# Perpendicular

### line lines compass planes

The term perpendicular describes a pair of lines or planes that intersect each other at a 90 degree **angle**. Perpendicularity is an important concept in **mathematics**, science, and **engineering**. A line *l1* is perpendicular to a line *l2* if the two intersect with congruent adjacent angles, which means that the angles are both equal to 90 degrees. Of course, a purely analytical definition of the term exists, also. If we define the slope m of a line as rise over run, then m = (y2 - y1)/(x2 - x1). A pair of nonvertical lines *l1* and *l2* are perpendicular if and only if *m1m2* = -1.

The concept of perpendicularity applies to any combination of lines and planes. Two or more planes can be perpendicular, or a line can be perpendicular to a **plane**, or to any number of **parallel** planes. Sometimes the term *orthogonal* is used with the same meaning, although orthogonal is also used outside of **geometry** and perpendicular is not. In science and engineering, a line perpendicular to another line or a plane is often referred to as being *normal* to the plane, or simply called the surface normal.

The concept of perpendicularity is a fundamental building block of geometry. It allows us to define figures such as squares and parallelpipeds, for example, and to draw conclusions about the relationships of the angles in certain types of figures such as triangles. The common *xy* or *xyz* coordinate system on which we plot geometrical figures and scientific data alike is defined by a set of perpendicular, or orthogonal, lines. The right triangles from which we define most of the basic relationships of **trigonometry** are based on a pair of perpendicular lines. Analytical geometry and vector **calculus**, which are an indispensible tool in engineering and science, make continual use of the concept.

Constructing a line perpendicular to another line is simple. Using a compass, measure equal distances both to the left (L_{L}) and to the right (L_{R}) of point P by putting the point of the compass on P and marking L_{L} and L R. Then place the compass point on L_{L} and scribe a short **arc**, then place the compass point on L_{R} and find the arc that intercepts the first arc. The line that connects point P to intercept X is perpendicular to the original line L.

Kristin Lewotsky

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