TITLE: Origin of the dot and cross product?
QUESTION [60 upvotes]: Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product equation as a determinant of a carefully-constructed matrix, 
but what I want to ask is... How did the dot and cross product come to be? When were they "invented"? Some detailed proofs? Did someone say: "Hey, wouldn't it be nice if we could construct a way to calculate a vector that is perpendicular to two given operands?"
Basically, how/why do they work?
I would appreciate explanations, links to other explanations, other web resources... I've been searching the Internet lately for explanations, but most of them are on how to use it and nothing that really gives substance to it.

REPLY [4 votes]: It is Sir William R. Hamilton who originally touched upon the concept. Near the end of his letter to John T. Graves about his recent discovery of quaternions in the same year (1843), he writes the following:

Multiplication will be easy if we are familiar with the rules for the product of two pure imaginaries. This product is, by (B.),
$$(0, b, c, d)(0, b′, c′, d′) = (−bb′ − cc′ − dd′, cd′ − dc′, db′ − bd′, bc′ − cb′);$$
the product-line is perpendicular to the plane of the factors; its length is the product of their lengths multiplied by the sine of the angle between them: and the real part of the product, with its sign changed, is the same product of the lengths of the factors multiplied by the cosine of their inclination.

In essence, he recognized that (using modern notation for quaternions):
$$(0, \vec{v})(0, \vec{w}) = (-\Vert \vec{v} \Vert \Vert \vec{w} \Vert \cos(\theta), \Vert \vec{v} \Vert \Vert \vec{w} \Vert \sin(\theta) \vec{k})$$
where $\theta$ is the angle between $\vec{v}$ and $\vec{w}$, and $\vec{k}$ is perpendicular to $\vec{v}$ and $\vec{w}$.
This discovery precedes Gibbs and Heaviside, who were born in 1839 and 1850 respectively.