Perpendicular Lines (Algebra)
Perpendicular lines intersect at a 90-degree angle, and in algebra their slopes are negative reciprocals of each other.
Formula
m_1 \cdot m_2 = -1
Definition
Perpendicular lines cross at a perfect right angle, like the corner of a square; algebraically, two lines with slopes $m_1$ and $m_2$ are perpendicular exactly when $m_1 \cdot m_2 = -1$, so $m_2$ is the negative reciprocal of $m_1$ (flip the fraction and change the sign). A horizontal line ($m = 0$) is perpendicular to any vertical line (undefined slope), and perpendicular lines always intersect. More generally, two lines are perpendicular exactly when their direction vectors have dot product zero: for $(1, m_1)$ and $(1, m_2)$, this requires $(1)(1) + m_1m_2 = 0$, giving $m_1m_2 = -1$, a condition that generalizes to orthogonality of subspaces in $\mathbb{R}^n$.
Example
$y = 2x + 1$ has slope $2$, so a perpendicular line has slope $-1/2$; check $2 \cdot (-1/2) = -1$, giving an example like $y = -\frac{1}{2}x + 3$. For $3x - 4y = 12$ with slope $3/4$, the perpendicular slope is $-4/3$, giving a perpendicular line through $(0, 5)$ of $y = -\frac{4}{3}x + 5$. Lines $2x - 3y = 5$ (direction vector $(3, 2)$) and $3x + 2y = 7$ (direction vector $(2, -3)$) have dot product $(3)(2) + (2)(-3) = 0$, confirming perpendicularity.
Key Insight
To find a perpendicular slope, flip the fraction and change the sign: slope $3$ becomes $-1/3$, slope $-4/5$ becomes $5/4$. This constant product rule lets you instantly test whether two lines meet at a right angle without measuring anything, and the same condition, orthogonality of direction vectors, is central to inner product spaces, Gram-Schmidt orthogonalization, and the QR decomposition in linear algebra.