Suppose I want to know the length of of the $n-$dimensional vector $(x_1,x_2,\dots,x_n)$. It goes as:
$$\sqrt{x_1^2 + x_2^2 +\dots+ x_n^2}\tag{1}$$
Which is analog to the way we can measure the length of a diagonal segment with the pythagorean theorem:
$$c^2=a^2+b^2 $$
$$ c=\sqrt{s^2+b^2}\tag{2}$$
But what always puzzled me was that we go from $(2)$ to $(1)$ in a pretty arbitrary way. It seemed like people added the remaining coordinates just because it was similar somehow. I always wanted to figure out what was the true justification for this. I thought about the following: Suppose we have the $3-$dimensional vector $(x,y,z)$, its length will be the expected one but it seems that the way we measure it is the following:
- First we measure the length of the first two components: $\lambda =\sqrt{x^2+y^2}$
- Now we think in the case of two dimensions and act as if $\lambda$ is one coordinate and $z$ is another and hence we have: $\sqrt{\lambda^2+z^2}=\sqrt{x^2+y^2+z^2}$.
So it seems that when we are taking the length of an $n-$dimensional vector, we are actually "recursively taking the length" of a $2-$dimensional vector. And for $(1)$, the procedure follows:
$$\sqrt{\left(\sqrt{\left(\sqrt{\left(\sqrt{x_1^2 + x_2^2}\right)^2 +\dots+x_{n-2}^2}\right)^2+x_{n-1}^2}\right)^2+ x_n^2} =\sqrt{x_1^2 + x_2^2 +\dots+ x_n^2} $$
Is it known if this is the original intention of this definition of distance? If yes, is this the original intention of this definition for distance? Sorry if the question is too stupid, but I never found about it on my textbooks, the first time I found this was at these lectures.