# What is the distance between (13,-14,1) and (12,-21,6)?

May 14, 2016

$d = 5 \sqrt{3} \text{ }$ Exactly

$d = 8.660 \text{ }$ to 3 decimal places

#### Explanation:

This is the coordinate system for 3-space. When it is broken down into projections onto the planes you end up with two triangles that are linked by a common side. Consequently we can turn to that good old favourite: Pythagoras.

Let the distance between points be $d$

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2} + {\left({z}_{2} - {z}_{1}\right)}^{2}}$

$d = \sqrt{{\left(12 - 13\right)}^{2} + {\left(- 21 - \left(- 14\right)\right)}^{2} + {\left(6 - 1\right)}^{2}}$

$d = \sqrt{75} = \sqrt{{5}^{2} \times 3}$
$d = 5 \sqrt{3} \text{ }$ Exactly

$d = 8.660 \text{ }$ to 3 decimal places