Definition:Quaternion/Multiplication

Definition

The product of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as:

\(\ds \mathbf x_1 \mathbf x_2\)

\(:=\)

\(\ds \paren {a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2} \mathbf 1\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \paren {a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2} \mathbf i\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \paren {a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2} \mathbf j\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \paren {a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2} \mathbf k\)

Also see

• Quaternion Multiplication, where this construction is proved to be consistent with the definition of quaternions.

Sources

• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem