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Konversi ke representasi orientasi lain > Matriks rotasi

Orientasi apa pun dapat dicapai dengan menyusun tiga rotasi elemen, dimulai dari orientasi standar yang diketahui. Secara ekivalen, setiap matriks rotasi  R  dapat terdekomposisi sebagai hasil kali dari tiga matriks rotasi elemen. Contohnya:
adalah matriks rotasi yang dapat digunakan untuk merepresentasikan komposisi rotasi ekstrinsik tentang sumbu z, y, x, (dalam urutan itu), atau komposisi rotasi intrinsik tentang sumbu x-y′-z″ (dalam urutan itu).
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 X_{1}Z_{2}X_{3}={\begin{bmatrix}c_{2}&-c_{3}s_{2}&s_{2}s_{3}\\c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}\\s_{1}s_{2}&c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{bmatrix}}:  X_{1}Z_{2}Y_{3}={\begin{bmatrix}c_{2}c_{3}&-s_{2}&c_{2}s_{3}\\s_{1}s_{3}+c_{1}c_{3}s_{2}&c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}\\c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}\end{bmatrix}}
 X_{1}Y_{2}X_{3}={\begin{bmatrix}c_{2}&s_{2}s_{3}&c_{3}s_{2}\\s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}\\-c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{bmatrix}}:  X_{1}Y_{2}Z_{3}={\begin{bmatrix}c_{2}c_{3}&-c_{2}s_{3}&s_{2}\\c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}\\s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}\end{bmatrix}}
 Y_{1}X_{2}Y_{3}={\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}&c_{1}s_{3}+c_{2}c_{3}s_{1}\\s_{2}s_{3}&c_{2}&-c_{3}s_{2}\\-c_{3}s_{1}-c_{1}c_{2}s_{3}&c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{bmatrix}}:  Y_{1}X_{2}Z_{3}={\begin{bmatrix}c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}\\c_{2}s_{3}&c_{2}c_{3}&-s_{2}\\c_{1}s_{2}s_{3}-c_{3}s_{1}&c_{1}c_{3}s_{2}+s_{1}s_{3}&c_{1}c_{2}\end{bmatrix}}
 Y_{1}Z_{2}Y_{3}={\begin{bmatrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}\\c_{3}s_{2}&c_{2}&s_{2}s_{3}\\-c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{bmatrix}}:  Y_{1}Z_{2}X_{3}={\begin{bmatrix}c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}\\s_{2}&c_{2}c_{3}&-c_{2}s_{3}\\-c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}\end{bmatrix}}
 Z_{1}Y_{2}Z_{3}={\begin{bmatrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}&c_{1}s_{2}\\c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}\\-c_{3}s_{2}&s_{2}s_{3}&c_{2}\end{bmatrix}}:  Z_{1}Y_{2}X_{3}={\begin{bmatrix}c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}&s_{1}s_{3}+c_{1}c_{3}s_{2}\\c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}\\-s_{2}&c_{2}s_{3}&c_{2}c_{3}\end{bmatrix}}
 Z_{1}X_{2}Z_{3}={\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}\\c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}\\s_{2}s_{3}&c_{3}s_{2}&c_{2}\end{bmatrix}}:  Z_{1}X_{2}Y_{3}={\begin{bmatrix}c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}\\c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}\\-c_{2}s_{3}&s_{2}&c_{2}c_{3}\end{bmatrix}}
Untuk mengubah rumus rotasi pasif (atau mencari rotasi aktif terbalik), ubah urutan matriks (kemudian setiap matriks mengubah koordinat awal dari sebuah vektor yang tetap menjadi koordinat dari vektor yang sama yang diukur dalam sistem referensi yang diputar; sumbu rotasi yang sama, sudut yang sama, tetapi sekarang sistem koordinat berputar, bukan vektor).