Quaternion Calculator
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Result
Quaternion Operations
Basic Operations
- Addition: (w₁ + x₁i + y₁j + z₁k) + (w₂ + x₂i + y₂j + z₂k) = (w₁ + w₂) + (x₁ + x₂)i + (y₁ + y₂)j + (z₁ + z₂)k
- Multiplication: (w₁ + x₁i + y₁j + z₁k) × (w₂ + x₂i + y₂j + z₂k) = (w₁w₂ - x₁x₂ - y₁y₂ - z₁z₂) + (w₁x₂ + x₁w₂ + y₁z₂ - z₁y₂)i + (w₁y₂ - x₁z₂ + y₁w₂ + z₁x₂)j + (w₁z₂ + x₁y₂ - y₁x₂ + z₁w₂)k
- Conjugate: (w + xi + yj + zk)* = w - xi - yj - zk
- Norm: |q| = √(w² + x² + y² + z²)
- Inverse: q⁻¹ = q*/|q|²
3D Rotation
A quaternion q = w + xi + yj + zk can represent a rotation around a unit vector (x, y, z) by an angle θ, where:
- w = cos(θ/2)
- x = sin(θ/2) * axis_x
- y = sin(θ/2) * axis_y
- z = sin(θ/2) * axis_z
To rotate a point p = (px, py, pz) using quaternion q:
p' = qpq⁻¹
Example Problems
Example 1: Quaternion Addition
q₁ = 1 + 2i + 3j + 4k
q₂ = 2 + 3i + 4j + 5k
q₁ + q₂ = 3 + 5i + 7j + 9k
Example 2: 90° Rotation around Z-axis
θ = 90°
axis = (0, 0, 1)
q = cos(45°) + sin(45°)k
q ≈ 0.7071 + 0.7071k
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