Chapter 04. What Are Quaternions?

If this book were a musical composition rather than a mathematical one, perhaps it would begin with a single melodic theme carried by a solo trumpet, followed by other instruments entering in harmony. It would then move to a full ensemble—echoing, developing, and exploring dozens of variations on the original theme, some barely recognizable—and finally reach a climactic crescendo, trailing with a lone trumpet reprising the original perfect melody. In this short chapter, we present that mathematical melody: the four quaternion equations from which every theme in the entire book is in some way derived.

Quaternions are four-vectors q = (q0, q1, q2, q3) = (q0, q) to which we assign the noncommutative multiplication ...

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