The post contains no geometry. Which is the only worthwhile content of dot products.

Explaining the dot product by its implementation over R^n is pointless. Conflating 1-forms and vectors is pointless.

▲ tptacek 3 days ago | parent [-]

The only worthwhile content of dot products is geometry?

	▲	nh23423fefe 3 days ago \| parent [-]
		of course. dot products are a symmetric form on vector spaces. they let you compute the spheres of radius r. given the sphere of radius r, for any pair of vectors v,w in the sphere `-r^2 <= dot(v,w)=dot(w,v) <= r^2` as w varies from v to -v the value moves from r^2 through 0 to -r^2 this is how we define parallel perpendicular and antiparallel. the dot product is only meaningful in a geometric context. by definition it projects vectors down to scalars. fixing the scalar value finds the spheres, and for a sphere we can vary the vectors to compute cosines.