# What is the distance between (1,-3,2) and (5,4,-3)?

Feb 22, 2017

The distance between the two points is $\sqrt{90}$ or $9.487$ rounded to the nearest thousandth.

#### 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:

$d = \sqrt{{\left(\textcolor{red}{5} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{4} - \textcolor{b l u e}{- 3}\right)}^{2} + {\left(\textcolor{red}{- 3} - \textcolor{b l u e}{2}\right)}^{2}}$

$d = \sqrt{{\left(\textcolor{red}{5} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{4} + \textcolor{b l u e}{3}\right)}^{2} + {\left(\textcolor{red}{- 3} - \textcolor{b l u e}{2}\right)}^{2}}$

$d = \sqrt{{4}^{2} + {7}^{2} + {\left(- 5\right)}^{2}}$

$d = \sqrt{16 + 49 + 25}$

$d = \sqrt{90} = 9.487$