Euclidean Distance

The Euclidean distance measures the length of a straight line connecting two points.

Pythagoras’ theorems says:

$$d(a,b)=(|x_b-x_a{|}^2+|y_b-y_a{|}^2{)}^{\frac{1}{2}}$$

In matrix notation we can write the same thing:

$$d(a,b)=\Vert a-b{\Vert }_2$$

The Euclidean distance can also be called the $L_2$-Norm.

The norm (or length) of a vector is simply its distance from the origin:

$$\Vert x-0{\Vert }_2=\Vert x{\Vert }_2=\sqrt{\sum _{i=1}^k|x_i{|}^2}=\sqrt{x^T\cdot x}$$