# What is the distance between (-2,1,-3) and (5,6,-2)?

Jan 9, 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:

$d = \sqrt{{\left(\textcolor{red}{5} - \textcolor{b l u e}{- 2}\right)}^{2} + {\left(\textcolor{red}{6} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{- 2} - \textcolor{b l u e}{- 3}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{5} + \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{6} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{- 2} + \textcolor{b l u e}{3}\right)}^{2}}$

$d = \sqrt{{7}^{2} + {5}^{2} + {1}^{2}}$

$d = \sqrt{49 + 25 + 1}$

$d = \sqrt{75}$

$d = \sqrt{25 \cdot 3}$

$d = \sqrt{25} \sqrt{3}$

$d = 5 \sqrt{3}$

Or

$d = 8.660$ rounded to the nearest thousandth.