Find a perpendicular line using reciprocal gradients

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Find a perpendicular line using reciprocal gradients

For two nonvertical perpendicular lines with nonzero gradients m₁ and m₂, the product m₁m₂ is −1. The second gradient is therefore the negative reciprocal of the first. This rule assumes ordinary Cartesian coordinates with compatible axis units, not a visually stretched contextual chart.

Find the required gradient first, then use a point to determine the intercept. Horizontal and vertical lines are the special pair: their equations are y = constant and x = constant, so the finite-gradient product rule is not used for them.

Perpendicularity is stronger than simply having opposite gradient signs. Gradients 2 and −2 do not have product −1, whereas 2 and −1/2 do. Substitution checks whether the final line also passes through the required point.

Worked example: Find the line through (0, 3) perpendicular to y = 2x + 1.

The original gradient is 2.

Its negative reciprocal is −1/2.

The required line has y-intercept 3.

Answer: y = −x/2 + 3.

Check your understanding: What gradient is perpendicular to 4/3?

Answer: −3/4.

Why: Their product is −1.