# What is the distance between (3,5,-2) and (-8,5,4)?

Nov 29, 2015

$d = \sqrt{157} \approx 12.53$

#### Explanation:

Recall the very useful formula to calculate the distance in 2 dimensions i.e: between 2 points:$\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right)$:
$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

In 3 dimensional space the distance between 3 points is calculated by adding 3rd dimension to the above formula, so now the distance between points:$\left({x}_{1} , {y}_{1} , {z}_{1}\right) , \left({x}_{2} , {y}_{2} , {z}_{2}\right)$ is:
$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2} + {\left({z}_{2} - {z}_{1}\right)}^{2}}$

In this case the points are: (3,5,−2),(−8,5,4) so we have:
$d = \sqrt{{\left(- 8 - 3\right)}^{2} + {\left(5 - 5\right)}^{2} + {\left(4 - \left(- 2\right)\right)}^{2}}$
$d = \sqrt{{\left(- 11\right)}^{2} + {\left(0\right)}^{2} + {\left(6\right)}^{2}}$
$d = \sqrt{121 + 0 + 36}$
$d = \sqrt{157}$
$d \approx 12.53$