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In mathematics, a plane is the fundamental two-dimensional object.
Intuitively, it may be visualized as a flat infinite piece of paper. Most of the fundamental work in geometry, trigonometry, and graphing is performed in
two dimensions, or in other words, in a plane.
Given a plane, one can introduce a Cartesian
coordinate system on it in order to label every point on the plane uniquely with two numbers, its coordinates.
In a three-dimensional x-y-z coordinate system, one can define a plane as the set of all solutions
of an equation ax + by + cz + d = 0, where a, b, c and d
are real numbers such that not all of a, b, c are
zero. Alternatively, a plane may be described parametrically as the set of all points of the form u + s
v + t w where s and t range over all real numbers, and
u, v and w are given vectors defining the plane.
A plane is uniquely determined by any of the following combinations:
- three points not lying on a line
- a line and a point not lying on the line
- a point and a line, the normal to the plane
- two lines which intersect in a single point or are parallel
In three-dimensional space, two different planes are either parallel or they intersect in a line. A line which is not parallel
to a given plane intersects that plane in a single point.
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