
Jadi, ΣFx = F1x + F2x + F3x = -2 + 8 + 0 = 6 N
Pada sumbu y:
- F1y = 0 N (angka nol (0) menandakan F1 tidak memiliki proyeksi vektor/komponen vektor pada sumbu y karena F1 tegak lurus terhadap sumbu y).
- F2y = -F2 cos θ = -(10) cos 53° = -(10)(0,6) = -6 N (tanda negatif menandakan arah vektor ke bawah).
- F3y = 6 N (tanda positif menandakan arah vektor ke atas).
Jadi, ΣFy = F1y + F2y + F3y = 0 + 6 – 6 = 0 N
Selanjutnya, setelah kita mengetahui komponen-komponen dari ketiga vektor di atas terhadap sumbu x dan y, maka kita dapat mencari resultan dari ketiga vektor tersebut.
Jadi, resultan dari vektor F1, F2, dan F3 adalah 6 N. Gimana? Ternyata nggak sesulit yang kamu kira, kan?
Baca Juga: Gerak Melingkar Beraturan (GMB): Besaran, Rumus & Contoh Soal
Oke, setelah kamu memahami ketiga metode penjumlahan vektor di atas, menurutmu, metode mana yang lebih mudah? Eits! Tapi ingat, jangan mentang-mentang kamu sreg dengan satu metode, terus metode yang lainnya tidak kamu pahami, deh. Kamu juga harus paham ketiga-tiganya. Siapa tahu keluar di ujian nanti.
Sampai sini, sudah mulai paham belum dengan materi metode penjumlahan vektor? Kalo masih ada poin-poin yang belum kamu mengerti, mending belajar sama ahlinya, deh. Belajar bareng kakak-kakak pengajar di Ruangguru Privat Fisika misalnya. Hehehe…
Belajar nggak cuma menyenangkan, tapi kamu juga bakal diajari konsepnya sampai paham! Para pengajar di Ruangguru Privat juga sudah terstandarisasi kualitasnya, loh. Kamu juga bisa pilih nih, mau diajarkan secara langsung (offline) atau daring (online). Fleksibel, kan? Untuk info lebih lanjut, cuss klik link berikut!
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<h1 class="content-title">
3 Metode Penjumlahan Vektor dan Contohnya | Fisika Kelas 10 </h1>
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<p class="content-author"><a href="https://www.ruangguru.com/blog/author/hani-ammariah" class="">Hani Ammariah</a> </p>
<p class="content-info"><span>July 24, 2025 • </span> <span>7 minutes read</span></p>
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<p style="text-align: justify;"><img decoding="async" class="aligncenter" src="https://cdn-web-2.ruangguru.com/landing-pages/assets/f502ff49-7892-41ef-8d97-c13fa903fa68.png" alt="3 metode mencari vektor" width="820" /></p>
<blockquote>
<p style="text-align: center;"><em>Pada <strong><a href="https://www.ruangguru.com/blog/c/fisika/fisika-sma-kelas-10">artikel Fisika kelas 10</a></strong> kali ini, kamu akan mengetahui cara menjumlahkan vektor menggunakan tiga metode, yaitu metode grafis, analisis, dan uraian.</em></p>
</blockquote>
<p style="text-align: center;">—</p>
<p> </p>
<p style="text-align: justify;">Siapa di antara kamu yang suka lari? Eits! Bukan lari dari masalah kehidupan loh, ya hehe. Tapi, olahraga lari, jogging gitu misalnya.</p>
<p style="text-align: justify;">Kamu tahu nggak nih, kalau jogging itu banyak manfaatnya, lho! Mulai dari meningkatkan kekebalan tubuh, fisik menjadi lebih fit dan segar, sampai menghilangkan stres. Wah, boleh juga tuh! Hitung-hitung, menghilangkan penat akibat banyaknya tugas di sekolah atau menyegarkan pikiran sebelum menghadapi ujian.</p>
<p style="text-align: justify;">Ngomong-ngomong masalah jogging, Rogu juga rutin melakukan jogging setiap Minggu pagi, lho. Biasanya, Rogu jogging di sekitar komplek tempat ia tinggal. Nah, berikut ini merupakan gambaran rute jogging yang biasa Rogu lewati.</p>
<p style="text-align: justify;"><img decoding="async" class="aligncenter" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Metode%20penjumlahan%20vektor%201.png" alt="Metode penjumlahan vektor 1" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Kira-kira nih, kamu bisa nggak menghitung berapa jarak yang ditempuh Rogu dari titik A ke titik D?</p>
<p style="text-align: justify;">Wah, kalau itu sih caranya mudah sekali, ya. Kita hanya tinggal menjumlahkan jarak dari titik AB ke titik BC, lalu ke titik CD. Sehingga,</p>
<p style="text-align: center;">AB + BC + CD = 550 m + 650 m + 700 m = 1.900 m</p>
<p> </p>
<p style="text-align: justify;">Simpel banget, kan? Tapi, bagaimana dengan perpindahan Rogu dari titik A ke titik D?</p>
<p style="text-align: justify;">Nah, jika kamu ingat, perpindahan itu termasuk besaran vektor. Perpindahan ditentukan oleh <strong>kedudukan awal dan kedudukan akhir</strong>, serta dapat <strong>bertanda positif maupun negatif</strong>, bergantung pada arah perpindahannya.</p>
<p><strong>Baca Juga: <a href="https://www.ruangguru.com/blog/perbedaan-kelajuan-dan-kecepatan">Apa Perbedaan antara Kelajuan dan Kecepatan?</a></strong></p>
<p style="text-align: justify;">Gambar rute jogging Rogu di atas bisa kita analogikan sebagai vektor nih, dengan memisalkan <strong>F1 merupakan vektor di titik AB, F2 merupakan vektor di titik BC, dan F3 merupakan vektor di titik CD</strong>.</p>
<p style="text-align: justify;"><img decoding="async" class="aligncenter" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Contoh%20kasus%20metode%20penjumlahan%20vektor.png" alt="Contoh kasus metode penjumlahan vektor" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Kemudian, perpindahan dari titik A ke titik D dapat ditentukan dengan mencari besar resultan vektornya saja. Apa itu resultan vektor? <strong>Resultan vektor adalah hasil dari penjumlahan dua atau lebih vektor</strong>.</p>
<p style="text-align: justify;">Terdapat beberapa metode yang bisa kita gunakan untuk mencari resultan vektor nih, di antaranya <strong>metode grafis, metode analisis vektor, atau metode uraian</strong>.</p>
<p style="text-align: justify;"><em>So</em>, kalau kamu mau tahu metode apa yang tepat untuk mencari besar perpindahan Rogu dari titik A ke titik D, yuk simak baik-baik artikel ini!</p>
<p> </p>
<h2 style="text-align: justify;"><span style="font-size: 18pt;">1. Metode Grafis</span></h2>
<p style="text-align: justify;">Metode yang pertama adalah metode grafis. Metode grafis adalah metode yang digunakan untuk <strong>menentukan besar resultan vektor dengan cara mengukurnya</strong>.</p>
<p style="text-align: justify;"><strong>Panjang resultan vektor</strong> dapat <strong>diukur menggunakan mistar (penggaris)</strong>, sedangkan <strong>besar sudut vektor</strong> (arah vektor) <strong>diukur menggunakan busur derajat</strong>. Perlu kamu ingat, pengukuran besar resultan vektor menggunakan metode grafis harus berdasarkan skala dan besar sudut yang tepat, ya.</p>
<p><img decoding="async" class="aligncenter" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/aturan%20menentukan%20metode%20vektor.png" alt="aturan menentukan metode vektor" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Nah, jika kamu menyimak cerita Rogu di atas, metode grafis ini merupakan metode yang tepat untuk mencari besar perpindahan Rogu dari titik A ke titik D.</p>
<p style="text-align: justify;">Langkah pertama yang bisa kamu lakukan adalah menetapkan skala dari masing-masing besaran vektor. Ingat! skala yang kita tentukan harus tepat dan juga sesuai ya.</p>
<p style="text-align: justify;">Berdasarkan cerita Rogu, besar vektor F1 = 550 m, besar vektor F2 = 650 m, dan besar vektor F3 = 700 m. Misalkan, untuk ketiga vektor, kita menetapkan skala 100 m = 1 cm.</p>
<p style="text-align: justify;">Artinya, setiap panjang 100 m kita gambar dengan 1 cm di kertas. Jadi, vektor F1 dapat digambar sepanjang 5,5 cm, vektor F2 digambar sepanjang 6,5 cm, dan vektor F3 digambar sepanjang 7 cm.</p>
<p style="text-align: justify;">Paham sampai di sini? Kita lanjut, ya.</p>
<p style="text-align: justify;">Kemudian, langkah kedua adalah menggambar besar dan arah masing-masing vektor seperti pada gambar di bawah ini. Panjang vektor R = F1 + F2 + F3 dapat dihitung menggunakan penggaris. Sementara itu, sudut arah vektor R dihitung menggunakan busur derajat.</p>
<p><img decoding="async" class="aligncenter" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/mencari%20resultan%20vektor.png" alt="mencari resultan vektor" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Sebelumnya, kita sudah tahu ya kalau untuk mencari perpindahan dari satu titik ke titik lain kita hanya tinggal menghitung besar resultan vektornya saja, jadi sudah dapat kita ketahui nih kalau perpindahan Rogu dari titik <strong>A ke titik D adalah sebesar 1.300 m</strong>.</p>
<p style="text-align: justify;">Jelas ya? Bagi yang belum paham, tulis saja pertanyaanmu di kolom komentar, oke?</p>
<p style="text-align: justify;">Oh iya, penggunaan metode grafis dalam menghitung jumlah dua atau lebih vektor ternyata memiliki kelemahan lho, yaitu dapat menimbulkan kesalahan sistematis. Nah, untuk menghindari kesalahan tersebut, kita dapat menggunakan metode yang akan kita bahas selanjutnya, yaitu metode analitis.</p>
<p><strong>Baca Juga: <a href="https://www.ruangguru.com/blog/besaran-satuan-dimensi-dalam-pengukuran-fisika">Pengertian Besaran, Satuan & Dimensi dalam Pengukuran Fisika</a></strong></p>
<p> </p>
<h2 style="text-align: justify;"><span style="font-size: 18pt;">2. Metode Analitis</span></h2>
<p style="text-align: justify;">Metode analitis adalah metode yang digunakan untuk menentukan besar resultan vektor secara matematis dengan <strong>menggunakan rumus</strong>. Adapun rumus yang digunakan merupakan rumus kosinus (<em>cos</em>) untuk menentukan besar resultan vektor dan rumus sinus (<em>sin</em>) untuk menentukan arah resultan vektor.</p>
<p><img decoding="async" class="aligncenter" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/rumus%20mencari%20besar%20resultan%20vektor.png" alt="rumus mencari besar resultan vektor" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Sekarang, supaya kamu lebih mudah untuk memahami cara mencari besar dan arah resultan vektor menggunakan metode ini, yuk, langsung saja kita simak contoh soal berikut ini.</p>
<p style="text-align: justify;"><strong>Contoh Soal</strong></p>
<p><img decoding="async" class="aligncenter" src="https://lh5.googleusercontent.com/3S36XB7GanT8h4frDXg0IFIq0UYrhM4P3BItqZzXWKQh_SNbmPjjsT9-nDfs6Bjoik6RaZTNy5dcZv5klldBFerJTKuK_4MRypiicW2vBIIFV7NxBK87A7IP9njjYtxDKg3IPHxx" alt="soal mencari arah resultan vektor" width="175" /></p>
<p> </p>
<p style="text-align: justify;">Hitunglah besar dan arah vektor resultannya terhadap sumbu x positif!</p>
<p style="text-align: justify;"><strong>Penyelesaian:</strong></p>
<p style="text-align: justify;"><strong>a. Besar resultan vektor</strong></p>
<p><img decoding="async" class="aligncenter" src="https://lh6.googleusercontent.com/WITvuma07aVxrCJUvGQV30TGd2QzKhkQjRwq65roXhfE9gislYVABcuCept5-kEoMR1kHeJ1-w2G_LroNw8dxvG29EOteW0lUFiaVOgOt5G4lUMkl1NXjc4GaWl0D2NdUvIBzZuX" alt="metode analitis mencari resultan vektor" width="213" /></p>
<p> </p>
<p><img decoding="async" class="alignnone" src="https://lh6.googleusercontent.com/TzIN5v0XbNd25MZLeCHzpdY1dLW-6B8tf63AsAzRuUw5uRYJNSPB-EMsCT4j5ekaaDFLaGH9r_Qzs2c2qFoffqLshl19gtaS6n8BgBQUKj9PWF3NBtnk8cd8IfAejzApI_stu5AW" alt="cara menghitung besar resultan vektor" width="296" /></p>
<p> </p>
<p style="text-align: justify;"><strong> b. Arah resultan vektor</strong></p>
<p><img decoding="async" class="aligncenter" src="https://lh6.googleusercontent.com/7vidpFr7XV-g4MzsMkyUYl5ADv9kwsBBYJ6wqcdndj94NhL40vNJHLy1o70fDQwG9eIAmdZaVHMGThANzZoysYY4o14kx4nanz8tG3fsWSBprLiaFXZWykSYfTZM_z5Ba2rniCyS" alt="cara mencari arah resultan vektor" width="400" /></p>
<p><img decoding="async" class="alignnone" src="https://lh6.googleusercontent.com/ZhdVZ2Zrmdw94mIdzb_zCqlyAbEr1hwzGih6Ilhwh3Tgrn0lZL7lDrOe5lVnJ1_D59bmKNL1sl--XVmgCHD4giN2zUE7eC93X9w3eJVlJp_gmcxvCuA2bsgqR1tv66CvM4HVLe8J" alt="cara mencari arah resultan vektor" width="199" /></p>
<p style="text-align: justify;">Jadi, besar resultan vektornya adalah 4√19 N dan arah resultan vektornya adalah 22,3o terhadap sumbu x positif. Gimana, mudah, kan? Oke, selanjutnya, kita masuk ke metode penjumlahan vektor yang terakhir, nih. Apakah itu? Yap! Metode uraian.</p>
<p> </p>
<h2 style="text-align: justify;"><span style="font-size: 18pt;"><strong>3. Metode Uraian</strong></span></h2>
<p style="text-align: justify;">Metode penjumlahan vektor yang terakhir adalah metode uraian. Pada materi sebelumnya, kamu telah mempelajari <strong><a href="https://www.ruangguru.com/blog/fisika-kelas-10-cara-mencari-komponen-vektor" target="_blank" rel="noopener">cara mencari komponen-komponen dari suatu vektor</a></strong> kan.</p>
<p style="text-align: justify;">Nah, pada metode uraian ini, sebelum kita mencari besar resultan vektor, kita <strong>uraikan</strong> terlebih dahulu vektor-vektor tersebut <strong>menjadi komponen vektor pada sumbu x dan komponen vektor pada sumbu y di koordinat kartesius</strong>.</p>
<p style="text-align: justify;">Kamu masih ingat kan cara mencari komponen vektor pada sumbu x dan y? Hayo, bagi yang sudah lupa, dipahami lagi ya materi sebelumnya.</p>
<p style="text-align: justify;">Setelah kita menguraikan vektor-vektor tersebut menjadi komponen vektor, barulah kita bisa mencari besar resultan vektornya, yaitu dengan menggunakan rumus <img decoding="async" class="alignnone" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/akar%20akaran%202.jpg" alt="rumus mencari besar resultan vektor" width="120" /> dan arah resultan vektornya dengan rumus <img decoding="async" class="alignnone" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/tan.jpg" alt="rumus mencari arah resultan vektor" width="90" />.</p>
<p style="text-align: justify;">Nah, ∑A<sub>x</sub> ini artinya jumlah komponen-komponen vektor pada sumbu x dan ∑A<span style="font-size: 13.3333px;">y</span> artinya jumlah komponen-komponen vektor pada sumbu y.</p>
<p><strong>Baca Juga: <a href="https://www.ruangguru.com/blog/konsep-dasar-vektor">Mengenal Vektor Bidang 2 Dimensi & Vektor Ruang 3 Dimensi</a></strong></p>
<p style="text-align: justify;">Perlu kamu perhatikan, <strong>besar suatu vektor akan selalu bernilai positif</strong>. Selain itu, dalam menentukan arah vektor, kita harus memperhatikan tanda Ax dan Ay yang nantinya akan menentukan kuadran dari vektor dalam sistem koordinat seperti pada tabel berikut ini:</p>
<p><img decoding="async" class="aligncenter" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/tabel-2.jpg" alt="tabel-2" width="400" /></p>
<p> </p>
<p style="text-align: justify;">Bingung? Tenang, nggak usah bingung-bingung, kita langsung coba kerjakan contoh soal di bawah ini saja, yuk! <em>Let’s go</em>!</p>
<p style="text-align: justify;"><strong>Contoh Soal</strong></p>
<p><img decoding="async" class="aligncenter" src="https://lh4.googleusercontent.com/72FcRlSSzypQWxqGyHd1UtSDYH64PgN1fYpf8ODlLxS7rei68vzhxKWwEfTW9gNbjVGG3Fd037t9Mw_HtCYayf0YBCvkTdNPZCbfXy9kI1nOSUb69PvHDEtM-CMRchJWinwCP5lG" alt="metode uraian penjumlahan vektor" width="206" /></p>
<p> </p>
<p style="text-align: justify;">Apabila F1 = 2 N, F2 = 10 N, dan F3 = 6 N, maka tentukan resultan dari ketiga vektor tersebut!</p>
<p style="text-align: justify;"><strong>Pembahasan:</strong></p>
<p style="text-align: justify;">Hal pertama yang bisa kita lakukan untuk mengerjakan soal di atas adalah dengan menguraikan vektor F1, F2, dan F3 terhadap sumbu x dan sumbu y.</p>
<p style="text-align: justify;"><strong>Pada sumbu x:</strong></p>
<ul>
<li style="text-align: justify;">F1x → -F1x = -2 N (tanda negatif menandakan arah vektor ke kiri).</li>
<li style="text-align: justify;">F2x = F2 sin θ → F2x = (10) sin 53°= (10)(0,8) = 8 N (tanda positif menandakan arah vektor ke kanan).</li>
<li style="text-align: justify;">F3x = 0 N (angka nol (0) menandakan F3 tidak memiliki proyeksi vektor/komponen vektor pada sumbu x karena F3 tegak lurus terhadap sumbu x).</li>
</ul>
<p> </p>
<p style="text-align: justify;">Jadi, ΣFx = F1x + F2x + F3x = -2 + 8 + 0 = 6 N</p>
<p> </p>
<p style="text-align: justify;"><strong>Pada sumbu y:</strong></p>
<ul>
<li style="text-align: justify;">F1y = 0 N (angka nol (0) menandakan F1 tidak memiliki proyeksi vektor/komponen vektor pada sumbu y karena F1 tegak lurus terhadap sumbu y).</li>
<li style="text-align: justify;">F2y = -F2 cos θ = -(10) cos 53° = -(10)(0,6) = -6 N (tanda negatif menandakan arah vektor ke bawah).</li>
<li style="text-align: justify;">F3y = 6 N (tanda positif menandakan arah vektor ke atas).</li>
</ul>
<p> </p>
<p style="text-align: justify;">Jadi, ΣFy = F1y + F2y + F3y = 0 + 6 – 6 = 0 N</p>
<p style="text-align: justify;">Selanjutnya, setelah kita mengetahui komponen-komponen dari ketiga vektor di atas terhadap sumbu x dan y, maka kita dapat mencari resultan dari ketiga vektor tersebut.</p>
<p style="text-align: justify;"><img decoding="async" class="aligncenter" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/akar%20akaran%203.jpg" alt="akar akaran 3" width="294" /></p>
<p> </p>
<p style="text-align: justify;">Jadi, resultan dari vektor F1, F2, dan F3 adalah 6 N. Gimana? Ternyata nggak sesulit yang kamu kira, kan?</p>
<p><strong>Baca Juga: <a href="https://www.ruangguru.com/blog/gerak-melingkar-beraturan-dan-penjelasannya">Gerak Melingkar Beraturan (GMB): Besaran, Rumus & Contoh Soal</a></strong></p>
<p style="text-align: justify;">Oke, setelah kamu memahami ketiga metode penjumlahan vektor di atas, menurutmu, metode mana yang lebih mudah? Eits! Tapi ingat, jangan mentang-mentang kamu sreg dengan satu metode, terus metode yang lainnya tidak kamu pahami, deh. Kamu juga harus paham ketiga-tiganya. Siapa tahu keluar di ujian nanti.</p>
<p style="text-align: justify;">Sampai sini, sudah mulai paham belum dengan materi metode penjumlahan vektor? Kalo masih ada poin-poin yang belum kamu mengerti, mending belajar sama ahlinya, deh. Belajar bareng kakak-kakak pengajar di <a href="https://www.ruangguru.com/privat/fisika"><strong>Ruangguru Privat Fisika</strong></a> misalnya. Hehehe…</p>
<p style="text-align: justify;">Belajar nggak cuma menyenangkan, tapi kamu juga bakal diajari konsepnya sampai paham! Para pengajar di Ruangguru Privat juga sudah terstandarisasi kualitasnya, loh. Kamu juga bisa pilih nih, mau diajarkan secara langsung (<em>offline</em>) atau daring (<em>online</em>). Fleksibel, kan? Untuk info lebih lanjut, cuss klik <em>link</em> berikut!</p>
<p><a href="https://cta.ruangguru.com/blog-rg-10-privat-b"><img decoding="async" class="aligncenter size-medium" src="https://cdn-web-2.ruangguru.com/landing-pages/assets/84b6f2c7-9c9c-480d-a218-7961334af9ce.png" alt="CTA ruangguru privat" width="820" /></a></p>
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<nav class="footer-menu">
<p class="title">Produk Lainnya</p>
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</nav>
</div>
<div class="col-6 col-md-3">
<nav class="footer-menu">
<p class="title"> Coba GRATIS Aplikasi Ruangguru</p>
<div class="download-list">
<ul>
<li>
<a target="_blank" rel="nofollow" href="https://apps.apple.com/id/app/ruangguru-solusi-belajar/id1099742206?l=id">
<img src="https://cdn-web-2.ruangguru.com/static/appstore.png" loading="lazy" alt="Button App Store Footer " />
</a>
</li>
<li>
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<img src="https://cdn-web-2.ruangguru.com/static/googleplay.png" loading="lazy" alt="Button Google Play Footer " />
</a>
</li>
<li>
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<img src="https://cdn-web-2.ruangguru.com/static/huawei.png" loading="lazy" alt="Button AppGallery Footer " />
</a>
</li>
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</div>
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</div>
<div class="d-lg-none text-center">
<nav class="footer-menu social">
<p class="title">Ikuti Kami</p>
<a target="_blank" href="https://www.instagram.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/instagram.png" loading="lazy" alt="Logo Instagram Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.facebook.com/ruanggurucom/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/facebook.png?height=60&convert=webp" type="image/webp">
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</picture>
</a>
<a target="_blank" href="https://twitter.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/twitter.png?height=60&convert=webp" type="image/webp">
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</picture>
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</picture>
</a>
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</picture>
</a>
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</picture>
</a>
</nav>
</div>
<p class="footer-copyright">
© 2026 All Rights Reserved PT. Ruang Raya Indonesia
</p>
</div>
<style>
.bantuan-container {
position: relative;
}
.bantuan-element {
position: fixed;
z-index: 999;
display: flex;
align-items: flex-end;
justify-content: flex-end;
cursor: pointer;
transition: all 0.6s ease;
}
.bantuan-top {
bottom: 11rem;
right: 2rem;
}
.bantuan-bottom {
bottom: 2rem;
right: 2rem;
}
.bantuan-top-2 {
bottom: 15rem;
right: 2rem;
}
.bantuan-top-3 {
bottom: 19rem;
right: 2rem;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 2rem;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 2rem;
}
@media (min-width: 768px) {
.bantuan-element img {
width: auto;
transition: all 0.4s ease;
}
.bantuan-top img {
width: 140px;
}
.bantuan-bottom img {
width: 140px;
}
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 140px;
}
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 140px;
}
.bantuan-element.clicked {
bottom: 9rem;
right: 2.5rem;
}
.bantuan-element.clicked img {
width: auto;
height: 64px;
}
}
@media (max-width: 767px) {
.bantuan-top {
bottom: 8rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom {
bottom: 2rem;
right: 1.25rem;
align-items: center;
}
.bantuan-top-2 {
bottom: 12rem;
right: 1rem;
align-items: center;
}
.bantuan-top-3 {
bottom: 16rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 1.25rem;
align-items: center;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 1.25rem;
align-items: center;
}
.bantuan-element img {
width: auto;
}
.bantuan-top img,
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 96px;
}
.bantuan-bottom img,
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 96px;
}
}
.bantuan-element:hover {
transform: scale(1.05);
}
.bantuan-element img {
max-width: 100%;
height: auto;
}
</style>
<div id="bantuan-divs-container" style="position: relative" data-product="ruangguru" data-table="rg_floating_icon"></div>
<script>
class FloatingDivsComponent {
constructor(product = 'ruangguru', table = 'rg_floating_icon') {
this.container = document.getElementById('bantuan-divs-container');
this.floatingElements = [];
this.product = product;
this.table = table;
// Configuration
this.config = {
baseApiUrl: 'https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/rg_banner/views/',
apiKey: 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
maxElements: 2,
zIndexBase: 100,
};
// Build dynamic API URL based on table parameter
this.config.apiUrl = this.buildApiUrl();
this.init();
}
buildApiUrl() {
const baseUrl = this.config.baseApiUrl;
const tableName = this.table || 'rg_floating_icon';
const queryParams =
'offset=0&limit=99999&sort=rankPosition&where=(type,eq,floating)~and(showIn,anyof,blog)~and(hide,is,false)~and(product,eq,' +
this.product +
')&nested%5Bcta%5D%5Bfields%5D=Id,name,ctaId,imageUrl,ctaLink,targetUrl';
return `${baseUrl}${tableName}?${queryParams}`;
}
async init() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to initialize floating divs component:', error);
}
}
async fetchFloatingElements() {
try {
const options = {
method: 'GET',
headers: {
'xc-token': this.config.apiKey || 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
},
};
const urlNoco = this.config.apiUrl;
const response = await fetch(urlNoco, options);
if (!response.ok) {
throw new Error(`HTTP error! status: ${response.status}`);
}
const data = await response.json();
this.floatingElements = this.processApiResponse(data);
} catch (error) {
console.error('Error fetching floating elements:', error);
throw error;
}
}
processApiResponse(data) {
if (!data || !data.list || !Array.isArray(data.list)) {
return [];
}
console.log(data.list);
// Ensure we only get maximum 2 elements
const limitedElements = data.list.slice(0, this.config.maxElements);
return limitedElements;
}
renderFloatingElements() {
if (!this.container || this.floatingElements.length === 0) {
return;
}
// Additional safeguard: ensure we never render more than maxElements
const elementsToRender = this.floatingElements.slice(0, this.config.maxElements);
// Clean up existing elements and their event listeners
this.cleanupExistingElements();
// Clear existing content
this.container.innerHTML = '';
elementsToRender.forEach((element, index) => {
const floatingDiv = this.createFloatingDiv(element, index);
this.container.appendChild(floatingDiv);
});
}
cleanupExistingElements() {
// Clear existing elements (no need for event listener cleanup with CSS approach)
const existingElements = this.container.querySelectorAll('[data-floating-element]');
existingElements.forEach(element => {
element.remove();
});
}
createFloatingDiv(element, index) {
const div = document.createElement('div');
const cta = element.cta[0]; // Use the first CTA
// Add data attribute for cleanup tracking
div.setAttribute('data-floating-element', 'true');
// Apply CSS classes based on rankPosition
const positionClass = this.getPositionClass(element.rankPosition, index);
div.className = `bantuan-element ${positionClass}`;
div.style.zIndex = this.config.zIndexBase + index;
// Create the link and image
const link = document.createElement('a');
link.href = 'https://cta.ruangguru.com/' + cta.ctaId || '#';
link.target = '_blank';
// Add tracking attributes
link.setAttribute('data-tracking-event-type', 'landing_page__core__item_click');
link.setAttribute(
'data-tracking-context',
JSON.stringify({
product_name: this.product,
cta_action: 'redirection',
cta_action_detail: link.href,
item_type: 'floating',
item_name: element.name || 'Floating element',
item_category: 'payment',
item_position: index,
item_serial: null,
section_type: 'floating_item',
section_position: 'body',
section_name: 'floating_item',
})
);
const img = document.createElement('img');
img.src = element.imageUrl + '?h=200&convert=webp';
img.alt = element.name || 'Floating element';
img.loading = 'lazy';
// Add click tracking
link.addEventListener('click', () => {
this.trackClick(element, cta);
});
link.appendChild(img);
div.appendChild(link);
return div;
}
getPositionClass(rankPosition, index) {
// Define positioning classes based on rankPosition
if (rankPosition === 1) {
return 'bantuan-top'; // Top position
} else if (rankPosition === 2) {
return 'bantuan-bottom'; // Bottom position
} else {
// For additional elements, distribute them evenly
return `bantuan-${index % 2 === 0 ? 'top' : 'bottom'}-${Math.floor(index / 2) + 1}`;
}
}
trackClick(element, cta) {
// Implement click tracking logic here
console.log('Floating element clicked:', {
elementId: element.Id,
elementName: cta.name,
ctaId: cta.ctaId,
targetUrl: cta.targetUrl,
});
}
// Public method to manually refresh
async refresh() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to refresh floating elements:', error);
throw error;
}
}
// Public method to get current floating elements
getFloatingElements() {
return [...this.floatingElements];
}
}
// Initialize the component when DOM is ready
document.addEventListener('DOMContentLoaded', () => {
// Get product and table parameters from data attributes
const container = document.getElementById('bantuan-divs-container');
const product = container.getAttribute('data-product') || 'ruangguru';
const table = container.getAttribute('data-table') || 'rg_floating_icon';
new FloatingDivsComponent(product, table);
});
// Export for use in other modules if needed
if (typeof module !== 'undefined' && module.exports) {
module.exports = FloatingDivsComponent;
}
</script> <div id="wp-install-nudge" style="
position: fixed;
bottom: 0rem;
right: 0;
z-index: 999;
cursor: pointer;
"></div>
<script>
(function (w, d, s, o, f, js, fjs) {
w['Sales-Nudge-Widget'] = o; w[o] = w[o] || function () { (w[o].q = w[o].q || []).push(arguments) };
js = d.createElement(s), fjs = d.getElementsByTagName(s)[0];
js.id = o; js.src = f; js.async = 1; fjs.parentNode.insertBefore(js, fjs);
}(window, document, 'script', 'wpInstallNudge', 'https://cdn-web-2.ruangguru.com/sales-nudge/6198a95130/dist/widget.js'));
wpInstallNudge('init', { targetElementId: 'wp-install-nudge', clientTarget: 'ruangguru_blog' });
</script>
<script>
// note: disabled due to auto reload page
// var lastScrollTop = 0;
// $(window).scroll(function(event){
// var st = $(this).scrollTop();
// if (st > lastScrollTop){
// $('#wp-install-nudge').slideDown( "slow", function() {});
// } else {
// $('#wp-install-nudge').slideUp( "slow", function() {});
// }
// lastScrollTop = st;
// });
// $(document).ready(function () {
// $("#wp-install-nudge").hide();
// });
$('#wp-install-nudge').click(function(){
Analytics.track("promotion__install_nudge__click",
{
user_email: "ruangguru@blog.com",
cta_link: widgetData.dataConfig.target_link
}).catch((err) => {
console.log("tracker error", err);
});
});
</script>
</footer>
<div id="__next"></div>
<script type="speculationrules">
{"prefetch":[{"source":"document","where":{"and":[{"href_matches":"\/blog\/*"},{"not":{"href_matches":["\/blog\/wp-*.php","\/blog\/wp-admin\/*","\/blog\/wp-content\/uploads\/*","\/blog\/wp-content\/*","\/blog\/wp-content\/plugins\/*","\/blog\/wp-content\/themes\/ruangguru\/*","\/blog\/*\\?(.+)"]}},{"not":{"selector_matches":"a[rel~=\"nofollow\"]"}},{"not":{"selector_matches":".no-prefetch, .no-prefetch a"}}]},"eagerness":"conservative"}]}
</script>
<script>
async function getCtaDetails(ctaIds) {
const endpoint =
" https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/CTA/views/CTA?offset=0&limit=10&where=(ctaId,in," + ctaIds.toString() + ")";
try {
const response = await fetch(endpoint, {
headers: {
"xc-token": "-hG2nBDJzSj0agSgNtcJx97M4F02y54nUMoY6plJ",
},
});
const data = await response.json();
const ctaList = data?.list || [];
const ctaMap = new Map();
for (const ctaData of ctaList) {
ctaMap.set(ctaData.ctaId, ctaData);
}
return ctaMap;
} catch (error) {
console.error("Error fetching CTA details:", error);
return new Map();
}
}
async function replaceCTA() {
const regex = /{{cta\((‘(.*?)’|“(.*?)”)\)}}/g;
const bodyHTML = document.body.innerHTML;
const matches = Array.from(bodyHTML.matchAll(regex));
const ctaIds = matches.map((match) => {
let ctaId = match[2] ?? match[3];
return ctaId;
});
const ctaMap = await getCtaDetails(ctaIds);
const replacedHTML = bodyHTML.replace(regex, (match, _inner, ctaId1, ctaId2) => {
const ctaId = ctaId1 ?? ctaId2;
const ctaData = ctaMap.get(ctaId);
if (ctaData) {
return `<a href="https://cta.ruangguru.com/${ctaData.ctaId}" target="_${ctaData.targetType}" style="position:relative;display:inline-block;">
<img src="${ctaData.imageUrl}" loading="lazy" alt="${ctaData.altText}" />
</a>`;
} else {
return match;
}
});
document.body.innerHTML = replacedHTML;
}
</script>
<meta name="author" content="Ruangguru Tech Team">
<!-- Google Tag Manager (noscript) -->
<noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PVNLZFT" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>
<!-- Tracker -->
<script type="text/javascript" src="https://cdn-web-2.ruangguru.com/landing-page-web/public/staticpages/global-assets/js/cta.js"></script>
<script src="https://unpkg.com/@dotlottie/player-component@latest/dist/dotlottie-player.mjs" type="module"></script>
</body>
</html>