Quaternion Calculator – Guide & Formulas
Perform quaternion arithmetic instantly. Add, subtract, multiply, find conjugate, norm, and inverse of quaternions with step-by-step solutions.
Perform quaternion arithmetic instantly with our free calculator. Enter two quaternions to add, subtract, multiply, or find the conjugate, norm, and inverse with step-by-step solutions.
Key Takeaway
Use the free Quaternion Calculator to perform quaternion arithmetic instantly. add, subtract, multiply, find conjugate, norm, and inverse of quaternions with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Quaternion Calculator
- Enter the real and imaginary parts (w, x, y, z) for each quaternion.
- Select the operation: Add, Subtract, Multiply, Conjugate, Norm, or Inverse.
- Review the result quaternion displayed in w + xi + yj + zk form.
- Check the step-by-step breakdown showing each component calculation.
The Formula
Variable Definitions
- w: The real (scalar) part of the quaternion
- x, y, z: The imaginary (vector) parts along i, j, k axes
- i, j, k: The three imaginary units satisfying i² = j² = k² = ijk = -1
- |q|: The norm (magnitude) of the quaternion
- q*: The conjugate: w - xi - yj - zk
- q⁻¹: The inverse: q*/|q|²
Multiplying (1 + 2i + 3j + 0k) × (4 + 0i + 1j + 2k)
Compute the Hamilton product of two quaternions.
- Step 1: Expand using distributive property: (1)(4) + (1)(j) + (1)(2k) + (2i)(4) + (2i)(j) + (2i)(2k) + (3j)(4) + (3j)(j) + (3j)(2k).
- Step 2: Apply quaternion rules: ij = k, ik = -j, ji = -k, jk = i, ki = j, kj = -i.
- Step 3: Compute each product: 4 + j + 2k + 8i + 2k - 4j + 12j + 3(-1) + 6i.
- Step 4: Combine like terms: (4 - 3) + (8 + 6)i + (1 - 4 + 12)j + (2 + 2)k = 1 + 14i + 9j + 4k.
Frequently Asked Questions
What is a quaternion?
A quaternion is a 4-dimensional number of the form w + xi + yj + zk, where w is a real number and x, y, z are real coefficients of the imaginary units i, j, k. They extend complex numbers and are used in 3D rotations, computer graphics, and aerospace.
Is quaternion multiplication commutative?
No. Quaternion multiplication is non-commutative: q₁ × q₂ ≠ q₂ × q₁ in general. For example, ij = k but ji = -k. This is a fundamental difference from complex number multiplication.
How do I find the conjugate of a quaternion?
The conjugate of q = w + xi + yj + zk is q* = w - xi - yj - zk. Simply negate the three imaginary components while keeping the real part unchanged.
What is the norm of a quaternion?
The norm |q| = √(w² + x² + y² + z²). It represents the magnitude of the quaternion and satisfies |q₁q₂| = |q₁||q₂|. A quaternion with norm 1 is called a unit quaternion.
When is a quaternion invertible?
A quaternion is invertible if and only if its norm is non-zero (i.e., it is not the zero quaternion). The inverse is q⁻¹ = q*/|q|² where q* is the conjugate.
What are unit quaternions used for?
Unit quaternions (norm = 1) are used to represent 3D rotations in computer graphics, robotics, and aerospace. They avoid gimbal lock and provide smooth interpolation between orientations (SLERP).
How do quaternions relate to complex numbers?
Quaternions extend complex numbers. A complex number a + bi is a quaternion with y = z = 0. The imaginary unit i in complex numbers corresponds to the i unit in quaternions.
Can I divide quaternions?
Yes. Quaternion division q₁/q₂ is defined as q₁ × q₂⁻¹ where q₂⁻¹ is the inverse of q₂. Since multiplication is non-commutative, left division and right division give different results.