How Josiah Willard Gibbs introduced the notions of dot product and vector product

Question

Before posting this question, I have realised that this question might be repeating the same topic which some users have asked before, but still I cannot find a satisfying answer. So please forgive me for repeating this question: How did J.W. Gibbs invent dot product and vector product? (Certainly the definition |v||w|cos θ cannot just come out from nowhere, the proof: ∥v−w∥^2=∥w∥^2+∥v∥^2−2∥v∥∥w∥ cos θ isn't helpful either, why dot product is used in this proof instead of cross product? v.w instead of v x w) Did he base on other people's works to introduce these notions? Can someone please briefly explain the history about the origin of these products. Are there any books that talk about the history of vector? Thanks.

Answer 1

Dot product and vector product are just a different presentation of the product of quaternions.

Quaternions were invented by Hamilton as a result of a twenty year long attempt at generalizing the $\mathbb R$-algebra $\mathbb C$ of complex numbers.
In modern notation , if two quaternions $q,q'$ are presented as the sum of a scalar in $\mathbb R $ and a vector in $3$-space, $q=a+\vec v, p=b+\vec {w}$, then we have for their quaternionic product : $$ q\cdot q'=(ab-\langle \vec v,\vec {w}\rangle) + ( a\vec {w} +b\vec {v}+\vec {v}\times\vec {w} ) $$ and in particular we have the fundamental relation for two pure quaternions, that is quaternions with zero scalar part: $$ \vec v\cdot \vec {w}= - \langle \vec v,\vec {w}\rangle + \vec {v}\times\vec {w} $$ [I have written the scalar product product as $\langle \vec v,\vec w\rangle $ in order not to confuse it with the quaternionic product $v\cdot w$]

So Gibbs's contribution was to notice that, given two pure quaternions, it is very convenient in their quaternionic product to isolate the scalar part, yielding the scalar product up to sign, and the vector part, yielding the cross product.
A great psychological advantage is that now everything takes place in $\mathbb R^3$ and there is no need to resort to $\mathbb R^4$, the underlying vector space of the $\mathbb R$-algebra of Hamilton's quaternions: I suppose that at the time four-dimensional space was regarded with some unease by physicists like Gibbs and by engineers.