Perpendicular Lines (Algebra)
AlgebraPerpendicular 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 each other at a perfect right angle (like the corner of a square). In algebra, if one line has a certain slope, the perpendicular line's slope is the "flipped and sign-changed" version.
Example
$y = 2x + 1$ has slope $2$. A perpendicular line has slope $-1/2$. Check: $2 \cdot (-1/2) = -1$. An example perpendicular: $y = -\frac{1}{2}x + 3$.
Key Insight
To find a perpendicular slope: flip the fraction and change the sign. Slope $3$ becomes $-1/3$. Slope $-4/5$ becomes $5/4$.
Definition
Two lines with slopes $m_1$ and $m_2$ are perpendicular if and only if $m_1 \cdot m_2 = -1$, equivalently $m_2 = -1/m_1$ ($m_2$ is the negative reciprocal of $m_1$). A horizontal line ($m = 0$) is perpendicular to any vertical line (undefined slope). Perpendicular lines always intersect.
Example
Line: $3x - 4y = 12$, slope $m_1 = 3/4$. Perpendicular slope: $m_2 = -4/3$. Perpendicular line through $(0, 5)$: $y = -\frac{4}{3}x + 5$.
Key Insight
Perpendicular slopes multiply to $-1$, meaning their product is always the same. This rule lets you instantly test whether two lines form a right angle without measuring.
Definition
Two lines with direction vectors $v_1$ and $v_2$ are perpendicular if their dot product is zero: $v_1 \cdot v_2 = 0$. For lines $y = m_1x + b_1$ and $y = m_2x + b_2$ with direction vectors $(1, m_1)$ and $(1, m_2)$, perpendicularity requires $(1)(1) + m_1m_2 = 0$, giving $m_1m_2 = -1$. This generalizes to orthogonality of subspaces in $\mathbb{R}^n$.
Example
Lines $2x - 3y = 5$ (direction vector $(3, 2)$) and $3x + 2y = 7$ (direction vector $(2, -3)$). Dot product: $(3)(2) + (2)(-3) = 6 - 6 = 0$. Perpendicular confirmed.
Key Insight
The perpendicularity condition $m_1m_2 = -1$ is equivalent to orthogonality of the direction vectors, a concept central to inner product spaces, Gram-Schmidt orthogonalization, and the QR decomposition in linear algebra.