# What is the distance between (2,-4,6) and (-9,-5,9)?

Jun 14, 2018

See a solution process below:

#### Explanation:

The formula for calculating the distance between two points is:

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

Substituting the values from the points in the problem gives:

$d = \sqrt{{\left(\textcolor{red}{- 9} - \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{- 5} - \textcolor{b l u e}{- 4}\right)}^{2} + {\left(\textcolor{red}{9} - \textcolor{b l u e}{6}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{- 9} - \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{- 5} + \textcolor{b l u e}{4}\right)}^{2} + {\left(\textcolor{red}{9} - \textcolor{b l u e}{6}\right)}^{2}}$

$d = \sqrt{{\left(- 11\right)}^{2} + {\left(- 1\right)}^{2} + {\left(3\right)}^{2}}$

$d = \sqrt{121 + 1 + 9}$

$d = \sqrt{131}$

Or, approximately:

$d \cong 11.4455$