# What is the distance between (–6, 3, 4)  and (4, –1, 2)?

Feb 23, 2016

I assume that you know the distance formula (square root of sum of corresponding coordinates squared)

Well, that formula can actually be EXTENDED to the third dimension. (This is a very powerful thing in future mathematics)

What that means is that instead of the known $\sqrt{{\left(a - b\right)}^{2} + {\left(c - d\right)}^{2}}$

We can extend this to be $\sqrt{{\left(a - b\right)}^{2} + {\left(c - d\right)}^{2} + {\left(e - f\right)}^{2}}$

This problem is beginning to look a lot easier huh?

We can just plug in the corresponding values into the formula

$\sqrt{{\left(- 6 - 4\right)}^{2} + {\left(3 - \left(- 1\right)\right)}^{2} + {\left(4 - 2\right)}^{2}}$

$\sqrt{{\left(- 10\right)}^{2} + {\left(4\right)}^{2} + {\left(2\right)}^{2}}$

$\sqrt{100 + 16 + 4}$

$\sqrt{120}$

which is equal to $2 \sqrt{30}$

And we are done.