Vector Lengths and Distances

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:

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

$L_p$ distance

The $L_p$ distance is a generalization with differnt $p$ coefficient of the Euclidean distance.

$$\Vert x-y{\Vert }_p={\left(\sum _{i=1}^d(x_i-y_i{)}^p\right)}^{\frac{1}{p}}$$

$L_p$ distance in $R^2$ with different values for $p$.

Note

Note that the number of dimensions $k$ for points in ${\mathbb{R}}^k$ and the $p$ coefficient are two different things.