# What is the distance between (-4,-3,4) and (-30,15,-16)?

May 31, 2018

$\quad \textcolor{red}{d = 10 \sqrt{14}}$ or $\textcolor{red}{\approx 37.417}$ (rounded to thousandth's place)

#### Explanation:

The distance between three dimensions is similar to the distance between two dimensions.

We use the formula:
$\quad \textcolor{red}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2} + {\left({z}_{2} - {z}_{1}\right)}^{2}}}$, where $x$, $y$, and $z$ are the coordinates.

Let's plug in the values for the coordinates into the formula. Pay attention to the negative signs:
$\quad d = \sqrt{{\left(- 30 - \left(- 4\right)\right)}^{2} + {\left(15 - \left(- 3\right)\right)}^{2} + {\left(- 16 - 4\right)}^{2}}$

And now simplify:
$\quad d = \sqrt{{\left(- 26\right)}^{2} + {\left(18\right)}^{2} + {\left(- 20\right)}^{2}}$

$\quad d = \sqrt{676 + 324 + 400}$

$\quad d = \sqrt{1400}$

$\quad d = \sqrt{100 \cdot 14}$

$\quad d = \sqrt{100} \sqrt{14}$

$\quad d = 10 \sqrt{14}$

$\quad \textcolor{red}{d = 10 \sqrt{14}}$ or $\textcolor{red}{\approx 37.417}$ (rounded to thousandth's place)

Hope this helps!