Kuaternion

Dalam matematika, Kuaternion merupakan perluasan dari bilangan-bilangan kompleks yang tidak komutatif, dan diterapkan dalam mekanika tiga dimensi. Kuaternion ditemukan oleh ahli matematika dan astronomi Inggris, William Rowan Hamilton, yang memperpanjang aritmatika kompleks nomor ke kuaternion.

Segera setelah itu penemuan Hamilton, matematikawan Jerman Hermann Grassmann mulai menyelidiki vektor. Meskipun karakter abstrak, fisikawan Amerika JW Gibbs diakui dalam aljabar vektor sistem utilitas besar bagi fisikawan, seperti Hamilton mengakui kegunaan kuaternion. Pengaruh luas dari pendekatan abstrak yang dipimpin George Boole untuk menulis Hukum Thought (1854), perawatan aljabar dasar logika.

Definisi

Sebagai himpunan, kuaternion, berlambang H, sama dengan R⁴ yang merupakan ruang vektor bilangan riil empat dimensi. H memiliki tiga macam operasi: pertambahan, perkalian skalar dan perkalian kuaternion. Elemen-elemen kuaternion ditandakan sebagai 1, i, j dan k (i, j dan k adalah komponen imaginer), dan dapat ditulis sebagai kombinasi linear, a + bi + cj + dk (a, b, c, dan d adalah bilangan riil).

Perkalian elemen dasar

Persamaan elemen kuaternion i, j, dan k adalah:

$i^{2}=j^{2}=k^{2}=ijk=-1,\$

Karena

$-1=ijk,\$

jika dua sisi dikalikan dengan k, maka

${\begin{aligned}-k&=ijkk=ij(k^{2})=ij(-1),\\k&=ij.\end{aligned}}$

Persamaan-persamaan yang lainnya juga bisa didapatkan dengan tahap aljabar:

${\begin{alignedat}{2}ij&=k,&\qquad ji&=-k,\\jk&=i,&kj&=-i,\\ki&=j,&ik&=-j,\end{alignedat}}$

Persamaan-persamaan ini lalu bisa ditampilkan dengan tabel di bawah ini:

Perkalian kuaternion

×	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1

Pertambahan kuaternion

Persamaan pertambahan dua kuaternion (a₁ + b₁i + c₁j + d₁k) dan (a₂ + b₂i + c₂j + d₂k) adalah:

${\begin{aligned}&(a_{1}+b_{1}i+c_{1}j+d_{1}k)+(a_{2}+b_{2}i+c_{2}j+d_{2}k)\\[8pt]&=(a_{1}+a_{2})+(b_{1}+b_{2})i+(c_{1}+c_{2})j+(d_{1}+d_{2})k\end{aligned}}$

Perkalian kuaternion

Persamaan perkalian dua kuaternion (a₁ + b₁i + c₁j + d₁k) dan (a₂ + b₂i + c₂j + d₂k) adalah:

${\begin{aligned}&(a_{1}+b_{1}i+c_{1}j+d_{1}k)\times (a_{2}+b_{2}i+c_{2}j+d_{2}k)\\[8pt]&=(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2})+\\&{}\qquad (b_{1}a_{2}+a_{1}b_{2}-d_{1}c_{2}+c_{1}d_{2})i+\\&{}\qquad (c_{1}a_{2}+d_{1}b_{2}+a_{1}c_{2}-b_{1}d_{2})j+\\&{}\qquad (d_{1}a_{2}-c_{1}b_{2}+b_{1}c_{2}+a_{1}d_{2})k\end{aligned}}$

Pembagian kuaternion

Persamaan pembagian dua kuaternion (a₁ + b₁i + c₁j + d₁k) dan (a₂ + b₂i + c₂j + d₂k) ditampilkan di bawah:

${\begin{aligned}&{\frac {a_{1}+b_{1}i+c_{1}j+d_{1}k}{a_{2}+b_{2}i+c_{2}j+d_{2}k}}\\[8pt]&={\frac {a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}}{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2}}}+\\&{}\qquad {\frac {b_{1}a_{2}-a_{1}b_{2}-d_{1}c_{2}+c_{1}d_{2}}{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2}}}i+\\&{}\qquad {\frac {c_{1}a_{2}+d_{1}b_{2}-a_{1}c_{2}-b_{1}d_{2}}{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2}}}j+\\&{}\qquad {\frac {d_{1}a_{2}-c_{1}b_{2}+b_{1}c_{2}-a_{1}d_{2}}{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2}}}k\end{aligned}}$

Konjugat

Suatu kuaternion a + bi + cj + dk memiliki konjugat q, dan didapatkan dengan rumus berikut:

${\begin{alignedat}{2}q=a-bi-cj-dk\end{alignedat}}$

Bentuk matriks

Kuaternion, seperti bilangan kompleks, bisa ditulis dalam bentuk matriks, yaitu matriks kompleks 2x2 atau matriks riil 4x4.

Bentuk matriks kompleks 2x2 untuk kuaternion a + bi + cj + dk adalah:

${\begin{bmatrix}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.$

Bentuk matriks riil 4x4 untuk kuaternion a + bi + cj + dk adalah:

${\begin{bmatrix}a&b&c&d\\-b&a&-d&c\\-c&d&a&-b\\-d&-c&b&a\end{bmatrix}}$

$=a{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}+b{\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}+c{\begin{bmatrix}0&0&1&0\\0&0&0&1\\-1&0&0&0\\0&-1&0&0\end{bmatrix}}+d{\begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&1&0&0\\-1&0&0&0\end{bmatrix}}.$

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