Perpendicular information

The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. The segment AB can be called *the perpendicular from A to the segment CD*, using "perpendicular" as a noun. The point B is called the *foot of the perpendicular from* A to segment CD, or simply, the *foot of* A on CD.^[1]

In geometry, two geometric objects are perpendicular if their intersection forms right angles (angles that are 90 degrees or π/2 radians wide) at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes.

Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal vector.

A line is said to be perpendicular to another line if the two lines intersect at a right angle.^[2] Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. A great example of perpendicularity can be seen in any compass, note the cardinal points; North, East, South, West (NESW) The line N-S is perpendicular to the line W-E and the angles N-E, E-S, S-W and W-N are all 90° to one another.

Perpendicularity easily extends to segments and rays. For example, a line segment ${\overline {AB}}$ is perpendicular to a line segment ${\overline {CD}}$ if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, ${\overline {AB}}\perp {\overline {CD}}$ means line segment AB is perpendicular to line segment CD.^[3]

A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.

Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle.

^ Kay (1969, p. 114)
^ Kay (1969, p. 91)
^ Kay (1969, p. 91)

[1] Kay (1969, p. 114)

[2] Kay (1969, p. 91)

[3] Kay (1969, p. 91)

Perpendicular information

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