Skill: Determine whether lines are perpendicular using slope

To determine whether two nonvertical lines are perpendicular:

1. Find the slope of each line.
2. Check whether the slopes are negative reciprocals.
3. Equivalently, multiply the slopes. If

$$m_1m_2=-1,$$

the lines are perpendicular.

Remember: negative reciprocals have signs that are opposite and fractions that are flipped. For example, the negative reciprocal of $\frac{2}{3}$ is $-\frac{3}{2}$, not $-\frac{2}{3}$.

Example: Are the lines

$$y=\frac{2}{3}x-1$$

and

$$y=-\frac{3}{2}x+4$$

perpendicular?

The equations are in slope-intercept form, $y=mx+b$, so the slopes are the coefficients of $x$:

$$m_1=\frac{2}{3}, \qquad m_2=-\frac{3}{2}.$$

Now multiply the slopes:

$$m_1m_2=\frac{2}{3}\left(-\frac{3}{2}\right)=-1.$$

Because the product of the slopes is $-1$, the lines are perpendicular.

A horizontal line and a vertical line are also perpendicular. A vertical line does not have a defined slope, so use this special case instead of multiplying slopes.