Quaternion product

The quaternion product is denoted by the symbol \( \otimes \). Consider two quaternions \( Q_1 \) and \( Q_2 \) :

$$ Q_1 = \left[ \begin{matrix} a_1 && b_1 && c_1 && d_1 \end{matrix} \right] $$ $$ Q_2 = \left[ \begin{matrix} a_2 && b_2 && c_2 && d_2 \end{matrix} \right] $$

The product is given by:

$$ Q_1 \otimes Q_2 = \left[ \begin{matrix} a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2 \\ a_1b_2 + b_1a_2 + c_1d_2 - d_1c_2 \\ a_1c_2 - b_1d_2 + c_1a_2 + d_1b_2 \\ a_1d_2 + b_1c_2 - c_1b_2 + d_1a_2 \end{matrix} \right]^\top $$

The quaternion product is not commutative :

$$ Q_1 \otimes Q_2 \neq Q_2 \otimes Q_1 $$

See also

• Quaternions and gyroscope
• Quaternion conjugate
• Quaternion normalization
• Quaternion to rotation matrix
• Quaternions and rotations

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