# What is the distance between (4, –1, 2)  and (4, –4, –2)?

Apr 6, 2016

$5$

#### Explanation:

In general, the distance between points $\left({x}_{1} , {y}_{1} , {z}_{1}\right)$ and $\left({x}_{2} , {y}_{2} , {z}_{2}\right)$ is given by the distance formula:

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

In our particular example, we have ${x}_{1} = {x}_{2}$ and this simplifies to what is basically a $3 , 4 , 5$ right angled triangle, but evaluating the formula directly we get:

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

$= \sqrt{{\left(4 - 4\right)}^{2} + {\left(- 4 - \left(- 1\right)\right)}^{2} + {\left(- 2 - 2\right)}^{2}}$

$= \sqrt{{0}^{2} + {\left(- 3\right)}^{2} + {\left(- 4\right)}^{2}}$

$= \sqrt{0 + 9 + 16}$

$= \sqrt{25}$

$= 5$