# What is the distance between (–1, 1, 3)  and (–5, –1, 1) ?

Mar 7, 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:

$d = \sqrt{{\left(\textcolor{red}{- 5} - \textcolor{b l u e}{- 1}\right)}^{2} + {\left(\textcolor{red}{- 1} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{1} - \textcolor{b l u e}{3}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{- 5} + \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{- 1} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{1} - \textcolor{b l u e}{3}\right)}^{2}}$

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

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

$d = \sqrt{24}$

$d = \sqrt{4 \cdot 6}$

$d = \sqrt{4} \sqrt{6}$

$d = 2 \sqrt{6}$

$d = 4.899$ rounded to the nearest thousandth