Quaternion and normalization

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I know that quaternions need to be normalized if I want to rotate a vector.

But are there any reasons to not automatically normalize a quaternion? And if there are, what quaternion operations do result in non-normalized quaternions?

  • Multiplying two quaternions?
  • Dot product?

Sorry, if this question is a little bit fuzzy. I'm still trying wrap my head around quaternions.

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