# What is the distance between (13,-23,-20) and (3,-17,-12)?

May 22, 2017

See a solution process below:

#### Explanation:

The formula for calculating the distance between two points is:

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

Substituting the values from the points in the problem gives:

$d = \sqrt{{\left(\textcolor{red}{3} - \textcolor{b l u e}{13}\right)}^{2} + {\left(\textcolor{red}{- 17} - \textcolor{b l u e}{- 23}\right)}^{2} + {\left(\textcolor{red}{- 12} - \textcolor{b l u e}{- 20}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{3} - \textcolor{b l u e}{13}\right)}^{2} + {\left(\textcolor{red}{- 17} + \textcolor{b l u e}{23}\right)}^{2} + {\left(\textcolor{red}{- 12} + \textcolor{b l u e}{20}\right)}^{2}}$

$d = \sqrt{{\left(- 10\right)}^{2} + {6}^{2} + {8}^{2}}$

$d = \sqrt{100 + 36 + 64}$

$d = \sqrt{200}$

$d = 10 \sqrt{2}$

$d = 14.142$ rounded to the nearest thousandth.