Quaternion Modulus in Terms of Conjugate

Theorem

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.

Let $\size {\mathbf x}$ be the modulus of $\mathbf x$.

Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.

Then:

$\size {\mathbf x}^2 \mathbf 1 = \mathbf x \overline {\mathbf x}$

Proof

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$.

Then:

\(\ds \mathbf x \overline {\mathbf x}\)

\(=\)

\(\ds \paren {a^2 + b^2 + c^2 + d^2} \mathbf 1\)

Product of Quaternion with Conjugate

\(\ds \)

\(=\)

\(\ds \size {\mathbf x}^2 \mathbf 1\)

Definition of Quaternion Modulus

$\blacksquare$