What Is a Perpendicular Slope?
Two lines are perpendicular when they meet at a right angle (90°). In coordinate geometry, the slope of a line perpendicular to a given line is the negative reciprocal of that line's slope. This calculator takes the slope of an original line and instantly returns the slope of any line perpendicular to it.
How to Use the Calculator
Enter the slope (m) of your original line — it can be a whole number, fraction-as-decimal, or negative value. The calculator returns \(m_{\perp} = -\frac{1}{m}\). If you enter a slope of 0 (a horizontal line), the perpendicular line is vertical and has an undefined slope, which the tool reports for you.
The Formula Explained
The relationship between perpendicular slopes is $$m_{\perp} = -\frac{1}{\text{Slope }(m)}$$ Equivalently, the product of the two slopes is always -1: \(m_1 \cdot m_2 = -1\). To find the perpendicular slope you do two things — flip the fraction (reciprocal) and change the sign. For example, the negative reciprocal of 3 is \(-\frac{1}{3}\), and the negative reciprocal of \(-\frac{2}{5}\) is \(\frac{5}{2}\).
Worked Example
Suppose a line has slope \(m = 4\). Its perpendicular slope is $$m_{\perp} = -\frac{1}{4} = -0.25$$ Any line with slope -0.25 will cross the original line at a perfect right angle. As a check: \(4 \times (-0.25) = -1\), confirming perpendicularity.
FAQ
What is the perpendicular slope of a horizontal line? A horizontal line has slope 0. Its perpendicular is a vertical line, whose slope is undefined (you cannot divide by zero).
What is the perpendicular slope of a vertical line? A vertical line has an undefined slope; lines perpendicular to it are horizontal with slope 0.
Do parallel lines have the same rule? No. Parallel lines share the same slope (\(m_1 = m_2\)), while perpendicular lines are negative reciprocals (\(m_1 \cdot m_2 = -1\)).
