Quaternion Calculator
Calculate quaternion operations, Hamilton products, 3D rotations, and transformations
Quaternion Operations
Quaternion Results
3D Vector Rotation
Rotation Results
[1.0000, 0.0000, 0.0000]
[0.0000, 0.0000, 1.0000]
Basic Unity Quaternions
× | 1 | i | j | k |
---|---|---|---|---|
1 | 1 | i | j | k |
i | i | -1 | k | -j |
j | j | -k | -1 | i |
k | k | j | -i | -1 |
Quaternion Properties
i² = j² = k² = ijk = -1
Multiplication is non-commutative
|q| = √(w² + x² + y² + z²)
q⁻¹ = q*/|q|² (conjugate/magnitude²)
Unit quaternions represent rotations
Understanding Quaternions
What are Quaternions?
Quaternions are a mathematical extension of complex numbers, discovered by William Rowan Hamilton in 1843. They consist of four real numbers: one real part (w) and three imaginary parts (x, y, z) with basis elements i, j, and k.
Quaternion Form
q = w + xi + yj + zk
Where w, x, y, z are real numbers and i, j, k are the fundamental quaternion units.
Hamilton Product
Quaternion multiplication follows the Hamilton product rules. The product of two quaternions q₁ and q₂ is calculated using the distributive property and the fundamental relations:
- i² = j² = k² = ijk = -1
- ij = k, jk = i, ki = j
- ji = -k, kj = -i, ik = -j
3D Rotations
Unit quaternions (magnitude = 1) can efficiently represent rotations in 3D space. The rotation quaternion is: q = cos(θ/2) + sin(θ/2)(xi + yj + zk) where (x,y,z) is the unit rotation axis and θ is the rotation angle.