perpendicularity

Perpendicularity is a geometric relation in which two objects meet at a right angle. The concept applies to lines, lines and planes, and planes within Euclidean space. When two lines intersect, they are perpendicular if the angle between them is 90 degrees. A line and a plane are perpendicular if the line is perpendicular to every line in the plane that passes through the intersection point. Two planes are perpendicular if their normals are perpendicular, equivalently if the angle between the planes is 90 degrees.

In three dimensions, perpendicularity between lines requires that the lines intersect and their direction vectors are orthogonal (dot product zero). Lines that do not intersect (skew lines) do not have a defined angle of intersection, though the shortest segment between them is perpendicular to both lines. A line can also be perpendicular to a plane, meaning it is perpendicular to every line in the plane that passes through the intersection point; equivalently, the line is parallel to the plane’s normal.

Mathematically, if lines have direction vectors a and b, they are perpendicular when a · b = 0. If planes have normal vectors n1 and n2, they are perpendicular when n1 · n2 = 0. In two dimensions, lines with slopes m1 and m2 are perpendicular when m1 m2 = −1 (provided neither line is vertical).

Perpendicularity is a common constraint in geometry, computer-aided design, and manufacturing. In geometric dimensioning and tolerancing (GD&T), perpendicularity specifies that a feature’s axis or surface be at right angle to a datum within a tolerance zone. Verification uses measuring tools such as squares, protractors, dial indicators, or coordinate measuring machines, and is informed by design intent and machining practices.