Yes, the slope of a perpendicular line is the opposite reciprocal of the original line’s slope. If one line has a slope of 2/3, any line perpendicular to it has a slope of -3/2. You flip the fraction and change the sign. This relationship holds for all lines except the special case of vertical and horizontal lines.
What “Opposite Reciprocal” Means
The term has two parts, and you apply both steps to find a perpendicular slope. “Reciprocal” means you flip the fraction: 2/3 becomes 3/2. “Opposite” means you switch the sign: positive becomes negative, or negative becomes positive. So the opposite reciprocal of 2/3 is -3/2.
You might also see this called the “negative reciprocal.” The two terms mean exactly the same thing and are used interchangeably in math courses. Both describe a number that, when multiplied by the original, gives a product of -1.
The Product Test: m₁ × m₂ = -1
There’s a quick way to check whether two lines are perpendicular: multiply their slopes together. If the product is -1, the lines are perpendicular. This works because the opposite reciprocal is, by definition, the number that produces -1 when multiplied by the original.
For example, take slopes of 2 and -1/2. Multiply them: 2 × (-1/2) = -1. That confirms the lines are perpendicular. If you had slopes of 3 and -3, the product would be -9, not -1. Those lines are not perpendicular. The signs are opposite, but the values aren’t reciprocals of each other.
Common Slope Pairs
Here are some examples that show up frequently in homework and exams:
- Slope of 2 → perpendicular slope is -1/2. Flip 2/1 to get 1/2, then switch the sign.
- Slope of -1/4 → perpendicular slope is 4. Flip 1/4 to get 4/1, then switch the negative to positive.
- Slope of 3/2 → perpendicular slope is -2/3.
- Slope of -4/5 → perpendicular slope is 5/4.
- Slope of 1 → perpendicular slope is -1. The reciprocal of 1 is still 1, so only the sign changes.
- Slope of -3/2 → perpendicular slope is 2/3. Flip the fraction and change negative to positive.
A common mistake is only doing one of the two steps. Changing 2/3 to -2/3 gives you the opposite but not the reciprocal, and those lines aren’t perpendicular. Changing 2/3 to 3/2 gives you the reciprocal but not the opposite, and those lines aren’t perpendicular either. You need both.
Why This Works Geometrically
The relationship isn’t an arbitrary rule. It comes from what happens when you rotate a line 90 degrees. Picture a line passing through the origin with slope a/b, meaning it rises a units for every b units it runs to the right. When you rotate that line 90 degrees counterclockwise, the rise and run swap roles: the new line runs a units but drops b units. The new slope is -b/a, which is the opposite reciprocal of a/b.
You can also see it through the Pythagorean theorem. If two lines meet at a right angle at the origin, you can draw a vertical reference line to form a right triangle. Working through the distance formula and the Pythagorean theorem confirms that the only way the angle between the lines equals 90 degrees is when their slopes multiply to -1.
The Exception: Vertical and Horizontal Lines
A horizontal line has a slope of 0, and a vertical line has an undefined slope. These two are perpendicular to each other, but the m₁ × m₂ = -1 test doesn’t work here because you can’t multiply by an undefined value. There’s no way to take the opposite reciprocal of 0 either, since flipping 0/1 to 1/0 is undefined.
So the opposite reciprocal rule applies to all perpendicular lines except this one pair. If you’re working with vertical and horizontal lines, you just need to recognize them by their equations: y = k is horizontal, and x = h is vertical. Whenever one of each appears together, they’re perpendicular by definition.
How to Find a Perpendicular Line’s Equation
If you need to write the equation of a line perpendicular to a given line and passing through a specific point, the process has three steps. First, identify the slope of the original line. Second, find the opposite reciprocal of that slope. Third, plug the new slope and the given point into point-slope form: y – y₁ = m(y – x₁).
Say you need a line perpendicular to y = 2x + 5 that passes through the point (4, 1). The original slope is 2. The opposite reciprocal is -1/2. Using point-slope form: y – 1 = -1/2(x – 4). Simplify to get y = -1/2x + 3. You can verify by multiplying the slopes: 2 × (-1/2) = -1.