Quaternion Multiplication Calculator

Quaternion Multiplication Formula:

\[ q_1 \times q_2 = (w_1 w_2 - \vec{v}_1 \cdot \vec{v}_2, w_1 \vec{v}_2 + w_2 \vec{v}_1 + \vec{v}_1 \times \vec{v}_2) \]

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):

Resulting Quaternion (q₁ × q₂):

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1. What is Quaternion Multiplication?

Quaternion multiplication is a non-commutative operation that combines two quaternions. A quaternion is a number system that extends complex numbers, represented as q = w + xi + yj + zk where w, x, y, z are real numbers and i, j, k are fundamental quaternion units.

2. How Does the Calculator Work?

The calculator uses the quaternion multiplication formula:

\[ q_1 \times q_2 = (w_1 w_2 - \vec{v}_1 \cdot \vec{v}_2, w_1 \vec{v}_2 + w_2 \vec{v}_1 + \vec{v}_1 \times \vec{v}_2) \]

Where:

\( w_1, w_2 \) — Scalar parts of the quaternions
\( \vec{v}_1, \vec{v}_2 \) — Vector parts of the quaternions (xi + yj + zk)
\( \cdot \) — Dot product
\( \times \) — Cross product

Explanation: The multiplication combines both the dot product and cross product of the vector parts, resulting in a new quaternion with both scalar and vector components.

3. Applications of Quaternions

Details: Quaternions are widely used in computer graphics, robotics, flight dynamics, and quantum mechanics. They provide an efficient way to represent and calculate 3D rotations without suffering from gimbal lock.

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 standard quaternion multiplication rules.

5. Frequently Asked Questions (FAQ)

Q1: Why is quaternion multiplication non-commutative?
A: The cross product operation in the multiplication formula is non-commutative, making the entire operation order-dependent (q₁ × q₂ ≠ q₂ × q₁ in general).

Q2: How are quaternions different from complex numbers?
A: Quaternions extend complex numbers to three imaginary dimensions (i, j, k) instead of just one (i), with specific multiplication rules between these units.

Q3: What are the multiplication rules for i, j, k?
A: i² = j² = k² = ijk = -1, ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j.

Q4: Why are quaternions used for rotations?
A: They provide a compact representation without singularities, allow smooth interpolation (slerp), and are computationally efficient compared to rotation matrices.

Q5: Can quaternions represent 4D rotations?
A: Yes, unit quaternions can represent both 3D and 4D rotations, though visualization becomes more complex in 4D.