# What is the distance between  (–2, 1, 3)  and (0, 4, –2) ?

Mar 3, 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:

$d = \sqrt{{\left(\textcolor{red}{0} - \textcolor{b l u e}{- 2}\right)}^{2} + {\left(\textcolor{red}{4} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{- 2} - \textcolor{b l u e}{3}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{0} + \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{4} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{- 2} - \textcolor{b l u e}{3}\right)}^{2}}$

$d = \sqrt{{2}^{2} + {3}^{2} + {\left(- 5\right)}^{2}}$

$d = \sqrt{4 + 9 + 25}$

$d = \sqrt{38}$

Or

$d = 6.164$ rounded to the nearest thousandth.