I am trying to figure out the relation between the euclidean distance and the norm term.
Given two points in the plane, this is the euclidean distance I am familiar with (a result of Pythagoras theorem):
$d=\sqrt{(y_{2}-y_{1})^{2}+(x_{2}-x_{1})^{2}}$
This is the norm term which should represent the same thing:
$\left \| x \right \|=\sqrt{\left | x_{1} \right |^{2}+...+\left | x_{n} \right |^{2}}$
I don't understand why the two formulas are identical. Where did the subtraction went to ?
Thank you (and apologies for the basic question).
The distance between two points $\vec x=(x_1,...,x_n)$ and $\vec y=(y_1,...,y_n)$ in $n$-dimensional space is defined as
$$d=\sqrt{(y_1-x_1)^2+(y_2-x_2)^2+\cdots+(y_n-x_n)^2}$$ Since $\vec y - \vec x = (y_1-x_1,...,y_n-x_n)$ , we have that $d=\Vert\vec y - \vec x \Vert$.
Therefore, $\Vert \vec x \Vert$ is the distance between the points $\vec x$ and $\vec 0 = (0,...,0)$, or in other words, the length of the vector $\vec x$.