Math101 · Independent Ontario learning library
Math101Perpendicular Lines
Perpendicular lines meet at a right angle; their nonzero finite slopes are negative reciprocals.
Perpendicular lines intersect at $90^\circ$. For nonvertical lines with nonzero slopes, $m_1m_2=-1$.
Negative reciprocal slopes
If one slope is $m$, a perpendicular slope is $-1/m$. Reverse numerator and denominator, then change the sign.
A line with slope $2/5$ is perpendicular to one with slope $-5/2$. Their product is $-1$.
Why the rule works
Direction vectors for slopes $m_1$ and $m_2$ can be written $(1,m_1)$ and $(1,m_2)$. Perpendicular vectors have dot product zero:
so $m_1m_2=-1$. This connects the slope rule to vector geometry.
Writing a perpendicular line
Substitute the point and multiply slopes to verify both requirements.
Horizontal and vertical lines
The negative-reciprocal calculation excludes slope $0$ because dividing by zero is undefined. Geometrically, every horizontal line $y=b$ is perpendicular to every vertical line $x=a$.
Recognize this special pair rather than saying their slope product equals $-1$.
Common mistakes
Changing only the sign. The slope perpendicular to $2/3$ is $-3/2$, not $-2/3$.
Taking a negative reciprocal for parallel lines. Parallel slopes are equal.
Applying $m_1m_2=-1$ to vertical lines. Use the horizontal–vertical special case.
Finding the correct slope but ignoring the required point. Both conditions define the line.
Quick self-check
- Did I reverse the fraction and change its sign?
- Is there a horizontal–vertical special case?
- Does the new equation pass through the given point?
- Can a slope product or geometric check verify $90^\circ$?
