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Determine equations of perpendicular lines

Perpendicular lines intersect at a right angle; in the coordinate plane, their nonzero, defined slopes are negative reciprocals, so m1m2=−1m_1m_2=-1. The learner determines an equation through a specified point by finding the perpendicular slope and expressing the line in point-slope, slope-intercept, or standard form, while handling the special case that horizontal lines are perpendicular to vertical lines and have undefined slope; three-dimensional and vector-based generalizations are not included.

Detailed Explanation: 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−y1=m(x−x1).y-y_1=m(x-x_1).
  1. Simplify if needed.

Example

Find the equation of the line perpendicular to

2x−3y=62x-3y=6

that passes through (4,1)(4,1).

Step 1: Find the slope of the given line.

Rewrite the equation in slope-intercept form:

2x−3y=6−3y=−2x+6y=23x−2\begin{aligned} 2x-3y&=6\\ -3y&=-2x+6\\ y&=\frac{2}{3}x-2 \end{aligned}

The slope is

m=23.m=\frac{2}{3}.

Step 2: Find the perpendicular slope.

Perpendicular slopes are negative reciprocals. The reciprocal of 23\frac{2}{3} is 32\frac{3}{2}, so change its sign:

m⊥=−32.m_{\perp}=-\frac{3}{2}.

Step 3: Use point-slope form.

The line must pass through (4,1)(4,1), so substitute x1=4x_1=4, y1=1y_1=1, and m=−32m=-\frac{3}{2}:

y−1=−32(x−4).y-1=-\frac{3}{2}(x-4).

This is an equation of the perpendicular line.

It can also be written in slope-intercept form:

y−1=−32x+6y=−32x+7\begin{aligned} y-1&=-\frac{3}{2}x+6\\ y&=-\frac{3}{2}x+7 \end{aligned}

Therefore, the equation is

y=−32x+7.\boxed{y=-\frac{3}{2}x+7}.

Check the slopes:

23(−32)=−1,\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=5y=5 is perpendicular to a vertical line such as x=2x=2. Vertical lines have undefined slope, so use their equations directly instead of finding a negative reciprocal.

Learn by doing: Determine equations of perpendicular lines

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