newtonian mechanics - How is the dot product a generalization of multiplication?

I've seen an interesting explanation for lots of what I previously thought were unmotivated definitions in Newtonian mechanics, namely that power is always defined as effort times flow. But when trying to define power in dynamics obviously you need to deal with vectors, hence my question : how is the dot product the good generalization of multiplication in $\mathbb{R}$ to space? Why not any other inner product on $\mathbb{R}^3$ that does reduce to multiplication on $\mathbb{R}$? I know it's sort of the canonical inner product on $\mathbb{R}^3$, has that something to do with it?