# What is the distance between (13,-23,-20) and (-3,-37,-22)?

Jun 23, 2018

See a solution process below:

#### Explanation:

The formula for calculating the distance between two points is:

$d = \sqrt{{\left(\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}\right)}^{2} + {\left(\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}\right)}^{2} + {\left(\textcolor{red}{{z}_{2}} - \textcolor{b l u e}{{z}_{1}}\right)}^{2}}$

Where $\left(\textcolor{b l u e}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}} , \textcolor{b l u e}{{z}_{1}}\right)$ and $\left(\textcolor{red}{{x}_{1}} , \textcolor{red}{{y}_{1}} , \textcolor{red}{{z}_{1}}\right)$ are two points.

Substituting the values from the points in the problem gives:

$d = \sqrt{{\left(\textcolor{red}{- 3} - \textcolor{b l u e}{13}\right)}^{2} + {\left(\textcolor{red}{- 37} - \textcolor{b l u e}{- 23}\right)}^{2} + {\left(\textcolor{red}{- 22} - \textcolor{b l u e}{- 20}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{- 3} - \textcolor{b l u e}{13}\right)}^{2} + {\left(\textcolor{red}{- 37} + \textcolor{b l u e}{23}\right)}^{2} + {\left(\textcolor{red}{- 22} + \textcolor{b l u e}{20}\right)}^{2}}$

$d = \sqrt{{\left(- 16\right)}^{2} + {\left(- 14\right)}^{2} + {\left(- 2\right)}^{2}}$

$d = \sqrt{256 + 196 + 4}$

$d = \sqrt{456}$

$d = \sqrt{4 \cdot 114}$

$d = \sqrt{4} \sqrt{114}$

$d = 2 \sqrt{114}$

Or, approximately:

$d \cong 21.354$