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 vect

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.