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:

$$ Q_{normalized}= \left[ \begin{matrix} \frac{a}{\sqrt {a^2 + b^2 + c^2 + d^2}} && \frac{b}{\sqrt {a^2 + b^2 + c^2 + d^2}} && \frac{c}{\sqrt {a^2 + b^2 + c^2 + d^2}} && \frac{d}{\sqrt {a^2 + b^2 + c^2 + d^2}} \end{matrix} \right] $$

See also

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

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