# Quaternion normalization

Let's consider the following quaternion:

$$Q=a+b.i+c.j+d.k = \left[ \begin{matrix} a && b && c && d \end{matrix} \right]$$

The normalization is given by the following formula:

$$Q_{normalized}=\frac{Q}{| Q | }$$

where:

$$| Q | = \sqrt{ Q.\overline{Q} } = \sqrt{ \overline{Q}.Q }=\sqrt {a^2 + b^2 + c^2 + d^2}$$

The general representation is given by:

$$Q_{normalized}= \left[ \begin{matrix} \dfrac{a}{| Q |} && \dfrac{b}{| Q |} && \dfrac{c}{| Q |} && \dfrac{d}{| Q |} \end{matrix} \right]$$

$$Q_{normalized}= \left[ \begin{matrix} A && B && C && D \end{matrix} \right]$$

with: $$A = \dfrac{a}{\sqrt {a^2 + b^2 + c^2 + d^2}}$$ $$B = \dfrac{b}{\sqrt {a^2 + b^2 + c^2 + d^2}}$$ $$C = \dfrac{c}{\sqrt {a^2 + b^2 + c^2 + d^2}}$$ $$D = \dfrac{d}{\sqrt {a^2 + b^2 + c^2 + d^2}}$$

\begin{multline} Q_{normalized}= \left[ \begin{matrix} \dfrac{a}{\sqrt {a^2 + b^2 + c^2 + d^2}} \end{matrix}\right. \\ \dfrac{b}{\sqrt {b^2 + b^2 + c^2 + d^2}} \quad \dfrac{c}{\sqrt {a^2 + b^2 + c^2 + d^2}} \\ \left. \begin{matrix} \dfrac{d}{\sqrt {a^2 + b^2 + c^2 + d^2}} \end{matrix} \right] \end{multline}