Perpendicular Slopes Condition
Imagine two roads crossing at a right angle — that's perpendicular lines. The question is: how do their slopes relate?
The Intuition
Take a line with slope 2. That means for every 1 unit you move right, you go up 2 units — a fairly steep climb. Now picture a line perpendicular to it. If the first line is climbing steeply, the perpendicular line must be falling gently, or climbing very shallowly in the opposite direction.
Why? Because a right angle means the two lines "flip" the rise and run. One line's steepness becomes the other's shallowness, but in the opposite sign.
Try this: a line with slope 2 (rise 2, run 1). A perpendicular line should have rise 1 and run −2 — that gives slope −21. Notice: 2×(−21)=−1.
That's the pattern: the slopes are negative reciprocals of each other.
The Precise Statement
m1⋅m2=−1
Two non-vertical lines are perpendicular if and only if the product of their slopes is −1.
Equivalently: m2=−m11 (provided m1=0).
What About Vertical and Horizontal Lines?
A vertical line has undefined slope. A horizontal line has slope 0. Their product? Undefined — not −1. Yet they are clearly perpendicular.
The formula m1⋅m2=−1 only works when both slopes are defined (neither line is vertical). For a vertical line (x=c) and a horizontal line (y=d), they are perpendicular by definition — no slope calculation needed.
Quick Check
Are y=3x+2 and y=−31x−5 perpendicular?
3×(−31)=−1. Yes.
Are y=4x and y=4x+1 perpendicular?
4×4=16=−1. No — they're parallel.
Why It Works (A Short Proof)
›Proof
Two lines with slopes m1 and m2 make angles θ1 and θ2 with the positive x-axis, where tanθ1=m1 and tanθ2=m2. …