# How do I find the slope of (1, 2, 3) and (3, 4, 5)?

Jul 20, 2017

See a solution process below:

#### Explanation:

Slope is defined as: $\text{rise"/"run}$

Calculate Run

We can first, find the run by finding the distance between the $x$ and $y$ points using the formula for distance:

$d = \sqrt{{\left(\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}\right)}^{2} + {\left(\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}\right)}^{2}}$

Substituting the $x$ and $y$ points gives the run as:

$d = \sqrt{{\left(\textcolor{red}{3} - \textcolor{b l u e}{1}\right)}^{2} + {\left(\textcolor{red}{4} - \textcolor{b l u e}{2}\right)}^{2}}$

$d = \sqrt{{2}^{2} + {2}^{2}}$

$d = \sqrt{4 + 4}$

$d = \sqrt{8}$

$d = \sqrt{4 \cdot 2}$

$d = \sqrt{4} \sqrt{2}$

$d$ or the run is $2 \sqrt{2}$

Calculate Rise

In 3-dimensions the Rise is the distance between the two $z$ points:

$\text{rise} = 5 - 3 = 2$

Calculate Slope

$\text{Slope" = "rise"/"run} = \frac{2}{2 \sqrt{2}} = \frac{1}{\sqrt{2}}$