Perpendicular
GeometryPerpendicular 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 (exactly $90^\circ$). We show this with a small square at the intersection point.
Example
The walls and the floor of a room are perpendicular. The lines on graph paper that cross are perpendicular. A plus sign (+) shows two perpendicular lines.
Key Insight
Perpendicular lines are found everywhere in construction because $90^\circ$ angles make buildings strong and stable. Builders use a tool called a "square" to check for perpendicularity.
Definition
Two lines (or segments, or rays) are perpendicular if they intersect at a right angle ($90^\circ$). In the coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes equals $-1$ (their slopes are negative reciprocals).
Example
Line 1 has slope $2/3$. Its perpendicular line has slope $-3/2$. Check: $(2/3) \times (-3/2) = -1$. The $x$-axis (slope $0$) is perpendicular to the $y$-axis (undefined slope - vertical). The altitude of a triangle is perpendicular to the base.
Key Insight
The negative reciprocal slope rule is the algebraic expression of geometric perpendicularity. It lets us find perpendicular lines using equations alone - no angle measurement needed - connecting geometry and algebra.
Definition
Two subspaces (or affine subspaces) are perpendicular if their direction vectors are orthogonal: $u \cdot v = 0$. In $\mathbb{R}^n$, two lines through the origin are perpendicular iff their direction vectors are orthogonal. The set of all lines through a point perpendicular to a given line forms the normal space at that point.
Example
The line $y = 2x - 3$ has direction vector $(1, 2)$. Its perpendicular lines have direction vectors satisfying $(1, 2) \cdot (a, b) = a + 2b = 0$, so $(a, b) = (-2, 1)$ (up to scaling), giving slope $b/a = -1/2$.
Key Insight
Perpendicularity is the geometric expression of orthogonality. In functional analysis, two functions are "perpendicular" (orthogonal) if their inner product (integral of their product) is zero. This generalization of geometric perpendicularity is the foundation of Fourier series and wavelet theory.