Perpendiculars

Perpendiculars describe the relation between lines, segments, or planes that meet at a right angle. In Euclidean geometry, two lines are perpendicular if they intersect and form a 90-degree angle. In coordinate form, lines with direction vectors u and v are perpendicular when their dot product is zero; in the plane this is equivalent to slopes m1 and m2 multiplying to -1, provided neither line is vertical. A vertical line is perpendicular to a horizontal line.

The perpendicular from a point to a line is the unique line through the point that meets the line at a right angle; its segment from the point to the line is the perpendicular distance from the point to the line.

The perpendicular bisector of a segment is the line that is perpendicular to the segment at its midpoint; any point on the bisector is equidistant from the segment’s endpoints.

An altitude of a triangle is a perpendicular line from a vertex to the opposite side (or its extension). The altitudes intersect at the orthocenter. In a right triangle, the two legs are themselves perpendicular.

In three-dimensional space, two lines are perpendicular if they intersect and their direction vectors are perpendicular. A line can be perpendicular to a plane if it is perpendicular to every line in the plane that passes through the point of intersection. If two lines do not meet, they are not perpendicular in the standard sense, though the shortest line segment joining skew lines is perpendicular to both.

Perpendiculars play a central role in construction, surveying, computer graphics, physics, and many areas of geometry, often indicated by the right-angle symbol.