# What is the distance between (3,-2,-12) and (5,-8,-16)?

Feb 8, 2017

The distance between the points is $\sqrt{56}$ or $7.48$ rounded to the nearest hundredth.

#### 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 and calculating gives:

$d = \sqrt{{\left(\textcolor{red}{5} - \textcolor{b l u e}{3}\right)}^{2} + {\left(\textcolor{red}{- 8} - \textcolor{b l u e}{- 2}\right)}^{2} + {\left(\textcolor{red}{- 16} - \textcolor{b l u e}{- 12}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{5} - \textcolor{b l u e}{3}\right)}^{2} + {\left(\textcolor{red}{- 8} + \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{- 16} + \textcolor{b l u e}{12}\right)}^{2}}$

$d = \sqrt{{2}^{2} + {\left(- 6\right)}^{2} + {\left(- 4\right)}^{2}}$

$d = \sqrt{4 + 36 + 16}$

$d = \sqrt{56} = 7.48$ rounded to the nearest hundredth.