# What is the distance between (2,-4) and (-10,1)?

Dec 30, 2015

The distance between $\left(2 , - 4\right)$ and $\left(- 10 , 1\right)$ is $13 u n i t s .$

Dec 30, 2015

$13$

#### Explanation:

Assuming these 2 points, call them $x \mathmr{and} y ,$ are in ${\mathbb{R}}^{2}$ which is a complete metric space and a complete normed space, we may use either the normal Euclidean metric or the metric induced by the norm to evaluate the distance.

Normal Euclidean metric:

$d \left(x , y\right) = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}^{2}\right)}^{2}}$

$= \sqrt{{\left(2 - \left(- 10\right)\right)}^{2} + {\left(- 4 - 1\right)}^{2}}$

$= 13$.

#Metric induced by the norm:

$d \left(x , y\right) = | | x - y | |$

$= \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}^{2}\right)}^{2}}$

$= 13$.