Perpendicular
Perpendicular lines or segments intersect at exactly 90 degrees, forming right angles at their point of intersection.
Formula
\text{slopes: } m_1 \times m_2 = -1 \text{ (for non-vertical perpendicular lines)}
Definition
Two lines are perpendicular when they cross at a right angle ($90^\circ$), shown with a small square at the intersection. In the coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes equals $-1$ (negative reciprocals), and more generally, two subspaces (or affine subspaces) are perpendicular if their direction vectors are orthogonal, $u \cdot v = 0$.
Example
The walls and floor of a room are perpendicular, as are crossing lines on graph paper, and a plus sign (+) shows two perpendicular lines. A line with slope $2/3$ has perpendicular lines of slope $-3/2$: check $(2/3)\times(-3/2) = -1$. The line $y = 2x - 3$ has direction vector $(1, 2)$; its perpendiculars have direction vectors satisfying $(1,2)\cdot(a,b) = a+2b = 0$, giving $(a,b) = (-2,1)$ and slope $-1/2$.
Key Insight
Perpendicular lines are found everywhere in construction because $90^\circ$ angles make buildings strong and stable, and builders use a "square" tool to check for them. The negative reciprocal slope rule lets us find perpendicular lines using equations alone, connecting geometry and algebra without ever measuring an angle. Perpendicularity is the geometric expression of orthogonality; in functional analysis two functions are "perpendicular" if their inner product (integral of their product) is zero, a generalization that is the foundation of Fourier series and wavelet theory.