# How do you find the slope of a line perpendicular to a slope of a line is 1/3?

May 20, 2015

In general, if the slope of a line is $m$, then the slope of any perpendicular line will be $- \frac{1}{m}$. So in your case the slope of any perpendicular line would be $- \frac{1}{\frac{1}{3}} = - 3$.

To see that, consider a line given by

$y = m x + c$

If you reflect that line in the line $y = x$, you get a line whose equation is the same, but with $x$ and $y$ swapped:

$x = m y + c$

If you then reflect that line in the $x$ axis, you are basically reversing the sign of the $y$ coordinate, so you get a line with equation:

$x = - m y + c$

The total result of these two geometric operations is to rotate the original line through a right angle (try it yourself with a square of paper).

Now we can rearrange this new line's equation into slope intercept form as follows:

Add $m y$ to both sides:

$m y + x = c$

Subtract $x$ from both sides:

$m y = - x + c$

Divide both sides by $m$:

$y = \left(- \frac{1}{m}\right) x + \frac{c}{m}$

Notice the new slope is $- \frac{1}{m}$.