Why do we use i to represent phasors and/or vectors?

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The representation of vectors and phasors using the imaginary unit "i" is rooted in the convention of complex numbers, where a vector is expressed as a sum of real and imaginary components, such as 3 + 4i. This notation facilitates the superposition of waves and simplifies calculations involving wave interactions. The historical evolution of this convention traces back to the use of quaternions and the eventual adoption of "i," "j," and "k" in vector mathematics, leading to the streamlined representation of vectors in modern applications.

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Somestimes, vectors (e.g <3,4>) are represented with a real and imaginary component (3 + 4i). When expressing wave phasors as vectors, this is usually the convention.

Insofar as phasors and waves go, what rationale does the convention have? If I had to superimpose two waves together, and perform an assortment of calculations, what role would the "i" term carry?
 
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I think the convention came from complex numbers use of i and then when quaternions were the rage the notion of i, j, k was used and later when vectors became more popular the real part was dropped and just the i, j, k remained.
 
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