Perpendicular

Definition · applies to lines, segments, and planes · also called ⟂

Elementary

Two lines are perpendicular if the lines cross and make square corners where they cross.

High school (Euclidean geometry)

Two lines in a plane or in space are perpendicular if the lines intersect and form a right angle at their intersection.

Undergraduate

Shapes are perpendicular at at a point $P$ if they intersect at $P$ at a right angle.

Examples

Two adjacent sides of a rectangle, as segments meeting at a corner Elementary
Two adjacent sides of a rectangle are perpendicular, because the sides meet at the corner and form a right angle there.
The lines $y = x$ and $y = -x$ in the plane, with direction vectors $(1, 1)$ and $(1, -1)$ High school
The lines $y = x$ and $y = -x$ are perpendicular, because the lines intersect at the origin and the product of their slopes is $1 \cdot (-1) = -1$.

Things that may be mistaken for examples.

The $x$-axis and the line through $(0, 0, 1)$ parallel to the $y$-axis in $\reals^3$ High school
The $x$-axis and the line through $(0, 0, 1)$ parallel to the $y$-axis are not perpendicular, because the lines never intersect: the first line has $z = 0$ and the second line has $z = 1$.
The vector $(1, 1)$ and the line $x + y = 1$ in the plane, whose direction vector is $(1, -1)$ High school
The vector $(1, 1)$ and the line $x + y = 1$ are not called perpendicular, because “perpendicular” describes two lines or planes that intersect; the normal line through $\left(\frac{1}{2}, \frac{1}{2}\right)$ in the direction $(1, 1)$ is perpendicular to the line $x + y = 1$.
The functions $\sin$ and $\cos$ on $[-\pi, \pi]$, with the $L^2$ inner product $\ip{f}{g} = \int_{-\pi}^{\pi} f(x) g(x) \, dx$ Undergraduate
The functions $\sin$ and $\cos$ are not called perpendicular, because perpendicularity describes lines and planes in geometry, and functions are not lines.
The vectors $(1, 0)$ and $(0, 0)$ in $\reals^2$ Undergraduate
The vectors $(1, 0)$ and $(0, 0)$ are not perpendicular, because the zero vector has no direction, so the vectors form no angle.

Two lines in the plane with slopes $m_1$ and $m_2$ are perpendicular exactly when $m_1 m_2 = -1$, or when one line is horizontal and the other is vertical.
Lines in space whose directions are at right angles but that do not intersect are usually called orthogonal rather than perpendicular.

Right angle vs. perpendicular vs. orthogonal vs. normal: Right angles, perpendicular lines, orthogonal vectors, and normal vectors all describe things that meet “squarely.” A right angle is an angle of $90^\circ$, so the term describes an angle. Perpendicular describes lines or planes that intersect and form right angles. Orthogonal describes vectors whose inner product is $0$, which extends the idea to vectors that do not meet, such as directions of skew lines, to the zero vector, and to vectors that are not arrows, such as functions. A normal vector is perpendicular to a line, curve, or surface at a point, so the term describes a vector relative to a shape.