Perpendicular Gradients

Statement

If two lines are perpendicular, their gradients multiply to give \(-1\):

\[ m_1 m_2 = -1 \]

Why it’s true

• The slope of a line measures its steepness.
• If two lines are perpendicular, one is the negative reciprocal of the other.
• That is, if one slope is \(m_1\), then the other must be \(m_2 = -1/m_1\).
• Multiplying them gives \(m_1 m_2 = -1\).

Recipe (how to use it)

1. Find slope of one line (\(m_1\)).
2. Use formula \(m_2 = -1/m_1\) for the perpendicular line.
3. If both slopes are known, check if their product is \(-1\).

Spotting it

This formula is used when checking whether two lines are perpendicular, or when finding the slope of a perpendicular line through a given point.

Common pairings

• Slope-intercept form of a line: \(y=mx+c\).
• Midpoint and perpendicular bisectors.
• Circle tangent and radius problems.

Mini examples

1. Given: Line slope = 2. Perpendicular slope: \(-1/2\).
2. Given: One line slope = -3, other line slope = 1/3. Product = -1, so they are perpendicular.

Pitfalls

• Forgetting to take the negative reciprocal.
• Confusing perpendicular with parallel (\(m_1 = m_2\)).

Exam strategy

• Check product of slopes quickly to confirm perpendicularity.
• Special case: vertical line (\(m=\infty\)) and horizontal line (\(m=0\)) are perpendicular.

Summary

Two lines are perpendicular if their gradients multiply to give \(-1\). Use \(m_2=-1/m_1\) to find the slope of a perpendicular line.