Quaternion Product Calculator

Quaternion Product Formula:

\[ q_3 = q_1 \times q_2 = (w_1 w_2 - \vec{v_1} \cdot \vec{v_2}, w_1 \vec{v_2} + w_2 \vec{v_1} + \vec{v_1} \times \vec{v_2}) \]

First Quaternion (q₁)

Scalar part (w₁):

Vector x (i):

Vector y (j):

Vector z (k):

Second Quaternion (q₂)

Scalar part (w₂):

Vector x (i):

Vector y (j):

Vector z (k):

Quaternion Product (q₃ = q₁ × q₂):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Quaternion Product?

The quaternion product combines two quaternions through a special multiplication operation that accounts for both scalar and vector parts. Quaternions are hypercomplex numbers used in 3D rotations and computer graphics.

2. How Does the Calculator Work?

The calculator uses the quaternion product formula:

\[ q_3 = q_1 \times q_2 = (w_1 w_2 - \vec{v_1} \cdot \vec{v_2}, w_1 \vec{v_2} + w_2 \vec{v_1} + \vec{v_1} \times \vec{v_2}) \]

Where:

\( w \) — Scalar part of the quaternion
\( \vec{v} \) — Vector part (x, y, z components)
\( \cdot \) — Dot product
\( \times \) — Cross product

Explanation: The product combines both the dot product and cross product of the vector parts with scalar multiplication.

3. Applications of Quaternion Multiplication

Details: Quaternion multiplication is essential for composing 3D rotations, computer graphics, aerospace navigation, and robotics. It avoids gimbal lock and provides smooth interpolation.

4. Using the Calculator

Tips: Enter the scalar (w) and vector (x, y, z) components for both quaternions. The calculator will compute the product using the proper quaternion multiplication rules.

5. Frequently Asked Questions (FAQ)

Q1: Why use quaternions instead of rotation matrices?
A: Quaternions are more compact (4 numbers vs 9), avoid gimbal lock, and interpolate more smoothly.

Q2: Is quaternion multiplication commutative?
A: No, q₁ × q₂ ≠ q₂ × q₁ in general due to the cross product term.

Q3: How are quaternions used in 3D rotations?
A: A rotation of θ radians around axis (x,y,z) is represented as (cos(θ/2), x·sin(θ/2), y·sin(θ/2), z·sin(θ/2)).

Q4: What's the identity quaternion?
A: (1, 0, 0, 0) - multiplying by this leaves any quaternion unchanged.

Q5: How is quaternion multiplication related to complex numbers?
A: Quaternions extend complex numbers to 3D, with three imaginary units (i, j, k) where i² = j² = k² = ijk = -1.