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

What is the distance between (13,-14,1) and (12,-21,6)?
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Yasminru
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
$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}$
$d=\sqrt{{\left(12-13\right)}^{2}+{\left(-21-\left(-14\right)\right)}^{2}+{\left(6-1\right)}^{2}}$
$d=\sqrt{75}=\sqrt{{5}^{2}×3}$
$d=5\sqrt{3}\phantom{\rule{1ex}{0ex}}\text{}\phantom{\rule{1ex}{0ex}}$
$d=8.660\phantom{\rule{1ex}{0ex}}\text{}\phantom{\rule{1ex}{0ex}}$ to 3 decimal places