Slopes of Perpendicular Lines
Imagine standing at the corner of a rectangular field. One side runs straight ahead of you — that's one line. The other side runs exactly to your right, making a perfect right angle (90°). These two sides are perpendicular.
Now think about walking along those sides. The first side might be flat (horizontal), and the other goes straight up (vertical). But what if the field is on a hill? Then both sides are tilted, yet they still meet at a right angle. The question is: how are their tilts — their slopes — related?
The Intuition: Flip and Reverse
Slope measures how steep a line is: rise over run. A line that goes up gently has a small positive slope; a line that plunges steeply has a large negative slope.
For two lines to be perpendicular, one must be the "opposite" of the other in a specific way. If one line rises 2 units for every 1 unit it runs (slope = 2), the perpendicular line must fall 1 unit for every 2 units it runs (slope = -1/2). The steepness gets flipped upside down, and the direction gets reversed.
Think of a seesaw. If one side goes up fast, the other must go down slowly to stay balanced at a right angle. The product of their slopes is always -1.
The Precise Statement
If two non-vertical lines are perpendicular, the product of their slopes is −1.
Let line L1 have slope m1 and line L2 have slope m2. Then:
m1⋅m2=−1
Equivalently, each slope is the negative reciprocal of the other:
m2=−m11andm1=−m21
This rule fails if one line is vertical. A vertical line has undefined slope (infinite). Its perpendicular is horizontal (slope 0). You cannot multiply undefined by 0 and get -1 — so the product rule only applies when both slopes are defined (neither line is vertical).
Why It Works
Consider two lines through the origin: y=m1x and y=m2x. They are perpendicular if the angle between them is 90°. Using the tangent formula for the angle between two lines:
tanθ=1+m1m2m2−m1
For θ=90∘, tan90∘ is undefined, which happens when the denominator is zero:
1+m1m2=0⇒m1m2=−1
That's the entire derivation. The denominator must vanish for the lines to be at right angles.
Quick Examples
| Slope of one line (m1) | Slope of perpendicular (m2) | Check: m1⋅m2 |
|---|
| 3 | −31 | 3⋅(−31)=−1 |
| −52 | 25 | (−52)⋅25=−1 |
| 0 (horizontal) | undefined (vertical) | Rule doesn't apply — special case |
| undefined (vertical) | 0 (horizontal) | Rule doesn't apply — special case |
Slopes of perpendicular lines are negative reciprocals of each other. Always check by multiplying: if the product is −1, they are perpendicular. If one slope is zero or undefined, the other must be the opposite extreme — horizontal meets vertical.
A Final Check
Suppose you're given two lines: y=4x+2 and y=−41x−3. Multiply their slopes: 4×(−41)=−1. They are perpendicular. You can verify by sketching — one rises steeply, the other falls gently, and they cross at a clean right angle.
That's the whole idea: flip the fraction, change the sign.