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| When | Method | m | m (dec) | m⊥ | m⊥ (dec) | Note |
|---|
Formula used
The perpendicular slope m⊥ to a line with slope m (when m ≠ 0 and finite) satisfies m · m⊥ = -1. Therefore,
m⊥ = -1 / m.
- If the given line is vertical (m undefined), the perpendicular line is horizontal (m⊥ = 0).
- If the given line is horizontal (m = 0), the perpendicular line is vertical (m⊥ undefined).
How to use this calculator
- Select an input method: slope, two points, standard form, slope-intercept, or angle.
- Enter values. Fractions and mixed numbers are accepted where relevant.
- Choose options: fraction simplification, decimal precision, and showing steps.
- Click Compute to see the perpendicular slope as fraction and decimal.
- Use Download CSV or Download PDF to export your results.
Example data table
| Given | Interpretation | m | m⊥ |
|---|---|---|---|
| m = 2 | slope | 2 | -1/2 |
| (x₁,y₁)=(0,0), (x₂,y₂)=(3,1) | two points | 1/3 | -3 |
| Ax+By+C=0 with A=3, B=4 | standard form | -3/4 | 4/3 |
| y = (-5/2) x + 7 | slope-intercept | -5/2 | 2/5 |
| θ = 45° | angle | 1 | -1 |
| m = 0 | horizontal | 0 | undefined |
| m = undefined | vertical | undefined | 0 |
Common slopes and their negative reciprocals
| m | m⊥ = -1/m | Check m·m⊥ |
|---|---|---|
| 2 | -1/2 | -1 |
| -3/5 | 5/3 | -1 |
| 4/7 | -7/4 | -1 |
| -1 | 1 | -1 |
| 3 | -1/3 | -1 |
| -2/3 | 3/2 | -1 |
| 5/2 | -2/5 | -1 |
| 0 | undefined | — |
| undefined | 0 | — |
Tip: Multiply your slope by the reported perpendicular slope. If the product equals -1, you’re correct for non-degenerate cases.
Perpendicular line through a given point
To find the perpendicular line through a point (x₀, y₀), first compute m⊥. Then write the point–slope form:
y - y₀ = m⊥(x - x₀)
Worked example:
- Given line slope m = 2 and point (3, -1).
- m⊥ = -1/2.
- Equation: y - (-1) = (-1/2)(x - 3).
- Simplified: y = (-1/2)x + 1/2.
Use the calculator to generate m⊥, then plug into this template.
Edge cases and quick checks
- Vertical lines: slope undefined ⇒ perpendicular slope 0.
- Horizontal lines: slope 0 ⇒ perpendicular slope undefined.
- Sign flip: the negative reciprocal changes sign relative to m.
- Fraction flip: swap numerator and denominator, then apply a negative.
- Product test: verify m · m⊥ = -1 whenever m is finite and nonzero.
When results look unusual, re-check mixed numbers and signs carefully.