Slope of Perpendicular Lines: The −1 Rule

Slope of Perpendicular Lines: The −1 Rule

TL;DR

The slope of perpendicular lines follows one rule: the product of the two slopes is −1, so m₁⋅m₂=−1. This article covers the negative-reciprocal rule, why it holds, how to find a perpendicular slope from any given slope, the vertical-horizontal exception, and six worked examples.

What Is the Slope of Perpendicular Lines?

Perpendicular lines are two lines that cross at a right angle, exactly 90°. The slope of perpendicular lines is governed by a single relationship: when you multiply the slope of one line by the slope of the other, the result is always −1. Slope here means the same thing it always does, the change in y for each unit of change in x, written m=\dfrac{\text{rise}}{\text{run}}.

That product rule has a second, equivalent name. Because m₁⋅m₂=−1 rearranges to m₂=−\dfrac{1}{m₁}, each slope is the negative reciprocal of the other. Reciprocal means "flipped fraction"; negative means "opposite sign."

The Negative Reciprocal Rule

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1:

m₁⋅m₂=−1.

To build a perpendicular slope from a given one, do two separate things, in either order: flip the fraction, then switch the sign.

Notice the sign always ends up opposite: a positive slope's perpendicular partner is negative, and a negative slope's partner is positive.

Why Does the Product Have to Be −1?

Rotating any line by exactly 90° does one clean thing to its rise and run: it swaps them and flips one sign. A line that goes up a for every run of b has slope \dfrac{a}{b}. Turn that direction a quarter-turn, and the same motion now goes b in one axis and a in the other, with one of them reversed, giving slope −\dfrac{b}{a}.

Multiply the original slope by this rotated slope:

\dfrac{a}{b} \cdot \left(-\dfrac{b}{a}\right) = -\dfrac{ab}{ab} = -1.

How Do You Find the Slope of a Line Perpendicular to a Given Line?

  1. Get the line into slope-intercept form y=mx+b if it is not already, so you can read the slope m as the coefficient of x.
  2. Read off that slope m₁.
  3. Flip it and switch the sign to get the perpendicular slope m₂=−\dfrac{1}{m₁}.
  4. Check by confirming m₁⋅m₂=−1.

What About Vertical and Horizontal Lines?

A horizontal line has slope 0; a vertical line has an undefined slope. A horizontal line and a vertical line clearly meet at a right angle. Yet you cannot verify that with m₁⋅m₂=−1, because 0×(undefined) is not a number at all. So treat this as a known exception: a horizontal line and a vertical line are always perpendicular.

Examples of the Slope of Perpendicular Lines

Example 1 - Find the slope of a line perpendicular to a line with slope 5.

Example 2 - Check the claim of a perpendicular slope. The intuitive read is "just flip the sign, so −4." The correct is to flip 4 to \dfrac{1}{4} and switch to −\dfrac{1}{4}. Check and confirm it.

Example 3 - For y=−\dfrac{3}{7}x+2, flip to -\dfrac{7}{3}, switch to \dfrac{7}{3}. Check: −\dfrac{3}{7}⋅\dfrac{7}{3} = −1.

Example 4 - A line is given by 3x+y=8. Rearranging gives y=−3x+8, so slope is −3. Flip to −\dfrac{1}{3}, switch to \dfrac{1}{3}. Check: −3⋅\dfrac{1}{3} = -1.

Example 5 - Write the equation of the line perpendicular to y=2x+1 that passes through the point (4,−3). The slope of the line is 2, so the perpendicular slope is −\dfrac{1}{2}. Use point-slope form: y+3=−\dfrac{1}{2}(x−4).

Example 6 - A vertical line x=5 and a horizontal line y=−2 are perpendicular; the product rule cannot be applied.

Where Students Trip Up on Perpendicular Slopes

  1. Mistake: Switching the sign but forgetting to flip.
  2. Mistake: Reading the slope before solving for y.
  3. Mistake: Mishandling the reciprocal of an integer or negative fraction.

Key Takeaways