Skill: Determine equations of perpendicular lines

To determine an equation of a line perpendicular to a given line:

1. Find the slope of the given line.
2. Find its negative reciprocal.
3. Use the new slope and the specified point in point-slope form:

$$y-y_1=m(x-x_1).$$

4. Simplify if needed.

Example

Find the equation of the line perpendicular to

$$2x-3y=6$$

that passes through $(4,1)$.

Step 1: Find the slope of the given line.

Rewrite the equation in slope-intercept form:

$$\begin{aligned} 2x-3y&=6\\ -3y&=-2x+6\\ y&=\frac{2}{3}x-2 \end{aligned}$$

The slope is

$$m=\frac{2}{3}.$$

Step 2: Find the perpendicular slope.

Perpendicular slopes are negative reciprocals. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$, so change its sign:

$$m_{\perp}=-\frac{3}{2}.$$

Step 3: Use point-slope form.

The line must pass through $(4,1)$, so substitute $x_1=4$, $y_1=1$, and $m=-\frac{3}{2}$:

$$y-1=-\frac{3}{2}(x-4).$$

This is an equation of the perpendicular line.

It can also be written in slope-intercept form:

$$\begin{aligned} y-1&=-\frac{3}{2}x+6\\ y&=-\frac{3}{2}x+7 \end{aligned}$$

Therefore, the equation is

$$\boxed{y=-\frac{3}{2}x+7}.$$

Check the slopes:

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

so the lines are perpendicular.

If the given line is horizontal, its perpendicular line is vertical. For example, a line with equation $y=5$ is perpendicular to a vertical line such as $x=2$. Vertical lines have undefined slope, so use their equations directly instead of finding a negative reciprocal.