Quaternion Modulus of Conjugate

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Theorem

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

Let $\overline z$ be the conjugate of $z$.

Let $\cmod z$ be the quaternion modulus of $z$.


Then:

$\cmod {\overline z} = \cmod z$


Proof

\(\displaystyle \cmod z\) \(=\) \(\displaystyle a^2 + b^2 + c^2 + d^2\) Definition of Quaternion Modulus
\(\displaystyle \cmod {\overline z}\) \(=\) \(\displaystyle \cmod {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}\) Definition of Quaternion Conjugate
\(\displaystyle \) \(=\) \(\displaystyle a^2 + \paren {-b}^2 + \paren {-c}^2 + \paren {-d}^2\) Definition of Quaternion Modulus
\(\displaystyle \) \(=\) \(\displaystyle a^2 + b^2 + c^2 + d^2\)

$\blacksquare$