# What is the distance between (10,5,-2) and (12,11,5)?

Apr 29, 2017

See the 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:

$d = \sqrt{{\left(\textcolor{red}{12} - \textcolor{b l u e}{10}\right)}^{2} + {\left(\textcolor{red}{11} - \textcolor{b l u e}{5}\right)}^{2} + {\left(\textcolor{red}{5} - \textcolor{b l u e}{- 2}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{12} - \textcolor{b l u e}{10}\right)}^{2} + {\left(\textcolor{red}{11} - \textcolor{b l u e}{5}\right)}^{2} + {\left(\textcolor{red}{5} + \textcolor{b l u e}{2}\right)}^{2}}$

$d = \sqrt{{2}^{2} + {6}^{2} + {7}^{2}}$

$d = \sqrt{4 + 36 + 49}$

$d = \sqrt{89} = 9.434$ rounded to the nearest thousandth.