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Plane (Mathematics)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 - ,
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. Plane determined by a point and a normal vector For a point and a vector , the plane equation is -
for the plane passing through the point and perpendicular to the vector . Plane after three points The equation for the plane passing through three points , and can be represented by the following determinant: -
x_2 - x_1 & y_2 - y_1& z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0 The distance from a point to a plane For a point and a plane , the distance from to the plane is: -
The angle between two planes The angle between the planes and is following - .
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