Formula Used
- If the original slope is m, then the perpendicular slope is m⊥ = −1/m (when m ≠ 0).
- If the original line is horizontal (m = 0), the perpendicular line is vertical (slope undefined).
- If the original line is vertical (slope undefined), the perpendicular slope is 0.
- From two points: m = (y2 − y1)/(x2 − x1).
- From Ax + By + C = 0: m = −A/B (when B ≠ 0).
How to Use This Calculator
- Select an input type: slope, two points, standard form, or slope-intercept.
- Enter your values. Fractions like -5/2 are accepted.
- Optionally enter a point (x0, y0) to get an equation.
- Press Calculate to see results above the form.
- Use the download buttons to export CSV or PDF.
Example Data Table
| Input Type | Inputs | Original Slope | Perpendicular Slope |
|---|---|---|---|
| Given slope | m = 3/4 | 0.75 | -1.3333… |
| Two points | (2, -1), (6, 5) | 1.5 | -0.6666… |
| Standard form | 3x + 2y − 6 = 0 | -1.5 | 0.6666… |
| Slope-intercept | y = -2x + 1 | -2 | 0.5 |
| Horizontal line | m = 0 | 0 | Undefined (vertical) |
| Vertical line | x = 4 | Undefined (vertical) | 0 |
Slope of a Perpendicular Line: Practical Guide
1) What this calculator returns
Perpendicular lines meet at a right angle (90°). This tool finds the perpendicular slope m⊥ from a line you describe using points, a known slope, or a standard-form equation. If you also enter a target point, it can build the full perpendicular line equation through that point.
2) The negative reciprocal rule
If the original slope is m, the perpendicular slope is m⊥ = −1/m. For example, m = 3/4 gives m⊥ = −4/3. This rule works for any nonzero finite slope and keeps the product m · m⊥ = −1.
3) From two points (x1, y1) and (x2, y2)
The calculator first computes m = (y2 − y1)/(x2 − x1). With points (2, −1) and (6, 5), the rise is 6 and the run is 4, so m = 6/4 = 1.5. The perpendicular slope becomes m⊥ = −2/3 ≈ −0.6667.
4) From standard form Ax + By + C = 0
Standard form converts to slope using m = −A/B (when B ≠ 0). For 3x + 2y − 6 = 0, m = −3/2, so m⊥ = 2/3. If you add a point, the tool outputs point-slope and slope-intercept forms.
5) Vertical and horizontal exceptions
A vertical line has undefined slope (run = 0), and its perpendicular line is horizontal with slope 0. A horizontal line has slope 0, and its perpendicular line is vertical with undefined slope. The calculator detects these cases and formats the equation as x = k or y = k.
6) Quick accuracy checks
Use two checks: (a) multiply slopes to see −1 for finite slopes, and (b) verify a 90° turn by comparing direction vectors like (1, m) and (1, m⊥), whose dot product should be 0. Small rounding differences are normal with decimals.
7) Real-world use and input tips
Perpendicular slopes appear in road cross-sections, roof framing, coordinate geometry, and drawing normal lines to curves. Prefer fractions for exact results, such as -5/2. If you enter decimals, choose a rounding level (2–6 places) for clean reporting and exports. It supports signed inputs and spaces.
FAQs
1) What is the slope of a perpendicular line?
For a nonzero finite slope m, the perpendicular slope is m⊥ = −1/m. This negative reciprocal makes the lines meet at 90°. If m is written as a fraction, swap numerator and denominator and change the sign.
2) What if the original line is vertical?
A vertical line has undefined slope because its run is zero. The perpendicular line is horizontal, so the perpendicular slope is 0 and the equation is y = k for some constant k.
3) What if the original line is horizontal?
A horizontal line has slope 0. The perpendicular line is vertical, which has undefined slope, and its equation is x = k. The calculator reports the slope as “undefined” in this case.
4) How do I get slope from Ax + By + C = 0?
Rearrange to y = (−A/B)x − C/B when B ≠ 0, so the slope is m = −A/B. Then compute m⊥ = −1/m = B/A, provided A ≠ 0.
5) Why doesn’t m·m⊥ equal −1 for my decimals?
Rounding can slightly change the product. Try increasing the decimal places or use fractional input like 7/3. For exact fractions, the product of the two slopes will be exactly −1.
6) Can it find the perpendicular line through a point?
Yes. Enter the original line (points, slope, or equation) and provide the point (x0, y0). The calculator uses y − y0 = m⊥(x − x0) and also shows slope-intercept or x = k / y = k when needed.